Discontinuity Defied

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(See Epistemic Probability Cheat Sheet, especially the Synopsis, for explanations of any unfamiliar terms and concepts.)

In Discontinuities Everywhere we found that the probability evaluation map 𝜋(𝐴, 𝜇) ≜ 𝜇(𝐴) is discontinuous at every point in its domain seq(𝜟⁰₂ × ℙ), the sequential product of 𝜟⁰₂ and ℙ. This sounds like a serious obstacle if we want to follow Jaynes’ finite sets policy, which implies that 𝘗𝘳(ℰ | 𝒳) must be the well-defined well-behaved limit of probabilities 𝘗𝘳(𝐸 | 𝑋) for finite queries 𝐸 and finite premises 𝑋.

To stay within the realm of well-defined and well-behaved limits when dealing with a discontinuity point (𝐴, 𝜇), we must eliminate the problematic joint trajectories (𝐴ᵢ) and (𝜇ᵢ) converging on 𝐴 and 𝜇 respectively; we must restrict the possible trajectories to those which yield the desired convergence of 𝜇ₙ(𝐴ₙ) to 𝜇(𝐴). We’ll look at several possibilities, and ultimately settle on the same solution used in computable measure theory.

Restricting the domain

Although not a general solution, one way of restricting trajectories is to restrict the domain of 𝜋, thus eliminating any trajectory that strays outside of the restricted domain:

1. If we fix 𝜇 to one specific value and only allow 𝐴 to vary, then 𝜋(⋅, 𝜇) is sequentially continuous, hence continuous, at all points in its domain, by Dominated Convergence for Sets.

2. If we fix 𝐴 to one specific value and only allow 𝜇 to vary, then 𝜋(𝐴, ⋅) is continuous at exactly those points 𝜇 for which the boundary of 𝐴 is assigned zero probability, 𝜇(𝜕𝐴) = 0. In measure theory, one says in this case that 𝐴 is a continuity set of 𝜇. See Corollary 3 and Theorem 4 in the Appendix of the PDF version.

Expanding on case 2 above, here are some examples where 𝜇(𝜕𝐴) = 0:

• If 𝐴 is clopen—expressible as [𝜑] for some propositional formula 𝜑—then 𝜕𝐴 = ∅ and 𝜇(𝜕𝐴) = 0 for all 𝜇.

• Let 𝐴 be a disc in the plane ℝ² and 𝜇 a probability measure on the plane. Then the boundary of 𝐴 is its circumference, which has an area of 0, and we will have 𝜇(𝜕𝐴) = 𝜇(circumference) = 0 for any 𝜇 that is definable by a probability density.

And here is a specific example of the discontinuity of 𝜋(𝐴, ⋅) at a point 𝜇 where 𝜇(𝜕𝐴) > 0:

Example 1. Let 𝑥₀ ∈ 𝕋 be an arbitrary truth assignment, let 𝜇 = 𝛿(𝑥₀) be the probability measure that concentrates its entire probability mass on 𝑥, and let 𝐴 = { 𝑥₀ }. Clearly, 𝜇(𝐴) = 1. Now define 𝜇ₙ to be the uniform probability measure on cyl(𝑥₀ ↾ 𝑛), the prefix cylinder of truth assignments that agree with 𝑥₀ on the first 𝑛 propositional symbols, and define 𝐴ₙ to be cyl(𝑥₀ ↾ (2𝑛)). Then (𝜇ᵢ) → 𝜇, (𝐴ᵢ) → 𝐴, and

Restricting the approximations

Another way of restricting the trajectories, one that fits in quite nicely with Jaynes’ finite sets policy, is to require (𝐴ᵢ) and (𝜇ᵢ) to be finitary:

Definition. A set 𝐴 ⊆ 𝜟⁰₂ is finitary if 𝐴 = [𝜑] for some finite query 𝜑. A probability measure 𝜇 ∈ ℙ is finitary if 𝜇 = pm 𝑋 for some finite premise 𝑋. A sequence (𝐴ᵢ) ⊆ 𝜟⁰₂ or (𝜇ᵢ) ⊆ ℙ is finitary if each element of the sequence is finitary.

Since every 𝐴 ∈ 𝜟⁰₂ and 𝜇 ∈ ℙ is the limit of a finitary sequence, requiring (𝐴ᵢ) and (𝜇ᵢ) to be finitary is a very reasonable restriction. Failure to converge to 𝜇(𝐴) even with this restriction imposed we’ll call strong discontinuity:

Definition. Let 𝐴 ∈ 𝜟⁰₂ and 𝜇 ∈ ℙ. If there exist finitary sequences (𝐴ᵢ) ⊆ 𝜟⁰₂ and (𝜇ᵢ) ⊆ ℙ with (𝐴ᵢ) → 𝐴 and (𝜇ᵢ) → 𝜇 but (𝜇ᵢ(𝐴ᵢ)) ↛ 𝜇(𝐴), we say that 𝜋 is strongly discontinuous at (𝐴, 𝜇), or that (𝐴, 𝜇) is a strong discontinuity of 𝜋.

One might hope that 𝜋 has no strong discontinuities, but alas, this is not so: The sequences (𝐴ᵢ) and (𝜇ᵢ) described in Example 1 are finitary and converge to 𝐴 = { 𝑥₀ } and 𝜇 = 𝛿(𝑥₀) respectively, but (𝜇ᵢ(𝐴ᵢ))ᵢ does not converge to 𝜇(𝐴) = 1.

That example can be generalized. An atom of a probability measure 𝜇 is a point 𝑥 such that 𝜇({ 𝑥 }) > 0. In Example 1, 𝑥₀ is an atom of 𝜇 that lies within 𝐴. In general, any point (𝐴, 𝜇) such that 𝐴 contains an atom of 𝜇 is a strong discontinuity of 𝜋. See Theorem 5 in the Appendix of the PDF version.

We can also have strong discontinuities at diffuse, non-atomic measures. Theorem 6 in the Appendix of the PDF version shows that (∅, 𝜆) is a strong discontinuity of 𝜋, where 𝜆 is the uniform probability measure on 𝕋, by constructing finitary sequences (𝐴ᵢ) → ∅ and (𝜇ᵢ) → 𝜆 with 𝜇ₙ(𝐴ₙ) > 1/2 for all 𝑛 ∈ ℕ, whereas 𝜆(∅) = 0.

I do not know whether every point in the domain of 𝜋 is a strong discontinuity, but I have been unable to find any counterexample to that conjecture. In any event, it’s clear that the strategy of restricting the approximations to be finitary doesn’t solve the discontinuity problem for us.

The solution: simulate a nested limit

Recall that for 𝒳 = (𝑋ᵢ) and ℰ = (𝐸ᵢ) we defined

that is, we have a nested limit:

Dominated Convergence for Sets and the weak topology on ℙ give us the analogous identity for sets and probability measures: if (𝜇ᵢ) → 𝜇 and (𝐴ᵢ) → 𝐴 then

Going back to Example 1, we then see what caused the pathology. With 𝜇ₙ defined as the uniform measure on cyl(𝑥₀ ↾ 𝑛) and 𝐴ₙ defined as cyl(𝑥₀ ↾ (2𝑛)), the 𝐴ₙ converged on { 𝑥₀ } much faster than the 𝜇ₙ converged on 𝛿(𝑥₀). Now consider what happens if we define

for 𝑓(𝑛) some strictly increasing function of 𝑛. If 𝑓(𝑛) < 2𝑛 for all 𝑛, then 𝜇ₙ(𝐴ₙ) ≤ 1/2 for all 𝑛 and so cannot converge to 1. But if 𝑓(𝑛) ≥ 2𝑛 for all 𝑛 then 𝜇ₙ(𝐴ₙ) = 1 and convergence to 1 is immediate. We just had to make sure that the 𝜇ₙ converged fast enough.

In this example we simulated the nested limit, replacing it with a single sequence of values obtained from finitary objects, as the Jaynesian policy on infinities requires. The good news is that we can always do this:

Theorem 2. For any (𝐴, 𝜇) ∈ 𝜟⁰₂ × ℙ and finitary sequences (𝐴ᵢ) → 𝐴 and (𝜈ᵢ) → 𝜇, we can construct a subsequence (𝜇ᵢ) of (𝜈ᵢ) such that (𝜇ᵢ) → 𝜇 and (𝜇ᵢ(𝐴ᵢ))ᵢ → 𝜇(𝐴).

Proof. First of all, (𝜇ᵢ) → 𝜇 because that is true of any subsequence (𝜇ᵢ) of (𝜈ᵢ), which itself converges to 𝜇. Now define the subsequence (𝜇ᵢ). For each 𝑛 ∈ ℕ:

1. Let 𝜖ₙ ≜ |𝜇(𝐴ₙ) - 𝜇(𝐴)|.

2. Choose 𝑘(𝑛) > 𝑘(𝑛 - 1) such that

   this is possible because 𝐴ₙ is clopen, so (𝜈ᵢ(𝐴ₙ)) → 𝜇(𝐴ₙ).

3. Defining 𝜇ₙ ≜ 𝜈ₖ₍ₙ₎ we have

Since (𝐴ᵢ) → 𝐴 we have (𝜖ᵢ) → 0, and hence (𝜇ᵢ(𝐴ᵢ))ᵢ → 𝜇(𝐴). ∎

This may be thought of as an adaptive scheme to compute an arbitrarily close approximation to 𝜇(𝐴): first find a suitable finite approximation 𝐴′ to 𝐴, then adaptively select a finite approximation 𝜇′ to 𝜇 that depends on 𝐴′, and finally return 𝜇′(𝐴′).

Final summary

We have found that not only is 𝜋(𝐴, 𝜇) = 𝜇(𝐴) discontinuous at every point in its domain, but it is also strongly discontinuous at many, possible all points in its domain. If 𝜇 is held fixed then 𝜋(⋅, 𝜇) is continuous in its first argument, and if 𝐴 is held fixed then 𝜋(𝐴, ⋅) is continuous in its second argument iff 𝜇(𝜕𝐴) = 0. A fully general solution requires acknowledging that we fundamentally have a nested limit here,

with convergence on 𝜇 required first. Fortunately this nested limit can be evaluated as a single limit by adaptively choosing how far into the approximating sequence for 𝜇 one must go for each approximation 𝐴ₙ to 𝐴.

Ep Cheat Sheet

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If you’re a new reader, your best bet is to start at the beginning with Introduction to This Blog and work your way forward from there… but if you’d prefer to dip in here and there, or if you’ve been reading Epistemic Probability for a while and need a refresher or reference, here’s a quick summary of what you need to know.

What It’s All About

These articles present the core ideas.

• Introduction to This Blog. A brief summary.

• The EPL Theorem. The result that kicked off this research program: the laws of probability are the uniquely determined logic of plausible reasoning, with the probability function 𝘗𝘳(⋅ | ⋅) serving as a generalization of the logical entailment relation ⊧ of classical propositional logic. This result assumes only that we wish to retain certain properties that classical propositional logic already has. The theorem also gives us an explicit formula for the required numeric value of 𝘗𝘳(𝐸 | 𝑋). You might want to stick to Parts 1–3 on a first reading, and leave the detailed proof for later.

• Applying the EPL Theorem. Some simple examples of encoding one’s information about a problem as a finite set of propositional formulas, and then applying the EPL Theorem to compute the desired probabilities.

• Turning Concrete Facts Into a Probability Distribution and Latent Symbols & Converging Approximations. Exercises in deriving the uniform distribution on the unit interval [0, 1], and the Poisson distribution on the natural numbers, as the limits of increasingly large finite collections of propositional formulas that serve as a sort of database of known concrete facts about the problem domain.

• Generalized Premises: A Redo. A generalized premise, this theory’s equivalent of a probability measure, is defined as a sequence of increasingly detailed propositional formulas that converge in a certain sense. This formalizes the notions presented informally in Turning Concrete Facts Into a Probability Distribution and Latent Symbols & Converging Approximations.

• The Epistemic Representation Theorem. In which we prove that every probability measure of any conceivable practical importance1 can be represented by a generalized premise. Informally, every probability measure of practical importance can be obtained as the limit of some sequence of increasingly large and detailed collection of concrete facts (propositional formulas).

• Generalized Queries. Generalized queries, this theory’s replacement for measurable sets, defined as a sequence of propositional formulas that converge pointwise. Part 1 argues that the full Borel hierarchy of measurable sets is too large. Part 2 argues that the class of Borel sets 𝜟⁰₂ is all we really need; it then gives the formal definition of a generalized query and shows that generalized queries correspond to the sets of 𝜟⁰₂.

Tutorials

I’ve written tutorials on various prerequisite topics:

• Review of Propositional Logic

• Equivalence Relations

• Metric Spaces Part 1, Part 2, and Part 3.

• Digression on Topology.

• A Brief Intro to Measure Theory.

• Some More Topology: Compactness.

• The Weak Topology on Probability Measures. (This is a mix of tutorial material and some results specific to Epistemic Probability.)

• A Brief Note on Quotient Maps.

• Sequential Spaces.

Many of the above deal with topology. Topology is the mathematics of continuity generalized to functions on arbitrary sorts of objects, not just numeric functions. Continuity is important because it allows us to work with “completed infinites” while still adhering to the spirit, if not the letter, of Jaynes’ finite sets policy:2

In probability theory, it appears that the only safe procedure known at present is to derive our results first by strict application of the rules of probability theory on finite sets of propositions; then, after the finite-set result is before us, observe how it behaves as the number of propositions increase indefinitely.

Synopsis of terminology and concepts

To help you keep track of the specialized notation, terminology, and concepts I’ve introduced in these articles, here is a quick synopsis.

Truth assignments and propositional formulas

• ℳ is the countably infinite set of (manifest) propositional symbols. (I originally defined a separate set of “latent” propositional symbols ℒ, but have since determined that the theory does not need them.)

• “Propositional formula” by default means “propositional formula on ℳ,” i.e. a formula constructed from the propositional symbols in ℳ and the logical operators.

• 𝔹 ≜ { 0, 1 } is the set of Boolean truth values, identifying 1 with true and 0 with false.

• 𝕋 is the set of truth assignments 𝛼 : ℳ → 𝔹. By default, “truth assignment” means “truth assignment on ℳ.”

• 𝛼⟦𝜑⟧ is the truth value obtained by evaluating 𝜑 after substituting in, for each propositional symbol 𝑠 occurring in 𝜑, its assigned truth value 𝛼(𝑠). For examples, if 𝛼(𝑠₁) = 1 and 𝛼(𝑠₃) = 0, then

  The truth assignment 𝛼 then satisfies 𝜑 iff 𝛼⟦𝜑⟧ = 1.

Truth assignments and binary sequences

• 𝗌 : ℕ → ℳ is some canonical unique enumeration of the propositional symbols: 𝗌(𝑖) is the 𝑖-th propositional symbol in the canonical ordering. 𝗌 is bijective (invertible), so 𝗌⁻¹(𝑠) is the position of propositional symbol 𝑠 in the enumeration.

• 𝛼∘𝗌 is the binary sequence corresponding to truth assignment 𝛼, obtained by listing the truth values assigned to the propositional symbols in ℳ in their canonical order. For example, if

  then (𝛼∘𝗌) = 0, 0, 1, ….

• 𝑥∘𝗌⁻¹ is the truth assignment corresponding to binary sequence 𝛼, obtained by assigning truth value 𝑥ᵢ to the 𝑖-th propositional symbol in the canonical order. For example, if 𝑥 = 1, 0, 1, … then

• We sometimes blur the distinction between truth assignments and binary sequences, identifying a truth assignment 𝛼 with its corresponding binary sequence 𝛼∘𝗌. Likewise, we may blur the distinction between the sets 𝕋 and 𝔹^𝜔.

(Finite) premises and (finite) queries

• In the conditional probability expression 𝘗𝘳(𝐸 | 𝑋), where 𝐸 is a propositional formula and 𝑋 is a satisfiable propositional formula, 𝐸 is called the query and 𝑋 the premise. The premise is often a large conjunction (AND) of many propositional formulas, representing a sort of propositional database of all that is known with certainty about the system being analyzed, including all logical relationships between the propositional symbols.

• We use “(finite) query” as a synonym for “propositional formula,” and “(finite) premise” as a synonym for “satisfiable propositional formula.”

• For finite query 𝐸 and finite premise 𝑋, the probability 𝘗𝘳(𝐸 | 𝑋) is necessarily

  where 𝐹 is any finite subset of ℳ that includes all propositional symbols appearing in either 𝐸 or 𝑋, and #_𝐹(𝜑) is the number of truth assignments on 𝐹 that satisfy 𝜑. This is a theorem, not a definition.

• We write [𝜑], where 𝜑 is any propositional formula, for the set of truth assignments that satisfy 𝜑.

• We write pm 𝑋, where 𝑋 is any finite premise, for 𝜆(⋅ | [𝑋]), the uniform probability measure conditional on [𝑋].

• We then have this identify: 𝘗𝘳(𝐸 | 𝑋) = (pm 𝑋)([𝐸]).

The canonical topology on 𝕋

• A cylinder set is any 𝐴 ⊆ 𝕋 defined by fixing the truth values of a finite set of propositional symbols:

  for some finite 𝐹 ⊆ ℳ and 𝑓 : 𝐹 → 𝔹.

• The cylinder sets form a base for the canonical (product) topology on 𝕋: an open set is any countable union of cylinder sets.

• 𝑥 ↾ 𝑛, for 𝑥 ∈ 𝔹^𝜔 a binary sequence and 𝑛 ∈ ℕ, is the bit-string comprising the first 𝑛 elements of 𝑥: 𝑥₀, …, 𝑥ₙ₋₁.

• 𝛼 ↾ 𝑛, for 𝛼 ∈ 𝕋 a truth assignment and 𝑛 ∈ ℕ, is (𝛼∘𝗌) ↾ 𝑛, i.e., the bit-string comprising the truth values 𝛼 assigns to the first 𝑛 propositional symbols in ℳ under the canonical enumeration.

• cyl 𝜎 is the set of truth assignments 𝛼 such that 𝛼 ↾ 𝑛 = 𝜎, where 𝑛 is the length of the finite binary string 𝜎 ∈ 𝔹^*. We say that cyl 𝜎 is a prefix cylinder of rank 𝑛. Prefix cylinders are a specific kind of cylinder set, and it turns out that the prefix cylinders by themselves form a base for the canonical (product) topology on 𝕋; in fact, the prefix cylinders of rank 𝑛 ≥ 𝑛₀, for any fixed 𝑛₀ ∈ ℕ, form a base.

• The clopen sets—those subsets of 𝕋 that are both open and closed—are exactly the finite unions of prefix cylinders, those that can be expressed as

  for some finite set 𝐹 ⊆ 𝔹*.

• The clopen sets can also be characterized as exactly those sets of the form [𝐸] for some finite query 𝐸.

Generalized premises

• A generalized premise is a sequence 𝒳 = (𝑋ᵢ) such that, for any finite query 𝐸, the sequence of probabilities (𝘗𝘳(𝐸 | 𝑋ᵢ))ᵢ converges. For a generalized premise we define

• 𝒫 is the set of generalized premises. A finite premise 𝑋 is identified with the generalized premise (𝑋, 𝑋, 𝑋, …).

• ℙ is the set of Borel probability measures on 𝕋.

• pm 𝒳, for 𝒳 a generalized premise, is the probability measure 𝜇 ∈ ℙ such that

  This uniquely defines 𝜇. The Epistemic Representation Theorem establishes that pm : 𝒫 → ℙ is surjective (onto): every 𝜇 ∈ ℙ can be represented by some 𝒳 ∈ 𝒫 that yields the same probabilities.

• The canonical topology for 𝒫 has as a sub-base all sets of the form

  for some finite query 𝐸 and real numbers 𝑎 < 𝑏. If we restrict 𝐸 to be a product term (conjunction of literals) then we still have a sub-base.

• The canonical topology for 𝒫 corresponds to the weak topology on ℙ, which has as a sub-base all sets of the form

  for some clopen set 𝐴 and real numbers 𝑎 < 𝑏. If we restrict 𝐴 to be a cylinder set then we still have a sub-base.

• Convergence in the canonical topology for 𝒫 corresponds to convergence in the weak topology for ℙ, which is the same as weak convergence of probability measures:

  when 𝜇ₙ = pm 𝒳ₙ and 𝜇 = pm 𝒳.

Generalized queries

• A generalized query is a sequence ℰ = (𝐸ᵢ) such that the sequence ([𝐸ᵢ])ᵢ converges pointwise to some 𝐴 ⊆ 𝕋. In this case we define [ℰ] = 𝐴. “Converges pointwise” means that for every 𝑥 ∈ 𝕋 there is some 𝑛₀ ∈ ℕ such that

• For a generalized query we define

  this limit is guaranteed to exist.

• 𝒬 is the set of generalized queries.

• We often use the Iverson bracket to convert truth-valued expressions into 0/1 numeric values: [𝑒] = 1 if 𝑒 is true, 0 otherwise. So the statement that the sequence (𝐴ᵢ) converges pointwise to 𝐴 could be written as

  Yes, the Iverson bracket conflicts with our notation [𝐸] for the set of truth assignments satisfying 𝐸. Sorry about that.

• 𝜟⁰₂ is the collection of sets 𝐴 ⊆ 𝕋 that are the pointwise limit of some sequence (𝐴ᵢ) ⊆ 𝕋 of clopen sets. It is a strict subset of the Borel measurable sets, but in keeping with our Jaynesian philosophy, we restrict our attention to these sets, and do not attempt to assign probabilities to sets lying outside of 𝜟⁰₂.

• 𝒬 and 𝜟⁰₂ are related as follows: 𝐴 ∈ 𝜟⁰₂ if and only if 𝐴 = [ℰ] for some ℰ ∈ 𝒬. Furthermore,

• sat : 𝒬 → 𝜟⁰₂ is a convient name for the [⋅] operation

• The pointwise topology on 𝜟⁰₂ is the product topology on indicator functions. It has as a sub-basis all sets of the form

  for some finite 𝐹 ⊆ 𝕋 and function 𝑓 : 𝐹 → 𝔹 specifying which truth assignments must belong to 𝐴 and which must not. In this topology, (𝐴ᵢ) → 𝐴 if and only if (𝐴ᵢ) converges pointwise to 𝐴.

• Our canonical topology for 𝜟⁰₂ is the sequentialization of the pointwise topology. This topology has exactly the same converging sequences and associated limits as the pointwise topology, but has just enough additional open sets to ensure that sequential continuity of a function 𝑓 : 𝜟⁰₂ → 𝑆 implies (topological) continuity. This extra step is necessary because the pointwise topology is not first countable.

• Our canonical topology on 𝒬 corresponds to the canonical topology on 𝜟⁰₂; in particular, it is the initial topology induced by [⋅] : 𝒬 → 𝜟⁰₂.

Topological terminology and notation

• We write 𝑥 ∼ 𝑦 to indicate that 𝑥 and 𝑦 are topologically indistinguishable: they belong to exactly the same open sets.

• We define a Kolmogorov quotient map to be a quotient map 𝑞 : 𝑆 → 𝑇 from a topological space 𝑆 to a topological space 𝑇 such that 𝑞(𝑥) = 𝑞(𝑦) if and only if 𝑥 ∼ 𝑦. Informally, a KQM collapses topologically indistinguishable points and then uniquely renames them, preserving topological properties.

• sat : 𝒬 → 𝜟⁰₂ is defined by sat ℰ ≜ [ℰ]. This is occasionally useful as an explicit name for the [⋅] operation.

• sat and pm are Kolmogorov quotient maps.

Discontinuities Everywhere

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We now look at the continuity of the probability function

Here’s the bad news up front: 𝘗𝘳 is discontinuous at every point in its domain. Upcoming posts will discuss what to do about this pathology. For now, let’s prove this claim.

Use of sequential product

By writing the domain of 𝘗𝘳 as seq(𝒬 × 𝒫) we are indicating that, for discussions of continuity, the topology of the domain is the sequential product of the topologies for 𝒬 and 𝒫. A sequence ((ℰᵢ, 𝒳ᵢ))ᵢ converges to (ℰ, 𝒳) in the sequential product topology if and only if (ℰᵢ) converges to ℰ and (𝒳ᵢ) converges to 𝒳. Using the sequential product ensures that 𝘗𝘳 is continuous at a point (ℰ, 𝒳) if and only if it is sequentially continuous at that point:

Continuity at (ℰ, 𝒳) means that, when writing 𝘗𝘳(ℰ | 𝒳), we don’t have to worry about the details of the limiting process that resulted in ℰ and 𝒳; any converging sequence of finite approximations (𝐸ᵢ, 𝑋ᵢ) will do.

The evaluation map 𝜋

The first thing we’ll do is reduce the continuity of 𝘗𝘳 to continuity of the corresponding measure-theoretic function.

Definition. The evaluation map 𝜋 : seq(𝜟⁰₂ × ℙ) → ℝ is defined by 𝜋(𝐴, 𝜇) ≜ 𝜇(𝐴).

Remark. Note that for the purpose of discussing continuity we take the topology of the domain of 𝜋 to be the sequentialization of the product of the topologies for 𝜟⁰₂ and ℙ, so that sequential continuity is the same as continuity, and 𝜋 is continuous at (𝐴, 𝜇) iff (𝜇ᵢ(𝐴ᵢ))ᵢ converges to 𝜇(𝐴) for any pair of sequences (𝐴ᵢ) and (𝜇ᵢ) converging to 𝐴 and 𝜇 respectively.

Theorem 1. The function 𝘗𝘳 : seq(𝒬 × 𝒫) → ℝ is continuous at (ℰ, 𝒳) if and only if 𝜋 : seq(𝜟⁰₂ × ℙ) → ℝ is sequential at ([ℰ], pm 𝒳).

Proof. The domains of 𝘗𝘳 and 𝜋 are sequential spaces, so sequential continuity and continuity are the same. Let 𝐴 = [ℰ] and 𝜇 = pm 𝒳. Using the facts that

• (ℰᵢ) → ℰ iff ([ℰᵢ])ᵢ → [ℰ],

• (𝒳ᵢ) → 𝒳 iff (pm 𝒳ᵢ)ᵢ → pm 𝒳,

• pm : 𝒫 → ℙ is onto, and

• [⋅] : 𝒬 → 𝜟⁰₂ is onto,

we find that 𝘗𝘳 is continuous at (ℰ, 𝒳) iff

iff

iff

iff 𝜋 is continuous at (𝐴, 𝜇). ∎

Prefix cylinders

As we’ve done before, we’ll find it useful to order the manifest symbols. A prefix cylinder of rank 𝑛 is a cylinder set in which the values of the first 𝑛 manifest symbols is fixed. An important property of prefix cylinders is that they never partially overlap: given any two prefix cylinders, either one is a subset of the other or they are disjoint.

Definition. 𝗌 : ℕ → ℳ is some arbitrary bijection from ℕ to the manifest symbols. As such it defines a total ordering on ℳ, with 𝗌(𝑛) being the 𝑛-th symbol.

Remark. Note that for any truth assignment 𝛼 ∈ 𝕋, 𝗌 defines a corresponding bit sequence 𝛼∘𝗌 ∈ 𝔹^𝜔.

Definition. For 𝛼 ∈ 𝕋, 𝑥 ∈ 𝔹^𝜔, 𝜎 ∈ 𝔹*, and 𝑛 ∈ ℕ:

• 𝑥 ↾ 𝑛 is the first 𝑛 bits of 𝑥, the bit-string 𝑥₀, …, 𝑥ₙ₋₁ ∈ 𝔹ⁿ.

• 𝛼 ↾ 𝑛 ≜ (𝛼∘𝗌) ↾ 𝑛, the first 𝑛 bits of the bit sequence corresponding to 𝛼.

• cyl 𝜎 ≜ { 𝛼 ∈ 𝕋 : 𝛼 ↾ 𝑛 = 𝜎 } where 𝑛 = |𝜎| is the length of 𝜎. We call cyl 𝜎 a prefix cylinder of rank 𝑛.

Remark. Recall that cylinder sets, which includes prefix cylinders, are clopen (both open and closed) in 𝕋. Furthermore, for any fixed 𝑛 the prefix cylinders of rank ≥ 𝑛 constitute a basis for the standard topology of 𝕋.

Proposition 2. Let 𝐶ₙ be a prefix cylinder of rank 𝑛. Then for every prefix cylinder 𝐶ₘ of rank 𝑚 ≤ 𝑛,

and hence for all 𝑥₁, 𝑥₂ ∈ 𝐶ₙ,

Proof. Let 𝐶ₘ = cyl 𝜎 and 𝐶ₙ = cyl 𝜎′. If 𝜎 = 𝜎′ ↾ 𝑚 then 𝐶ₙ ⊆ 𝐶ₘ, otherwise 𝐶ₙ has no elements in common with 𝐶ₘ. ∎

A theorem of Varadarajan

We’ll make use of a theorem of Varadarajan that says that any probability measure on a separable metric space can be approximated by a sequence of empirical probability measures. The space of truth assignments 𝕋 is a separable metric space.

In this section,

• 𝔐 is an arbitrary separable metric space;

• ℬ is the Borel 𝜎-algebra induced by 𝔐;

• 𝜇 is an arbitrary probability measure on ℬ.

• 𝛺 is the set of infinite sequences 𝑤 = (𝑤ᵢ) ⊆ 𝔐.

Definition. For each 𝑤 ∈ 𝛺 and 𝑛 ∈ ℕ, 𝑛 ≠ 0, the empirical distribution 𝜇ʷₙ is the measure with masses 1/𝑛 at each of the points 𝑤₀, …, 𝑤ₙ₋₁, i.e.,

We shall use the following specialization of a theorem of Varadarajan1:

Theorem 3. Let (𝛺, 𝒮, 𝑃) be the probability space that is the product of countably many copies of the probability space (𝔐, ℬ, 𝜇). Then 𝑃(𝜇ʷₙ ⇒ 𝜇) = 1.

What the theorem is saying is that if 𝑤 is an infinite sequence of independent and identically distributed draws from 𝜇, then 𝜇ʷₙ converges weakly to 𝜇 with probability 1. This has the following consequence:2

Corollary 4. Let 𝔐=𝕋. Then there exists a sequence (𝑥ᵢ) ⊆ 𝔐 such that 𝜇ˣₙ ⇒ 𝜇 and 𝑥ⱼ ≠ 𝑥ₖ for all 𝑗 ≠ 𝑘.

Proof. From Theorem 3 we have 𝑃(𝜇ʷₙ ⇒ 𝜇) = 1, so there must exist some specific 𝑤 ∈ 𝛺 such that 𝜇ʷₙ ⇒ 𝜇; otherwise we would have 𝑃(𝜇ʷₙ ⇒ 𝜇) = 0. However, the sequence (𝑤ᵢ) could have some repeats. To avoid repeats, we inductively define the sequence (𝑥ᵢ) as follows:

So (𝑥ᵢ) perturbs (𝑤ᵢ) to avoid repeats, while ensuring that 𝑥ₙ lies within the same rank-𝑛 prefix cylinder as 𝑤ₙ.

Clearly, 𝑥ⱼ ≠ 𝑥ₖ for all 𝑗 ≠ 𝑘. We must show that 𝜇ˣₙ ⇒ 𝜇. We do this by showing that, for any 𝑛₀ and any rank-𝑛₀ prefix cylinder 𝑈, (𝜇ˣᵢ(𝑈))ᵢ → 𝜇(𝑈).

First note that by Proposition 2,

whenever 𝑗 ≥ 𝑛₀. Therefore, for 𝑛 ≥ 𝑛₀,

where |𝛥| ≤ 𝑛₀. But |𝛥|/𝑛 ≤ 𝑛₀/𝑛 → 0 and 𝜇ʷₙ(𝑈) → 𝜇(𝑈) , so 𝜇ˣₙ(𝑈) → 𝜇(𝑈), as 𝑛 → ∞. ∎

Discontinuity of 𝜋 and 𝘗𝘳 everywhere

Now for the main result.

Theorem 5. 𝜋 is discontinuous at every point (𝐴, 𝜇) ∈ 𝜟⁰₂ × ℙ.

Proof. Let (𝑥ᵢ) ⊆ 𝕋 be the sequence of points and (𝜇ᵢ) ⊆ ℙ be the sequence of empirical probability measures weakly converging to 𝜇, whose existence Corollary 4 asserts:

and 𝑥ⱼ ≠ 𝑥ₖ for all 𝑗 ≠ 𝑘. Recall that weak convergence (𝜇ₙ ⇒ 𝜇) and convergence in the weak topology ((𝜇ᵢ) → 𝜇) are the same thing.

For all 𝑛 ∈ ℕ define

Singleton sets are closed in 𝕋, thus 𝐹ₙ is closed, as it is the union of a finite collection of closed sets, and hence 𝐹ₙ ∈ 𝜟⁰₂. Since 𝐴, 𝐹ₙ ∈ 𝜟⁰₂ this then ensures that 𝐴ₙ ∈ 𝜟⁰₂ for all 𝑛.

Since (𝜈ᵢ) is a subsequence of (𝜇ᵢ) we have (𝜈ᵢ) → 𝜇.

For any 𝑦 ∈ 𝕋 and sufficiently large 𝑛 we have 𝑦 ∉ 𝐹ₙ, hence 𝑦 ∈ 𝐴ₙ iff 𝑦 ∈ 𝐴:

• If 𝑦 ≠ 𝑥ⱼ for all 𝑗 ∈ ℕ then 𝑦 ∉ 𝐹ₙ for all 𝑛.

• If 𝑦 = 𝑥ⱼ for some 𝑗 ∈ ℕ then 𝑦 = 𝑥ⱼ for exactly one 𝑗 ∈ ℕ, and 𝑦 ∉ 𝐹ₙ for 𝑛 > 𝑗.

Thus (𝐴ᵢ) → 𝐴.

If 𝜋 is continuous at (𝐴, 𝜇) then it is sequentially continuous at (𝐴, 𝜇), and (𝜇ᵢ(𝐴ᵢ))ᵢ must converge to 𝜇(𝐴). We will show that it does not, and therefore 𝜋 is not continuous at (𝐴, 𝜇).

Suppose that 𝜇(𝐴) > 0; then 𝐴ₙ = 𝐴 ∖ 𝐹ₙ, and

If 𝜇(𝐴) = 0, then 𝐴ₙ = 𝐴 ∪ 𝐹ₙ, and so

Either way, (𝜈ᵢ(𝐴ᵢ))ᵢ does not converge to 𝜇(𝐴). ∎

Theorem 6. 𝘗𝘳 : seq(𝒬 × 𝒫) → ℝ is discontinuous at every point in its domain.

Proof. Follows directly from Theorems 5 and 1. ∎

Commentary

This seems like a rather discouraging result, but there are ways of working around it. In the next article I’ll discuss the options.

No Latent Vars

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No, this isn’t about faster response when playing online games; it’s about simplifying the theory of epistemic probability I’m developing here by getting rid of latent variables. It turns out we don’t need them.

Before proving that, let me relate how I realized that we could do without latent variables. I had been working on a post about joint continuity of the probability evaluation map 𝘗𝘳(⋅ | ⋅), with both arguments allowed to vary. Here’s the bad news: it’s not just discontinuous, it’s discontinuous at every single point in its domain. Yikes. It really does matter what trajectory you take in jointly approximating the generalized query and generalized premise. This is not a showstopper, though; there are ways of dealing with the issue, and in fact it’s one that the computational measure theory people have had to deal with.

My proof that 𝘗𝘳(⋅ | ⋅) is discontinuous at every point in its domain uses a 1958 theorem by V. S. Varadarajan about empirical distributions. (ChatGPT pointed me at this theorem, by the way.) One consequence of his theorem is that every measure on a separable metric space can be approximated arbitrarily closely by empirical probability distribution. An empirical probability distribution assigns weight 1/𝑛 to each of 𝑛 points in the space, for some finite nonzero 𝑛. Specifically, for any Borel probability measure 𝜇 on a separable metric space 𝔐, there exists a sequence (𝑥ᵢ) ⊆ 𝔐 such that 𝜇ˣₙ ⇒ 𝜇, where 𝜇ˣₙ is the empirical probability distribution created from the first 𝑛 members of the sequence (𝑥ᵢ). Note that 𝕋, the space of truth assignments, is a separable metric space. These fact can be used to construct, for any 𝐴 ∈ 𝜟⁰₂ and 𝜇 ∈ ℙ, sequences (𝐴ᵢ) ⊆ 𝜟⁰₂ and (𝜇ᵢ) ⊆ ℙ such that (𝐴ᵢ) → 𝐴, (𝜇ᵢ) → 𝜇, but (𝜇ᵢ(𝐴ᵢ))ᵢ converges to a value different than 𝜇(𝐴).

At this point one might ask: but what if we restrict the sets 𝐴ᵢ to correspond to finite queries 𝐸ᵢ and the probability measures 𝜇ᵢ to correspond to finite premises 𝑋ᵢ? Might this restriction be sufficient to prevent the pathology of (𝜇ᵢ(𝐴ᵢ))ᵢ not converging to 𝜇(𝐴)?

I haven’t finished the proof, but it appears not. You can modify the original discontinuity proof to replace each empirical distribution 𝜇ˣₙ with a distribution of the form

where

• 𝐶(𝑛,𝑖) is a prefix cylinder set that contains 𝑥ᵢ, fixes the values of at least the first 𝑛 manifest symbols, and has no overlap with 𝐶(𝑛,𝑗) for 𝑗 ≠ 𝑖; and

• 𝜆(𝐴) is the uniform distribution over the set 𝐴.

This new sequence of distributions also weakly converges to 𝜇: (𝜈ˣₙ) ⇒ 𝜇.

And a lightbulb goes off in my head. Wait a minute! Cylinder sets correspond to finite queries, 𝐶(𝑛,𝑖) = [𝜑(𝑛,𝑖)] for some propositional formula 𝜑(𝑛,𝑖), and 𝜆(𝐶(𝑛,𝑖)) = pm(𝜑(𝑛,𝑖)), and

But 𝑋ₙ is a finite premise that makes no mention of any latent symbol. We have constructed a generalized premise 𝒳 = (𝑋ᵢ) with pm 𝑋ₙ ⇒ 𝜇, and hence pm 𝒳 = 𝜇, without using any latent symbols. We did this for arbitrary 𝜇 ∈ ℙ. So for any generalized premise 𝒴 we can find an equivalent generalized premise 𝒳, i.e. pm 𝒳 = pm 𝒴, constructed entirely without the use of latent symbols.

Once I knew this was possible I was able to construct a simpler proof—the above is just a rough proof sketch—and that is what I present to you now:

Theorem. For every generalized premise 𝒳 ∈ 𝒫 there is another 𝒴 = (𝑌ᵢ) ∈ 𝒫 such that 𝒴 ∼ 𝒳 (pm 𝒴 = pm 𝒳) and each 𝑌ₙ contains no latent propositional symbols.

Proof. The idea is this: with regards to a specific finite query 𝐸, any propositional symbol not occurring in 𝐸 functions as a latent symbol. Any such symbol in 𝑋 can be consistently renamed to a different symbol not already appearin in 𝑋 or 𝐸 without changing 𝘗𝘳(𝐸 | 𝑋). In particular, if we obtain 𝑌 from 𝑋 by replacing each latent symbol in 𝑋 with a new, uniquely chosen manifest symbol not already occurring in 𝐸 or 𝑋, then 𝘗𝘳(𝐸 | 𝑌) = 𝘗𝘳(𝐸 | 𝑋). Now enumerate the manifest symbols in some arbitrary ordering. If, as 𝑛 increases, the replacement symbols for the latent symbols in 𝑋ₙ are chosen from later and later in this ordering, then for any given finite query 𝐸 and large enough 𝑛, none of the replacement symbols will occur in 𝐸.

Now to formalize that. Define the following:

• 𝗋 : ℕ → ℒ and 𝗌 : ℕ → ℳ are arbitrary bijections uniquely enumerating the latent symbols ℒ and the manifest symbols ℳ, respectively. That is, the 𝑛-th latent symbol is 𝗋(𝑛) and the 𝑛-th manifest symbol is 𝗌(𝑛)

• Let 𝒳 = (𝑋ᵢ).

• 𝑙ₙ ≜ max { 𝑗 ∈ ℕ : 𝗋(𝑗) appears in 𝑋ₙ }. Thus 𝑟(𝑗) does not appear in 𝑋ₙ for any 𝑗 > 𝑙ₙ.

• 𝑚ₙ ≜ max { 𝑗 ∈ ℕ : 𝗌(𝑗) appears in 𝑋ₙ }. Thus 𝗌(𝑗) does not appear in 𝑋ₙ for any 𝑗 > 𝑚ₙ.

• 𝑌ₙ is obtained from 𝑋ₙ by replacing each occurrence of a latent symbol 𝗋(𝑗) with the manifest symbol 𝗌(𝑚ₙ + 𝗋(𝑗) + 𝑛 + 1). Note that the replacement symbol is guaranteed not to occur in 𝑋ₙ.

• 𝒴 = (𝑌ᵢ).

We then have the following:

• 𝑌ₙ contains no latent symbols.

• 𝘗𝘳(𝐸 | 𝑌ₙ) = 𝘗𝘳(𝐸 | 𝑋ₙ) if 𝐸 contains no symbol 𝗌(𝑗) for 𝑗 > 𝑚ₙ + 𝑙ₙ + 𝑛 + 1 > 𝑛.

• Thus for any fixed finite query 𝐸 and sufficiently large 𝑛, 𝘗𝘳(𝐸 | 𝑌ₙ) = 𝘗𝘳(𝐸 | 𝑋ₙ); this implies that the sequence (𝘗𝘳(𝐸 | 𝑌ᵢ))ᵢ converges to the same limit as the sequence (𝘗𝘳(𝐸 | 𝑋ᵢ))ᵢ.

• Thus 𝒴 is a generalized premise and 𝒴 ∼ 𝒳.

∎

Generalized Queries Part 4

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In this article we’ll define and investigate the topology of generalized queries. In short, we’ll construct a topology in which convergence is the same as pointwise convergence of sets, and continuity is the same as sequential continuity.

Some nomenclature and a quick review

Recall that a generalized query ℰ can be thought of as representing a set of truth assignments on ℳ, the set of manifest propositional symbols: it is a sequence of finite queries (𝐸ᵢ) ⊆ 𝛷(ℳ) such that1

where [𝐸] is the set of truth assignments on ℳ that satisfy the propositional formula 𝐸, and 𝕋 ≜ (ℳ → 𝔹) is the set of truth assignments on ℳ. We then extended the definition of [⋅] to generalized queries by defining [ℰ] to be this set 𝐴, and showed that2

We also showed that [ℰ] ∈ 𝜟⁰₂, those Borel sets that can be expressed both as the intersection of a countable number of open sets and as the union of a countable number of closed sets.

It’s convenient to give this operation [⋅] a name:

Definition. sat : 𝒬 → 𝜟⁰₂ is defined by sat ℰ ≜ [ℰ].

By way of analogy, recall that the Kolmogorov quotient map pm : 𝒫 → ℙ maps a generalized premise to the probability measure it represents, and links the topology we defined for generalized premises to the weak topology on probability measures: 𝑉 ⊆ ℙ is open iff pm⁻¹[𝑉] ⊆ 𝒫 is open, and pm 𝒳₁ = pm 𝒳₂ iff 𝒳₁ ∼ 𝒳₂ (𝒳₁ and 𝒳₂ are topologically indistinguishable.)

We likewise will define linked topologies on 𝒬 and 𝜟⁰₂, such that sat is a Kolmogorov quotient map between the respective topological spaces.

Initial topology

We approach this via the idea of an initial topology. We’ll define a topology on 𝜟⁰₂, and our topology on 𝒬 will be the initial topology induced by sat.

Definition. Given a set 𝑆 and topological space 𝑇, the initial topology induced by function 𝑓 : 𝑆 → 𝑇 on 𝑆 is the coarsest (smallest) topology 𝜎 on 𝑆 that makes 𝑓 continuous.

Why the coarsest topology? Because if 𝑓 is continuous for a topology 𝜎 on 𝑆, it is also continuous for any refinement of 𝜎, i.e. any topology 𝜎′ ⊇ 𝜎; in particular it is continuous for the discrete topology 𝜎′ = ℘(𝑆), the finest (largest) possible topology containing all subsets of 𝑆.

(Note: more generally, one can talk about the initial topology induced by an entire collection of functions, but we only need the single-function case.)

The following standard result gives a simple characterization of the initial topology induced by a single function 𝑓:

Proposition 1. If 𝑆 is a set and (𝑇, 𝜏) a topological space, the initial topology induced by 𝑓 : 𝑆 → 𝑇 on 𝑆 is

Proof. See the PDF. ∎

If 𝑓 is surjective (onto) and distinct elements of 𝑇 are always topologically distinct, then the initial topology also makes 𝑓 a Kolmogorov quotient map:

Proposition 2. Let 𝑆 be a set, (𝑇, 𝜏) a 𝑇₀-space, and 𝑓 : 𝑆 → 𝑇 be surjective. The initial topology induced by 𝑓 on 𝑆 makes 𝑓 a Kolmogorov quotient map from (𝑆, 𝜎) to (𝑇, 𝜏).

Proof. See the PDF. ∎

So from this point on we will focus on choosing an appropriate topology for 𝜟⁰₂, and this will give us our topology on 𝒬.

Some notes on pointwise convergence

We want our topology 𝜎 on 𝜟⁰₂ to have this property:

A sequence (𝐴ᵢ) ⊆ 𝜟⁰₂ converges pointwise to a set 𝐴 ∈ 𝜟⁰₂ if and only if (𝐴ᵢ) → 𝐴 in 𝜎.

With that in mind, we first need to discuss some general properties of pointwise convergence.

Convergence to a point outside of 𝜟⁰₂

We have to be careful with taking pointwise limits: although the pointwise limit of a sequence of clopen sets always belongs to 𝜟⁰₂ (when it is pointwise convergent), an arbitrary sequence (𝐴ᵢ) ⊆ 𝜟⁰₂ may converge pointwise to a a set 𝐴 ⊆ 𝕋 that is not itself in 𝜟⁰₂.

Here is an example. Order the elements of ℳ as 𝑠₀, 𝑠₁, 𝑠₂, … and define

Each 𝐴ₙ is open, as it is the countable union (over all 𝑚 ≥ 𝑛) of single-coordinate cylinder sets. So 𝐴ₙ ∈ 𝜮⁰₁ ⊆ 𝜟⁰₂. Since this is a decreasing sequence (𝐴ₙ₊₁ ⊆ 𝐴ₙ) the pointwise limit is

which, as we saw in Part 2, is not a 𝜟⁰₂ set.

So when we speak of sequences in 𝜟⁰₂ that converge pointwise, we will be careful to specify that they converge pointwise to a set in 𝜟⁰₂, as these are the only pointwise-convergent sequences of interest to us.

Pointwise convergence and convergence of probabilities

Theorem 4 of Part 3 gave a result for sequences of clopen subsets of 𝕋. Here we generalize that result to arbitrary sequences in 𝜟⁰₂(𝕋); the proof is nearly identical.

Theorem 3. A sequence (𝐴ᵢ) ⊆ 𝜟⁰₂ converges pointwise to set 𝐴 iff for every 𝜇 ∈ ℙ, (𝜇(𝐴ᵢ))ᵢ → 𝜇(𝐴).

Proof. See the PDF. ∎

Theorem 3 has several consequences. First we get another way of seeing that 𝘗𝘳(ℰ | 𝒳) is well defined for a generalized query ℰ, in that the limit limₙ 𝘗𝘳(𝐸ₙ | 𝒳) exists when ℰ = (𝐸ᵢ), and we also get the explicit expression for that limit.

Corollary 4. Let ℰ = (𝐸ᵢ) ∈ 𝒬, 𝒳 ∈ 𝒫, and 𝜇 = pm 𝒳. Then (𝘗𝘳(𝐸ᵢ | 𝒳))ᵢ → 𝜇([ℰ]).

Proof. Apply Theorem 3 using

• 𝐴ᵢ = [𝐸ᵢ], 𝐴 = [ℰ],

• 𝘗𝘳(𝐸ᵢ | 𝒳) = 𝜇([𝐸ᵢ]), and

• ([𝐸ᵢ])ᵢ converges pointwise to [ℰ].

∎

This in turn lets us extend the identity 𝘗𝘳(𝜑 | 𝒳) = (pm 𝒳)([𝜑]) for finite queries 𝜑, to generalized queries ℰ:

Corollary 5. For all ℰ ∈ 𝒬 and 𝒳 ∈ 𝒫, 𝘗𝘳(ℰ | 𝒳) = 𝜇([ℰ]) where 𝜇 ≜ pm 𝒳.

Proof. Let ℰ = (𝐸ᵢ) and use 𝘗𝘳(ℰ | 𝒳) ≜ limₙ 𝘗𝘳(𝐸ₙ | 𝒳) = 𝜇([ℰ]). ∎

We can also restate Theorem 3 in terms of generalized premises and generalized queries:

Corollary 6. For every sequence (ℰᵢ) ⊆ 𝒬 and ℰ ∈ 𝒬, the sequence ([ℰᵢ])ᵢ converges pointwise to [ℰ] iff for all 𝒳 ∈ 𝒫, (𝘗𝘳(ℰᵢ | 𝒳))ᵢ → 𝘗𝘳(ℰ | 𝒳).

Proof. Use 𝐴ᵢ = [ℰᵢ], 𝐴 = [ℰ], and 𝜇 = pm 𝒳 in Theorem 3. ∎

The pointwise topology on 𝜟⁰₂

If we identify a set 𝐴 ∈ 𝜟⁰₂ with its indicator function 𝜒_𝐴, then the obvious topology to ensure that pointwise convergence and topological convergence are the same is the product topology on 𝜟⁰₂; this topology has as a sub-base all sets of the form

for some 𝛼 ∈ 𝕋 and 𝑏 ∈ 𝔹. This defines a base consisting of all sets of form

for some function 𝑎 : 𝐹 → 𝔹 where 𝐹 is a finite subset of 𝕋. That is, 𝑎 defines a finite set of truth assignments (𝑎⁻¹[{ 1 }]) that every member of 𝐵(𝑎) must contain, and a finite set of truth assignments (𝑎⁻¹[{ 0 }]) that every member of 𝐵(𝑎) must not contain.

Definition. We call the above topology on 𝜟⁰₂ the pointwise topology, 𝜏ₚₜ.

We call this the pointwise topology because it has the desirable property that a sequence (𝐴ᵢ) ⊆ 𝜟⁰₂ converges to 𝐴 ∈ 𝜟⁰₂ in this topology if and only if (𝐴ᵢ) converges pointwise to 𝐴:3

Theorem 7. Suppose 𝐴ₙ, 𝐴 ∈ 𝜟⁰₂ for all 𝑛. Then (𝐴ᵢ) → 𝐴 in 𝜏ₚₜ if and only if (𝐴ᵢ) converges pointwise to 𝐴.

Proof. See the PDF. ∎

Although the collection of all sets 𝐵(𝑎) is a base for 𝜏ₚₜ, it is not countable: there are uncountably many truth assignments 𝛼 ∈ 𝕋, hence uncountably many finite-domain functions 𝑎 :⊆ 𝕋 → 𝔹. In fact, no countable base for this topology exists. This is a problem, because we have been relying on the existence of a countable base to ensure that the topologies we use are sequential, and therefore sequential limits/continuity are the same as (topological) limits/continuity.

Recall the following:

• A function 𝑓 : 𝑆 → 𝑇 is sequentially continuous at 𝑠 ∈ 𝑆 iff (𝑓(𝑠ᵢ))ᵢ → 𝑓(𝑠) whenever (𝑠ᵢ) → 𝑠. The function 𝑓 is sequentially continuous iff it is sequentially continuous at all points 𝑠 ∈ 𝑆.

• If 𝑓 : 𝑆 → 𝑇 is (topologically) continuous at 𝑠 then it is sequentially continuous at 𝑠.

• A sequential topology on 𝑆 is defined to be one for which the reverse implication also holds: for every topological space 𝑇 and sequentially continuous function 𝑓 : 𝑆 → 𝑇, 𝑓 is also (topologically) continuous.

Unfortunately, the topology 𝜏ₚₜ is not sequential.

Theorem 8. 𝜏ₚₜ is not a sequential topology.

Proof. To show this we demonstrate a function 𝑓 : 𝜟⁰₂ → ℝ that is sequentially continuous in 𝜏ₚₜ at all points in its domain, and yet (topologically) discontinuous at ∅. Let 𝜇 ∈ ℙ be the “fair-coin” measure that assigns 𝜇([𝑠]) = 1/2 independently for all 𝑠 ∈ ℳ, and let 𝑓(𝐴) ≜ 𝜇(𝐴). It is straightforward to show that 𝑓 is sequentially continuous, but (topologically) discontinuous at 𝐴 = ∅: although 𝑓(∅) = 0, for any 𝜀 > 0 and any open neighborhood of ∅ we can construct some 𝐵 ∈ 𝜟⁰₂ with 𝜇(𝐵) > 1 - 𝜀. Details in the PDF for the specific case of 𝜀 = 0.1, which suffices to make 𝑓 discontinuous. ∎

The sequential topology on 𝜟⁰₂

As discussed in Sequential Spaces, there is an easy fix for the above problem: instead of using 𝜏ₚₜ, we will use its sequentialization 𝜏ₛ.

Definition. 𝜏ₛ ≜ seq 𝜏ₚₜ, the sequentialization of the pointwise topology on 𝜟⁰₂.

As the sequentialization of 𝜏ₚₜ, the topology 𝜏ₛ has the following properties:

• 𝜏ₛ ⊇ 𝜏ₚₜ, that is, 𝜏ₛ is finer than 𝜏ₚₜ; it only adds additional open sets.

• A sequence (𝐴ᵢ) ⊆ 𝜟⁰₂ converges to 𝐴 ∈ 𝜟⁰₂ according to the pointwise topology 𝜏ₚₜ iff it converges to 𝐴 according to the sequential topology 𝜏ₛ. (The two topologies have exactly the same converging sequences and limits of those sequences.)

• 𝜏ₛ is a sequential topology: sequential limits/continuity are the same as (topological) limits/continuity.

See also Theorems 2 and 3 of Sequential Spaces for other properties 𝜏ₛ has as a sequential topology.

There is no easily-described basis for 𝜏ₛ, but it is straightforward to generate examples of open sets in 𝜏ₛ: for any topological space 𝑇 and continuous function 𝑓 : 𝜟⁰₂ → 𝑇, every set 𝑓⁻¹[𝑉] is open in (𝜟⁰₂, 𝜏ₛ). This works because we can prove that 𝑓 is continuous by merely showing that it is sequentially continuous; we don’t have to work with the definition in terms of open sets.

Our first collection of open sets is parameterized by a probability measure 𝜇 ∈ ℙ and nonempty real interval (𝑎, 𝑏): the collection of sets whose probability under 𝜇 lies in the interval (𝑎, 𝑏).

Theorem 9. 𝑈 ≜ { 𝐴 ∈ 𝜟⁰₂ : 𝜇(𝐴) ∈ (𝑎, 𝑏) } is an open set in 𝜏ₛ for any 𝑎 < 𝑏 and Borel probability measure 𝜇 on 𝕋.

Proof. 𝜇 is sequentially continuous in 𝜏ₛ: if (𝐴ᵢ) → 𝐴 in 𝜏ₛ then (𝐴ᵢ) converges pointwise to 𝐴, hence by Dominated Convergence for sets we have (𝜇(𝐴ᵢ))ᵢ → 𝜇(𝐴). But 𝑈 = 𝜇⁻¹[(𝑎, 𝑏)], and (𝑎, 𝑏) is open in ℝ, so 𝑈 is open in 𝜏ₛ. ∎

Our second collection of open sets is parameterized by a probability measure 𝜇 ∈ ℙ, set 𝐵 ∈ 𝜟⁰₂, and positive real 𝑟 > 0: the collection of sets lying within 𝑑_𝜇-distance 𝑟 of 𝐵.

Theorem 10. 𝑈 ≜ { 𝐴 ∈ 𝜟⁰₂ : 𝑑_𝜇(𝐴, 𝐵) < 𝑟 } is an open set in 𝜏ₛ for all 𝑟 > 0, set 𝐵 ∈ 𝜟⁰₂, and Borel probability measure 𝜇 on 𝕋.

Proof. Let 𝑓 : 𝜟⁰₂ → ℝ be defined by 𝑓(𝐴) ≜ 𝑑_𝜇(𝐴, 𝐵). Then

and (-1, 𝑟) is open in ℝ, so if 𝑓 is continuous then 𝑈 is an open set.

In the PDF it is proven that 𝑓 is sequentially continuous, hence continuous. ∎

The induced topology on 𝒬

We now have our topology for generalized queries:

Definition. 𝜏_𝒬 is the initial topology on 𝒬 induced by sat : 𝒬 → (𝜟⁰₂, 𝜏ₛ).

Thus the open sets of 𝜏_𝒬 are exactly those sets of the form sat⁻¹[𝑉] for some open set 𝑉 of 𝜏ₛ. For example, applying Theorems 9 and 10 we find that

• { ℰ ∈ 𝒬 : 𝘗𝘳(ℰ | 𝒳) ∈ (𝑎, 𝑏) } is an open set in 𝜏_𝒬 for any generalized premise 𝒳 ∈ 𝒫 and pair of real numbers 𝑎 < 𝑏.

• { ℰ ∈ 𝒬 : 𝑑_𝜇([ℰ], [ℰ₀]) < 𝑟 } is an open set in 𝜏_𝒬, where 𝜇 = pm 𝒳, for any generalized premise 𝒳 ∈ 𝒫, generalized query ℰ₀ ∈ 𝒬, and real number 𝑟 > 0.

Another property is that convergent sequences map directly between 𝒬 and 𝜟⁰₂ as expected, linked by sat. This is in fact true for the domain and codomain of any Kolmogorov quotient map.

Proposition 11. Let 𝑆 and 𝑇 be topological spaces and 𝑞 : 𝑆 → 𝑇 a Kolmogorov quotient map. For any (𝑥ᵢ) ⊆ 𝑆 and 𝑥 ∈ 𝑆,

Proof. See the PDF. ∎

Corollary 12. For every sequence (ℰᵢ) ⊆ 𝒬 and ℰ ∈ 𝒬,

For every sequence (𝒳ᵢ) ⊆ 𝒫 and 𝒳 ∈ 𝒫,

Proof. Use 𝑞 = sat and 𝑞 = pm in Proposition 11. ∎

Up next

Now that we have a topology on 𝒬 and tools for proving sequential continuity (and hence topological continuity), we will in upcoming articles investigate various operations on generalized queries and their continuity. These operations include

• taking probabilities of GQs: 𝘗𝘳(ℰ | 𝒳) for 𝒳 ∈ 𝒫;

• logical operations on GQs: ℰ₁ ∨ ℰ₂ and ¬ℰ;

• conditionalization using GQs: ℰ ∧ 𝒳 for 𝒳 ∈ 𝒫.

Recall that every pseudo-metrizable space is first countable, as is every second-countable space; hence, for the sorts of spaces of interest to us, (topological) [limits]/continuity and sequential [limits]/continuity will always be the same.1

As we’ll discuss in the next post, with generalized queries we have left the realm of first countable spaces and so we can no longer take for granted the equivalence of certain topological notions and their sequential analogs. In this post we’ll see how to ensure that the most important equivalences are retained, although some equivalences will still be lost.

Preliminaries: some terminology and notation

We’ll start by introducing or reviewing some standard terminology and notation that will facilitate the discussion.

• We write (𝑥ᵢ) ⊆ 𝑆 to mean that 𝑥ₙ ∈ 𝑆 for all 𝑛 ∈ ℕ.

• We have been taking 𝑖 implicitly to be a bound variable indicating the index of the sequence when writing (𝑥ᵢ) or even expressions such as (𝑥ₙ₊ᵢ) or (𝑓(𝑥ᵢ)); when we want to be explicit, we will indicate the bound variable as a subscript to the parenthesized expression, writing (𝑥ᵢ)ᵢ, (𝑥ₙ₊ᵢ)ᵢ, (𝑓(𝑥ᵢ))ᵢ, etc.

• We say that (𝑥ᵢ) is eventually in 𝐴 if there is some 𝑛 ∈ ℕ such that (𝑥ₙ₊ᵢ)ᵢ ⊆ 𝐴; that is, some tail of the sequence lies entirely in 𝐴.

• We may simply write 𝜏 for the topological space (𝑆, 𝜏) when discussing multiple topological spaces having the same underlying set of values 𝑆. (This echoes the usual practice of simply writing 𝑆 when the associated topology 𝜏 is clear.)

• A reminder: (𝑥ᵢ) → 𝑦 (the sequence converges to 𝑦) means that (𝑥ᵢ) is eventually in every open neighborhood of 𝑦. When discussing multiple topologies on the same set of values, we may append “in 𝜏” to disambiguate what we mean by “open set.”

• The phrase “an open 𝑈 ∋ 𝑥” means “an open neighborhood 𝑈 of 𝑥,” i.e. an open set 𝑈 such that 𝑥 ∈ 𝑈.

The results presented here are all standard, and I omit the longer, more complex proofs.

Sequentially open/closed

We start by defining sequential versions of some topological concepts:

Definition. Let 𝑆 be a topological space and 𝐴 ⊆ 𝑆.

• 𝐴 is sequentially open if for every (𝑥ᵢ) ⊆ 𝑆 and 𝑦 ∈ 𝐴, (𝑥ᵢ) → 𝑦 ⟹ (𝑥ᵢ) is eventually in 𝐴.

• 𝐴 is sequentially closed if for every (𝑥ᵢ) ⊆ 𝐴 and 𝑦 ∈ 𝑆, (𝑥ᵢ) → 𝑦 ⟹ 𝑦 ∈ 𝐴.

• The sequential closure of 𝐴 is the set of points to which some sequence in 𝐴 converges:

  This notation leaves the topology implicit; if we want to make it explicit that “(𝑥ᵢ) → 𝑦” above means (𝑥ᵢ) → 𝑦 in 𝜏, we write scl_𝜏 𝐴.

Here are some basic properties of these concepts:

Theorem 1. For any topological space 𝑆 and 𝐴 ⊆ 𝑆,

1. 𝐴 is sequentially open ⟺ ∼𝐴 is sequentially closed;

2. 𝐴 is open ⟹ 𝐴 is sequentially open;

3. 𝐴 is closed ⟹ 𝐴 is sequentially closed;

4. 𝐴 ⊆ scl 𝐴 ⊆ cl 𝐴;

5. 𝐴 is sequentially closed ⟺ scl 𝐴 = 𝐴.

Proof. 1. ( ⇒ ) Suppose that 𝐴 is sequentially open. Then for every (𝑥ᵢ) ⊆ ∼𝐴 and 𝑦 ∈ 𝑆 such that (𝑥ᵢ) → 𝑦 we cannot have 𝑦 ∈ 𝐴, as that would imply that (𝑥ᵢ) is eventually in 𝐴, a contradiction; therefore 𝑦 ∈ ∼𝐴.

1. (⇐) Suppose that ∼𝐴 is sequentially closed but 𝐴 is not sequentially open. Then there exists (𝑥ᵢ) ⊆ 𝑆 and 𝑦 ∈ 𝐴 such that (𝑥ᵢ) → 𝑦 but 𝑥ₙ ∈ ∼𝐴 for infinitely many 𝑛. Define 𝑤ₙ to be the 𝑛-th element of (𝑥ᵢ) that belongs to ∼𝐴; then (𝑤ᵢ) is a subsequence of (𝑥ᵢ), hence (𝑤ᵢ) → 𝑦, and since (𝑤ᵢ) ⊆ ∼𝐴 this implies 𝑦 ∈ ∼𝐴, a contradiction.

2. Suppose that 𝐴 is open and (𝑥ᵢ) → 𝑦 ∈ 𝐴; then there is an open neighborhood 𝑈 of 𝑦 such that 𝑈 ⊆ 𝐴, and (𝑥ᵢ) is eventually in 𝑈, hence eventually in 𝐴.

3. Suppose that 𝐴 is closed; then ∼𝐴 is open, hence ∼𝐴 is sequentially open, hence 𝐴 is sequentially closed.

4. Let 𝑦 ∈ 𝐴. If 𝑥ₙ = 𝑦 for all 𝑛 then (𝑥ᵢ) ⊆ 𝐴 and (𝑥ᵢ) → 𝑦, so 𝑦 ∈ scl 𝐴. So 𝐴 ⊆ scl 𝐴.

Suppose 𝑦 ∈ scl 𝐴. Then there exists (𝑥ᵢ) ⊆ 𝐴 such that (𝑥ᵢ) → 𝑦. For any open 𝑈 ∋ 𝑦 we have that (𝑥ᵢ) is eventually in 𝑈, and hence 𝑈 ∩ 𝐴 ≠ 0; this implies 𝑦 ∈ 𝖼𝗅 𝐴. So scl 𝐴 ⊆ 𝖼𝗅 𝐴.

5. 𝐴 is sequentially closed iff every sequence in 𝐴 that converges, converges to a point in 𝐴. This is the same as saying that scl 𝐴 adds no new points not already in 𝐴. ∎

For the closure operator we have 𝖼𝗅 𝐴 is closed and 𝖼𝗅(𝖼𝗅 𝐴) = 𝖼𝗅 𝐴, but scl lacks the corresponding properties: scl 𝐴 is not guaranteed to be sequentially closed, nor is scl(scl 𝐴) = scl 𝐴 guaranteed to hold. The additional points added by the sequential closure may create new converging sequences that converge to points outside of the sequential closure.

If we iterate scl we can obtain an operator that behaves more like the closure operator. For any ordinal 𝛼, define

• scl⁰(𝐴) = 𝐴

• scl^{𝛼+1}(𝐴) = scl(scl^𝛼(𝐴))

• scl^𝛼(𝐴) = ⋃_{𝛽 < 𝛼} scl^𝛼(𝐴) if 𝛼 is a limit ordinal.

This transfinite sequence stabilizes by 𝛼 = 𝜔₁, the first uncountable ordinal, so

• scl^{𝜔₁}(scl^{𝜔₁}(𝐴)) = scl^{𝜔₁}(𝐴), and

• scl^{𝜔₁}(𝐴) is sequentially closed.

Review: limits and continuity

In A Digression on Topology (4): Limits and Continuity we discussed topological vs. sequential limits and continuity. Recall the following definitions, where 𝑆₁ and 𝑆₂ are topological spaces, 𝑓 : 𝑆₁ → 𝑆₂, 𝑥 ∈ 𝑆₁ and 𝑣 ∈ 𝑆₂:

• 𝑓(𝑦) → 𝑣 as 𝑦 → 𝑥 (𝑣 is a (topological) limit of 𝑓(𝑦)) if

• 𝑓(𝑦) →ₛ 𝑣 as 𝑦 → 𝑥 (𝑣 is a sequential limit of 𝑓(𝑦)) if

• 𝑓 is (topologically) continuous at 𝑥 if 𝑓(𝑦) → 𝑓(𝑥) as 𝑦 → 𝑥. Equivalently,

• 𝑓 is sequentially continuous at 𝑥 if 𝑓(𝑦) →ₛ 𝑓(𝑥) as 𝑦 → 𝑥. Equivalently,

• 𝑓 is (topologically) continuous if it is continuous at every point in its domain. Likewise, 𝑓 is sequentially continuous if it is sequentially continuous at every point in its domain.

Analogous to the facts that an open set is sequentially open and a closed set is sequentially closed, we have the following:

• 𝑓(𝑦) → 𝑣 as 𝑦 → 𝑥 ⟹ 𝑓(𝑦) →ₛ 𝑣 as 𝑦 → 𝑥.

• 𝑓 is continuous at 𝑥 ⟹ 𝑓 is sequentially continuous at 𝑥.

The converses do not hold in general for arbitrary topological spaces.

Definitions of “sequential space”

A sequential space is one in which continuity and sequential continuity are the same. There are, however, a variety of equivalent ways of characterizing a sequential space.

Theorem 2. The following are equivalent for any topological space 𝑆:

1. For every topological space 𝑇 and function 𝑓 : 𝑆 → 𝑇, 𝑓 is continuous ⟺ 𝑓 is sequentially continuous.

2. Every 𝐴 ⊆ 𝑆 that is sequentially open is open.

3. Every 𝐴 ⊆ 𝑆 that is sequentially closed is closed.

4. 𝑆 is a quotient of a first-countable space.

5. 𝑆 is a quotient of a metric space.

Proof. Standard results, proof omitted. ∎

When we say “𝑆 is a quotient of 𝑅” above, we mean “there is a quotient map from 𝑅 to 𝑆”. Think of the points of 𝑅 as representations of the points in 𝑆, with every point in 𝑆 having one or more representations in 𝑅. Recall that if 𝑓 : 𝑅 → 𝑆 is a quotient map then the topology of 𝑅 defines the topology of 𝑆: a set 𝑉 is open in 𝑆 iff 𝑓⁻¹[𝑉] is open in 𝑅.

Definition. A sequential space is a topological space (𝑆, 𝜏) satisfying any of the equivalent conditions 1–5 above. For an arbitrary topology 𝜏 on an arbitrary set 𝑆, we say that 𝜏 is sequential if (𝑆, 𝜏) is a sequential space.

We are most interested in property 1 above, but the other equivalent characterizations also come in handy. For instance, 2 and 3 are often easier to prove than 1; and 4 and 5 are often a more convenient characterization when dealing with computable representations.

Some properties of sequential spaces

In a sequential space, sequentially open/closed is the same as open/closed:

Theorem 3. Let 𝑆 be a sequential space and 𝐴 ⊆ 𝑆. Then

• 𝐴 is open iff 𝐴 is sequentially open;

• 𝐴 is closed iff 𝐴 is sequentially closed.

Proof. Combine items 2 and 3 of Theorem 1 with items 2 and 3 of Theorem 2.

Continuity and sequential continuity are also the same for a sequential space, according to item 1 of Theorem 2: if 𝑆 is a sequential space then for every topological space 𝑇 and function 𝑓 : 𝑆 → 𝑇, f is continuous ⟺ f is sequentially continuous . But beware! It is sequential continuity and continuity on the entire domain of the function that are equivalent, not equivalence of sequential continuity and continuity at a single point. More on this below.

Sequential topologies are entirely defined by their convergent sequences. By working only with sequential topologies we are in effect saying that what matters to us, topologically, is which sequences are convergent (and to what values):

Theorem 4. Let 𝜏 and 𝜏′ be sequential topologies on the same set 𝑆 such that (𝑥ᵢ) → 𝑦 in 𝜏 ⟺ (𝑥ᵢ) → 𝑦 in 𝜏′, for all (𝑥ᵢ) ⊆ 𝑆 and 𝑦 ∈ 𝑆. Then 𝜏 = 𝜏′.

Proof. Writing “e.i.” for “is eventually in,” a set 𝐴 ⊆ 𝑆 is sequentially open in 𝜏 iff

But (𝑥ᵢ) → 𝑦 in 𝜏 iff (𝑥ᵢ) → 𝑦 in 𝜏′, so the above condition is equivalent to

which is true iff 𝐴 is sequentially open in 𝜏′. Furthermore, 𝜏 and 𝜏′ are sequential topologies, so

∎

Being sequential is a property that is not inherited by arbitrary subspaces of a sequential space. There are, however, two important cases where sequentiality is inherited:

Theorem 5. Let 𝑆 be a sequential space and let 𝑅 ⊆ 𝑆.

1. If 𝑅 is open then 𝑅 (as a topological space) is also sequential.

2. If 𝑅 is closed then 𝑅 (as a topological space) is also sequential.

Proof. Standard result; proof given in PDF version. ∎

This will be important when dealing with the continuity of functions that are defined on only a subset of a sequential space: as long as the function’s domain is either an open or a closed subset of the sequential space, the domain forms a sequential space.

Some non-properties of sequential spaces

Now let’s consider some properties you might expect sequential spaces to have, and which first countable spaces do have, but which sequential spaces are not guaranteed to have. In the following we take 𝑆 and 𝑆′ to be sequential spaces, 𝑇 to be a topological space, 𝑓 : 𝑆 → 𝑇, 𝑥 ∈ 𝑆, and 𝑣 ∈ 𝑇.

• A subspace 𝐴 ⊆ 𝑆 is not guaranteed to be sequential, except in the special cases where 𝐴 is either open or closed.

• 𝑆 × 𝑆′ with the usual product topology is not guaranteed to be sequential.

• scl 𝐴 = 𝖼𝗅 𝐴 is not guaranteed; put another way, 𝑥 being in the closure of 𝐴 does not guarantee that there is a sequence (𝑥ᵢ) ⊆ 𝐴 converging to 𝑥. On the other hand,

  • scl 𝐴 ⊆ 𝐴 as previously mentioned;

  • (𝐴 = 𝖼𝗅 𝐴) ⟺ (𝐴 = scl 𝐴), since 𝐴 is closed iff 𝐴 = 𝖼𝗅 𝐴, and 𝐴 is sequentially closed iff 𝐴 = scl 𝐴.

• (𝑓 is continuous at 𝑥) ⟺ (𝑓 is sequentially continuous at 𝑥) is not guaranteed. On the other hand,

  • (𝑓 is continuous at 𝑥) ⟹ (𝑓 is sequentially continuous at 𝑥);

  • the global properties are equivalent: (𝑓 is continuous) ⟺ (𝑓 is sequentially continuous).

• (𝑓(𝑦) → 𝑣 as 𝑦 → 𝑥) ⟺ (𝑓(𝑦) →ₛ 𝑣 as 𝑦 → 𝑥) is not guaranteed. On the other hand,

  • (𝑓(𝑦) → 𝑣 as 𝑦 → 𝑥) ⟹ (𝑓(𝑦) →ₛ 𝑣 as 𝑦 → 𝑥) does hold.

Sequentialization

Fortunately, any non-sequential topology can be refined to a sequential topology that has exactly the same convergent sequences.

Definition. Let (𝑆, 𝜏) be a topological space. The sequentialization of (𝑆, 𝜏) is (𝑆, seq 𝜏), where

Since all open sets are sequentially open, we immediately see that 𝜏 ⊆ seq 𝜏: we have only added additional open sets. It’s only a bit more work to verify that seq 𝜏 does in fact satisfy the requirements of a topology, and that it is in fact sequential. We start with a simple lemma.

Lemma 6. If (𝑥ᵢ) → 𝑦 in 𝜏 then (𝑥ᵢ) → 𝑦 in seq 𝜏.

Proof. Assume that (𝑥ᵢ) → 𝑦 in 𝜏. Let 𝑈 be any open neighborhood of 𝑦 in seq 𝜏; that is, 𝑦 ∈ 𝑈 and 𝑈 is sequentially open in 𝜏. (𝑥ᵢ) is eventually in 𝑈, by definition of “sequentially open.” This holds for any open neighborhood of 𝑦 in seq 𝜏, therefore (𝑥ᵢ) → 𝑦 in seq 𝜏.

Theorem 7. If 𝜏 is a topology on 𝑆 then seq 𝜏 is a sequential topology on 𝑆.

Proof. Let’s check that seq 𝜏 is in fact a topology:

1. ∅ ∈ seq 𝜏 and 𝑆 ∈ seq 𝜏: these sets are in 𝜏, hence in seq 𝜏.

2. Let 𝒰 be any subset of seq 𝜏 and let 𝑈 = ⋃𝒰 be the union of the sets in 𝒰. If (𝑥ᵢ) → 𝑦 in 𝜏 and 𝑦 ∈ 𝑈, then 𝑦 ∈ 𝑉 for some 𝑉 ∈ 𝒰, which is sequentially open in 𝜏, hence (𝑥ᵢ) is eventually in 𝑉 ⊆ 𝑈. Thus 𝑈 is sequentially open in 𝜏 and therefore belongs to seq 𝜏.

3. Let 𝑈, 𝑉 ∈ seq 𝜏. If (𝑥ᵢ) → 𝑦 in 𝜏 and 𝑦 ∈ 𝑈 ∩ 𝑉, then since 𝑈 and 𝑉 are both sequentially open in 𝜏, there exist 𝑚 and 𝑛 such that 𝑥ₖ ∈ 𝑈 for 𝑘 ≥ 𝑚 and 𝑥ₖ ∈ 𝑉 for 𝑘 ≥ 𝑛, hence 𝑥ₖ ∈ 𝑈 ∩ 𝑉 for 𝑘 ≥ max(𝑚,𝑛). So 𝑈 ∩ 𝑉 is sequentially open in 𝜏 and therefore belongs to seq 𝜏.

To show that seq 𝜏 is in fact sequential, we have to check that every set that is sequentially open in seq 𝜏 is open in seq 𝜏, i.e., sequentially open in 𝜏. So suppose 𝐴 is sequentially open in seq 𝜏. Let (𝑥ᵢ) → 𝑦 ∈ 𝐴 in 𝜏; then by Lemma 6, (𝑥ᵢ) → 𝑦 in seq 𝜏, and hence (𝑥ᵢ) is eventually in 𝐴 (which is sequentially open in seq 𝜏). This holds for arbitrary (𝑥ᵢ) → 𝑦 ∈ 𝐴 in 𝜏, so 𝐴 is sequentially open in 𝜏. ∎

An important property of seq 𝜏 is that it has exactly the same converging sequences as 𝜏:

Theorem 8. Let (𝑆, 𝜏) be a topological space. Then (𝑥ᵢ) → 𝑦 in 𝜏 ⟺ (𝑥ᵢ) → 𝑦 in seq 𝜏.

Proof. (⇐) Follows directly from 𝜏 ⊆ seq 𝜏. (⇒) See Lemma 6. ∎

Another important property of seq is that it leaves sequential topologies unchanged; thus if you’re not sure whether 𝜏 is sequential, and you only want to expand it if this is necessary to make it sequential, it is always safe to take seq 𝜏:

Theorem 9. Let (𝑆, 𝜏) be a sequential space; then seq 𝜏 = 𝜏.

Proof. For any 𝐴 ⊆ 𝑆,

∎

Finally, we’ll note that seq may be defined in terms of the transfinitely iterated sequential closure scl^{𝜔₁}:

Theorem 10. Let (𝑆, 𝜏) be a topological space and define 𝜏′ to be the unique topology on 𝑆 whose closed sets are exactly the fixed points of scl_𝜏^{𝜔₁}:

𝜏′ is well defined (there is exactly one topology on 𝑆 satisfying this condition) and 𝜏′ = seq 𝜏.

Proof. Too involved to give here. It involves the notion of a Kuratowski closure operator, the properties of such operators, and showing that scl_𝜏^{𝜔₁} is a Kuratowski closure operator. ∎

Product spaces and multi-argument functions

When investigating the joint continuity of all arguments to a multi-argument function, such as the AND (∧) of two generalized queries, we are dealing with the product of the spaces for each argument. But, as mentioned above, even though 𝑆₁,…,𝑆ₙ may all be sequential spaces, this does not guarantee that the product space2

is sequential.

Sequentialization comes to the rescue here: rather than assume the standard product space topology, we work with the sequential space

Sequentialization and product play together nicely in certain ways:

• The sequential product is associative, up to homeomorphism:

  meaning that, even though the two topological spaces are not strictly the same, they are homeomorphic (via the map ((𝑥,𝑦),𝑧) ↦ (𝑥,(𝑦,𝑧)).) This echoes the same property of the topological product.

• The sequential product yields the same result whether or not you sequentialize any of its arguments:

• All of this generalizes to 𝑛-fold products seq(𝑆₁ × ⋯ × 𝑆ₙ) for 𝑛 > 2.

Upcoming

Now that we have in hand the notion of a sequential space and the sequentialization of a topology, in the next post we can define the topology of generalized premises. It will amount to the sequential topology in which the sequence of generalized queries (ℰᵢ) converges iff the sequence of sets of truth assignments ([ℰᵢ]) converges pointwise.

Generalized Queries Part 3

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Introduction

A quick review of where we are on generalizing queries:

• The goal is to generalize finite queries—the propositional formulas 𝜑 in 𝘗𝘳(𝜑 | 𝒳), containing only manifest propositional symbols—much as we generalized finite premises—the satisfiable propositional formulas 𝑋 in 𝘗𝘳(𝜑 | 𝑋). The guiding principle is the Jaynesian policy on infinite objects: they should only arise as the well-defined and well-behaved limit of finite objects.

• In Part 1 we considered the Borel hierarchy, a decomposition of 𝜎-algebra of measurable sets generated by the open sets of a topological space. Our concern is with the specific topological space (𝓜 → 𝔹), the set of truth assignments on manifest symbols, with cylinder sets as the basic open sets. The hierarchy for this space (and many others) is breathtakingly large, having uncountably many levels, with the level for each ordinal number 𝛼 comprising classes 𝜮⁰_𝛼 and 𝚷⁰_𝛼 (and 𝚫⁰_𝛼 = 𝜮⁰_𝛼 ∩ 𝚷⁰_𝛼).

• In Part 2 I argued that, in line with the Jaynesian policy on infinite objects, a generalized query should correspond to the pointwise limit of sequence of clopen sets. Clopen sets in (𝓜 → 𝔹) are exactly those sets of form [𝜑], the set of satisfying truth assignments for some finite query 𝜑. We found that 𝚫⁰₂, the class of sets that can be expressed as either a countable union of closed sets (𝜮⁰₂) or countable intersection of open sets (𝚷⁰₂), is exactly the class of sets that correspond to some such pointwise limit. Therefore most of the Borel hierarchy is unnecessary for our purpose; we never go past the second level.

• I therefore defined a generalized query ℰ (think “event” as in the usual probability theory terminology) to be a sequence of finite queries (𝜑ᵢ) such that the corresponding sequence of sets ([𝜑ᵢ]) converges pointwise.

In this post we investigate pointwise convergence some more and strengthen the case for the definition of a generalized query as a pointwise limit:

• We find that pointwise convergence is equivalent to the lim inf and lim sup of a sequence of sets being the same.

• We find that pointwise convergence of (𝐴ᵢ) to set 𝐴 is equivalent to convergence of (𝜇(𝐴ᵢ)) to 𝜇(𝐴) for every probability measure 𝜇; hence generalized queries are also those sequences of queries (𝜑ᵢ) such that (𝘗𝘳(𝜑ᵢ | 𝒳)) converges for every generalized query 𝒳.

• We find that pointwise convergence of (𝐴ᵢ) to set 𝐴 is also equivalent to convergence of the Fréchet-Nikodym distance (𝜇(𝐴ᵢ△ 𝐴)) to 0 for every probability measure 𝜇.

Pointwise convergence as lim inf = lim sup

For a bounded sequence (𝑥ᵢ) of real numbers, lim inf and lim sup are defined as follows:1

The value inf {𝑥ₘ : 𝑚 ≥ 𝑛} increases monotonically as 𝑛 increases, and is bounded, which is why the required limit for lim inf exists and is just the supremum of the infima. A similar comment applies to lim sup.

This definition works not just for sequences of real numbers, but for sequences of values from any 𝜎-complete lattice: a set of values that 1) has a partial order relation and 2) for which every countable set has both a supremum and an infimum. The reals ℝ are one example, but the subsets of any set 𝑆 are another example, with ⊆ serving as the partial order relation, ⋃ₙ 𝐴ₙ serving as the supremum of a sequence of sets (𝐴ᵢ), and ⋂ₙ 𝐴ₙ serving as the infimum. Therefore,

Using the Iverson bracket to convert truth values to 0/1 values, it’s straightforward to see that

and so

This gives us another way of characterizing pointwise convergence, and and hence generalized queries:

Theorem 1. A sequence of sets (𝐴ᵢ) converges pointwise to a set 𝐴 if and only if lim infₙ 𝐴ₙ = lim supₙ 𝐴ₙ = 𝐴.

Proof. From real analysis we have the following standard result: a bounded sequence of reals (𝑥ᵢ) converges if and only if lim infₙ 𝑥ₙ = lim supₙ 𝑥ₙ. Then

∎

Theorem 2 of Part 2 states that if (𝐴ᵢ) is a sequence of clopen sets that converges to 𝐴, then 𝐴 ∈ 𝚫⁰₂. The proof is very similar to this one, and that result could have been obtained as an immediate consequence of this post’s Theorem 1. Another immediate consequence is that we obtain a means of bounding the pointwise limit:

Corollary 2. If the sequence of sets (𝐴ᵢ) converges pointwise to 𝐴, then for all 𝑛 ∈ ℕ we have

Proof. Using Theorem 1,

∎

Universal convergence of probabilities

In Part 2 we defined a generalized query to be a sequence of finite queries (𝜑ᵢ) such that the corresponding sequence of sets ([𝜑ᵢ]) converges pointwise:

for any truth assignment 𝛼 : 𝓜 → 𝔹, the sequence ([𝛼 ∈ [𝜑ᵢ]]) = (𝛼⟦𝜑ᵢ⟧ ) converges, either to 1 or to 0.

There is, however, a different direction we could have gone. Recall that a generalized premise is defined to be a sequence of finite premises (𝑋ᵢ) such that the sequence of probabilities (𝘗𝘳(𝜑 | 𝑋ᵢ)) converges for all queries 𝜑. We might therefore have defined a generalized query to be a sequence of finite queries 𝜑ᵢ such that

for any (finite) premise 𝑋, the sequence of probabilities (𝘗𝘳(𝜑ᵢ | 𝑋)) converges.

This doesn’t work. To see why, enumerate the manifest symbols as 𝑠₀, 𝑠₁, 𝑠₂, … and let 𝜑ₙ be 𝑠ₙ. Since 𝘗𝘳(𝑠 | 𝑋) = 1/2 for any manifest symbol 𝑠 not appearing in 𝑋, and only a finite number of manifest symbols appear in 𝑋, we conclude that 𝘗𝘳(𝜑ₙ | 𝑋) = 1/2 for all but a finite number of 𝑛, hence

This gives us convergence of probabilities for any finite premise 𝑋, but if we want to define

then we need convergence of probabilities for arbitrary generalized premises. Now consider 𝒳 = (𝑋ᵢ) where

That is, even-numbered propositional symbols are false, odd-numbered symbols are true. We have 𝘗𝘳(𝜑₂ₙ | 𝒳) = 0 but 𝘗𝘳(𝜑₂ₙ₊₁ | 𝒳) = 1 for all 𝑛, so no convergence of probabilities in this case.

The obvious remedy is to expand “all premises” to “all generalized premises”:

Definition. A sequence of queries (𝜑ᵢ) is universally probability-convergent if the sequence of probabilities (𝘗𝘳(𝜑ᵢ | 𝒳)) converges for all generalized premises 𝒳.

We then find that universal probability-convergence is in fact equivalent to pointwise convergence. This is good news: it means that Equation (1), which we gave in Part 2 to define the probability of a generalized query, is valid, as the limit does in fact exist. Furthermore, if generalized premises 𝒳 and 𝒴 are topologically indistinguishable, 𝒳 ∼ 𝒴, then we get the same limit, because we get the same sequence of probabilities: 𝘗𝘳(𝜑ₙ | 𝒳) = 𝘗𝘳(𝜑ₙ | 𝒴).

To prove this claim we will use a result from measure theory. I won’t give it in its most general form, just a restricted (and more easily described) form we need for our purposes.

Lemma 3. (Dominated Convergence Theorem for sets.) Let 𝑆 be a topological space, (𝐴ᵢ) be a sequence of Borel subsets of 𝑆, and 𝜇 be a Borel probability measure on 𝑆. Suppose that (𝐴ᵢ) converges pointwise to 𝐴. Then (𝜇(𝐴ᵢ)) → 𝜇(𝐴).

Proof. The Dominated Convergence Theorem is a standard result in measure theory. See, for example, here. For those who know some measure theory, we have specialized the DCT as follows: “complex-valued measurable functions” we restricted to the special case of indicator functions of Borel sets; “Lebesgue-integrable dominating function” we restricted to the special case of the constant function 𝑔(𝑥) = 1; and “nonnegative measure on a measurable space” we restricted to the special case of a Borel probability measure. ∎

Now we can show that universal probability-convergence is equivalent to pointwise convergence.

Theorem 4. Let 𝜑ₙ ∈ 𝛷(ℳ) for all 𝑛. The sequence ([𝜑ᵢ]) converges pointwise iff it is universally probability-convergent.

Proof. Recall that 𝘗𝘳(𝜑 | 𝒳) = 𝜇([𝜑]), where 𝜇 = pm 𝒳 is the probability measure on (𝓜 → 𝔹) corresponding to 𝒳, for all finite queries 𝜑 and generalized premises 𝒳. Furthermore, pm is surjective: every 𝜇 ∈ ℙ is pm 𝒳 for some 𝒳 ∈ 𝒫. Therefore it suffices to show that ([𝜑ᵢ]) converges pointwise iff (𝜇([𝜑ᵢ])) converges for every probability measure 𝜇 ∈ ℙ.

(⇐). Suppose that (𝜇([𝜑ᵢ])) converges for every probability measure 𝜇 ∈ ℙ. Given any 𝛼 : ℳ → 𝔹, choose 𝜇₀ to be the probability measure concentrated on 𝛼; that is, 𝜇₀(𝐴) = [𝛼 ∈ 𝐴]. By hypothesis the sequence (𝜇₀([𝜑ᵢ])) converges, which implies that the sequence ([𝛼 ∈ [𝜑ᵢ]]) must also converge, to either 0 or 1.

This reasoning holds for every 𝛼:ℳ→𝔹, and so ([𝜑ᵢ]) converges pointwise.

(⇒) Suppose that ([𝜑ᵢ]) converges pointwise to 𝐴 ⊆ (ℳ→𝔹). By Theorem 2 of Generalized Queries, Part 2, 𝐴 ∈ 𝚫⁰₂, which implies that 𝐴 is a Borel set. Now let 𝜇 ∈ ℙ. Then by Lemma 3 we have

∎

The Fréchet-Nikodym pseudometric

The symmetric set difference of two sets is defined as

that is, everything that is in one of the two sets but not in the other. Given a probability measure 𝜇 on a space 𝑆, the Fréchet-Nikodym pseudometric quantifies how different two sets 𝐴, 𝐵 ⊆ 𝑆 are:

Extending the notation, we can also define

where 𝜇 = pm 𝒳, when 𝜑 and 𝜓 are queries rather than sets.

Following the lead of Weihrauch and Tavana2 in their work on computability for measurable sets, we then could have defined a generalized query to be a sequence of queries (𝜑ᵢ) that is Cauchy under the pseudo-metric 𝑑_𝒳, with two generalized queries (𝜑ᵢ) and (𝜓ᵢ) being equivalent if the corresponding distance between the two sequences,

is zero. This then guarantees that the probability of a set is computable, as long as the sequences themselves are computable and the Cauchy sequences converge sufficiently rapidly (with a known, computable bound). This approach is commonly used in the academic literature on computable measure theory.

The downside of such an approach is that it makes the definition of “generalized query” depend on one’s state of information (the generalized premise), an odd situation that I wanted to avoid. Nonetheless, it’s worth exploring the relationship between this concept and generalized queries as pointwise limits of finite queries. What we find is encouraging: pointwise convergence is equivalent to convergence in the 𝑑_𝜇 pseudometric for all probability measures 𝜇.

Theorem 5. Let (𝐴ᵢ) be a sequence of measurable sets in some measurable space 𝑆. Then (𝐴ᵢ) converges pointwise to 𝐴 ⊆ 𝑆 if and only if (𝐴ᵢ) converges to 𝐴 in the Fréchet-Nikodym pseudometric 𝑑_𝜇 for every probability measure 𝜇 on 𝑆.

Proof. (⇒) Suppose that (𝐴ᵢ) converges pointwise to 𝐴. Then the sequence (𝐴ᵢ△𝐴) converges pointwise to ∅. In addition, 𝐴 = lim supₙ 𝐴ₙ and hence 𝐴 is measurable, as is each set 𝐴ₙ△𝐴. Then by Lemma 3, Dominated Convergence for sets, we have

(⇐) Suppose that (𝐴ᵢ) converges to 𝐴 in the Fréchet-Nikodym pseudometric 𝑑_𝜇, (𝜇(𝐴ᵢ△ 𝐴)) → 0, for every probability measure 𝜇 on 𝑆. Then for every 𝑥 ∈ 𝑆 consider 𝜇 = 𝛿_𝑥, the Dirac probability measure defined by 𝛿_𝑥(𝐵) = [𝑥 ∈ 𝐵]. Writing ⊕ for exclusive-or, we find that

and hence ([𝑥 ∈ 𝐴ᵢ]) → [𝑥 ∈ 𝐴]. Since this holds for every 𝑥 ∈ 𝑆, we then have that (𝐴ᵢ) converges pointwise to 𝐴. ∎

For each generalized premise 𝒳, there is a set 𝒬_𝒳 of sequences of queries (𝜑ᵢ) that converge under the 𝑑_𝒳 pseudo-metric. What Theorem 5 tells us is that the set of generalized queries 𝒬 is the intersection (greatest lower bound) of all sets 𝒬_𝒳 as 𝒳 ranges over all generalized premises.

• I changed “quasi-homeomorphism” to “Kolmogorov quotient map” in line with the terminology change to the previous post.

• I fixed two places where I mistakenly asserted that 𝑥 ∼ 𝑦 implies 𝑓(𝑥) = 𝑓(𝑦) for a continuous function 𝑓; this is only true if the codomain is a 𝑇₀-space, and in general all we can assert is that 𝑓(𝑥) ∼ 𝑓(𝑦).

• I added Corollary 2, which I think sheds some more light on what a Kolmogorov quotient map is. It also reduces to a one-liner the proof that the codomain of a KQM is 𝑇₀.

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Quotient Maps

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In the previous post I introduced Kolmogorov quotient maps. As you might suspect, these are a special kind of quotient map, an important topological concept we’ll discuss in this article; they are quotient maps that look like homeomorphisms if you consider topologically indistinguishable points to be the same.

Definition. A quotient map is a surjective (onto) function 𝑞 : 𝑆 → 𝑇 between topological spaces such that any set 𝑉 ⊆ 𝑇 is open if and only if 𝑞⁻¹[𝑉] ⊆ 𝑆 is open.

This definition ensures that the topology on 𝑇 is precisely the one “induced” by 𝑞 from 𝑆, with no extra or missing open sets. This definition implies that quotient maps are continuous: a subset 𝑉 of 𝑇 is open only if 𝑞⁻¹[𝑉] is open in 𝑆.

Quotient maps and equivalence relations

Quotient maps are related to equivalence relations. Recall that an equivalence relation ≈ is a sort of poor-man’s equality, one that throws away some of the information in its arguments but otherwise behaves like equality in that it is reflexive (𝑥 ≈ 𝑥), commutative (𝑥 ≈ 𝑦 iff 𝑦 ≈ 𝑥), and transitive (𝑥 ≈ 𝑦 and 𝑦 ≈ 𝑧 implies 𝑥 ≈ 𝑧). Examples we’ve seen include equality ( = ), logical equivalence of propositional formulas ( ≡ ), and topological indistinguishability ( ∼ ).

The set of values equivalent to 𝑥 is the equivalence class of 𝑥, written

or just [𝑥].1 The entire collection of equivalence classes for an equivalence relation ≈ on a set 𝑆 is known as the quotient space that ≈ induces on 𝑆, written 𝑆/≈. (It’s called the quotient space because it divides up—partitions—the space 𝑆 into equivalence class.) The quotient topology on the quotient space defines a set 𝑈 of equivalence classes to be open iff their union is an open set in 𝑆.

For a given equivalence relation ≈ on 𝑆, if we define 𝜋 : 𝑆 → (𝑆/≈) such that 𝜋(𝑥) = [𝑥], then 𝜋 is a quotient map:

• 𝜋 is surjective: every 𝐸 ∈ (𝑆/≈) is an equivalence class, and every equivalence class 𝐸 has at least one member 𝑥 ∈ 𝑆, so 𝐸 = 𝜋(𝑥).

• 𝑉 ⊆ (𝑆/≈) is open iff the union of its elements is open in 𝑆. But the union of the elements of 𝑉 is the set of all 𝑥 ∈ 𝑆 whose equivalence class is in 𝑉; that is, the set of all 𝑥 ∈ 𝑆 such that 𝜋(𝑥) ∈ 𝑉, and this is just the definition of 𝜋⁻¹[𝑉]. So 𝑉 is open iff 𝜋⁻¹[𝑉] is open.

Conversely, any quotient map 𝑞 : 𝑆 → 𝑇 induces an equivalence relation on 𝑆: define 𝑥 ≈ 𝑦 iff 𝑞(𝑥) = 𝑞(𝑦). Then 𝑇 is homeomorphic to (𝑆/≈), that is, there is a bijection 𝑔 : (𝑆/≈) → 𝑇 that pairs up corresponding elements of (𝑆/≈) and 𝑇, such that both 𝑔 and its inverse 𝑔⁻¹ are continuous. In fact, it can be shown that 𝑞 = 𝑔 ∘ 𝜋.

For a geometric example of a quotient map, consider the set 𝕊¹ ⊆ ℝ², defined to be the unit circle:

Then the function 𝑞 : ℝ → 𝕊¹ defined by

is a quotient map, assuming the standard topologies on ℝ and ℝ², and assuming for 𝕊¹ the subset topology induced by ℝ². If we think of 𝑡 as time, and 𝑞(𝑡) as the position of a dot on the plane at time 𝑡, then the dot moves counterclockwise in a circle at a constant speed of one unit of distance per unit of time (see figure below). Since 𝑞(𝑠) = 𝑞(𝑡) iff 𝑠 and 𝑡 differ by a multiple of 2𝜋, the equivalence relation associated with 𝑞 is then 𝑠 ≈ 𝑡 iff 𝑠 ≡ 𝑡 (mod 2𝜋).

Note that if we had written 𝑞 : ℝ → ℝ² then 𝑞 would not be a quotient map, because it would not be surjective. The codomain 𝑇 (set of potential values) of a function 𝑓 : 𝑆 → 𝑇 must be considered part of its identity, not just its range (set of actual values attained).

Although we have defined a quotient map 𝑞 : 𝑆 → 𝑇 in terms of the topology on 𝑇, one often goes in the other direction: start with a surjective function 𝑞 and then define the topology on 𝑇 to be the one that makes 𝑞 a quotient map:

That is exactly how the quotient topology on a quotient space was defined, starting with the function 𝜋.

Kolmogorov quotient maps as quotient maps

Recall the definition we gave for a Kolmogorov quotient map: a function 𝑓 : 𝑆 → 𝑇 between topological spaces having the following properties:

1. 𝑓 is surjective (onto).

2. 𝑓(𝑥) = 𝑓(𝑦) iff 𝑥 ∼ 𝑦 (𝑥 and 𝑦 are topologically indistinguishable).

3. 𝑓 preserves open sets both ways:

   1. If 𝑈 ⊆ 𝑆 is open then 𝑓[𝑈] ⊆ 𝑇 is open.

   2. If 𝑉 ⊆ 𝑇 is open then 𝑓⁻¹[𝑉] ⊆ 𝑆 is open.

Once we have the concept of a quotient map, we can further simplify the definition:

Theorem 1. A Kolmogorov quotient map 𝑓 is a quotient map such that 𝑓(𝑥) = 𝑓(𝑦) iff 𝑥 ∼ 𝑦.

Proof. ( ⇒ ) Suppose 𝑓 is a Kolmogorov quotient map. Then it is surjective (Property 1). If 𝑉 ⊆ 𝑇 is open then 𝑓⁻¹[𝑉] ⊆ 𝑆 is open (Property 3b). If 𝑓⁻¹[𝑉] is open then, by Property 3a, 𝑓[𝑓⁻¹[𝑉]] is open, and by surjectivity, 𝑉 = 𝑓[𝑓⁻¹[𝑉]], so 𝑉 is open. Therefore 𝑓 is a quotient map. And, of course, Property 2 guarantees that 𝑓(𝑥) = 𝑓(𝑦) iff 𝑥 ∼ 𝑦.

(⇐) Suppose that 𝑓 is a quotient map such that 𝑓(𝑥) = 𝑓(𝑦) iff 𝑥 ∼ 𝑦. Since quotient maps are surjective, Property 1 is satisfied; since they are continuous, Property 3b holds; furthermore, continuity implies that 𝑓(𝑥) = 𝑓(𝑦) if 𝑥 ∼ 𝑦; and by hypothesis, Property 2 holds.

It remains only to show that Property 3a holds. Suppose that 𝑈 ⊆ 𝑆 is open. Since 𝑓 is a quotient map, 𝑓[𝑈] is open iff 𝑓⁻¹[𝑓[𝑈]] is open. If we can show that 𝑈 = 𝑓⁻¹[𝑓[𝑈]], then Property 3a does indeed hold. But 𝑈 ≠ 𝑓⁻¹[𝑓[𝑈]] only if there is some 𝑥 ∈ 𝑈 and 𝑦 ∉ 𝑈 with 𝑓(𝑥) = 𝑓(𝑦). But 𝑓(𝑥) = 𝑓(𝑦) implies that 𝑥 and 𝑦 are topologically indistinguishable, and 𝑈 is open, so they either both belong to 𝑈 or neither belongs to 𝑈. ∎

Theorem 1 yields another simple characterization of Kolmogorov quotient maps:

Corollary 2. 𝑓 : 𝑆 → 𝑇 is a Kolmogorov quotient map iff 𝑓 = 𝑔 ∘ 𝜂 for some homeomorphism 𝑔 : (𝑆/∼) → 𝑇, where 𝜂 : 𝑆 → (𝑆/∼) is defined by

Proof. (⇐) 𝑔 is a quotient map because it is a homeomorphism, 𝜂 is a quotient map because ∼ is an equivalence relation, and the composition of two quotient maps is a quotient map; hence 𝑓 is a quotient map. Furthermore, since 𝑔 is a homeomorphism,

(⇒) Define 𝑥 ≈ 𝑦 iff 𝑓(𝑥) = 𝑓(𝑦); then since 𝑓 is a quotient map we can decompose it as 𝑓 = 𝑔 ∘ 𝜋 for some homeomorphism 𝑔, where 𝜋 : 𝑆 → (𝑆/≈) is defined by

But 𝑓(𝑥) = 𝑓(𝑦) iff 𝑥 ∼ 𝑦, so the relation ≈ is just ∼ , the space (𝑆/≈) is (𝑆/∼), and the map 𝜋 is 𝜂. ∎

So a KQM just collapses topologically distinct points into one and then renames them. Corollary 2 is the reason for the name “Kolmogorov quotient map”: (𝑆/∼) is known as the Kolmogorov quotient of 𝑆.

The codomain of a Kolmogorov quotient map is 𝑇₀

In the previous article I claimed that if 𝑓 : 𝑆 → 𝑇 is a Kolmogorov quotient map, then 𝑇 must be a 𝑇₀ space: all distinct points of 𝑇 are topologically distinguishable. I did not give a proof, thinking it was a simple one-liner. Turns out it wasn’t, not if you start with the definition I gave, but Corollary 2 makes it a one-liner: the Kolmogorov quotient (𝑆/∼) is a 𝑇₀-space, and 𝑔 : (𝑆/∼) → 𝑇 is a bijection, so 𝑇 is also a 𝑇₀-space.

One last note

One thing I may not have made clear enough is the relation between homeomorphisms and Kolmogorov quotient maps. If every pair of points in space 𝑆 is topologically distinguishable—𝑆 is a 𝑇₀ space—then a Kolmogorov quotient map from 𝑆 to 𝑇 is exactly the same thing as a homeomorphism from 𝑆 to 𝑇. And if 𝑆 does contain any pair of points that are topologically indistinguishable, then it is impossible to have a homeomorphism from 𝑆 to 𝑇, because any continuous function 𝑓 : 𝑆 → 𝑇 must map topologically indistinguishable points in 𝑆 to the same point in 𝑇, and hence cannot be one-to-one in this case.

In summary, for any topological space 𝑆, either

• 𝑆 is a 𝑇₀ space, so homeomorphisms 𝑆 → 𝑇 and Kolmogorov quotient maps 𝑆 → 𝑇 are the same thing; or

• 𝑆 is not a 𝑇₀ space, so there exists no homeomorphism 𝑆 → 𝑇.

Before continuing our discussion of generalized queries, we need to to tie up some loose ends with generalized premises; specifically, we need to connect the topology of generalized premises to the most commonly employed topology on probability measures, known as the weak topology. Combined with the Epistemic Representation Theorem and related results, this will allow us to simply import various continuity results from measure theory instead of deriving them from scratch for the space of generalized premises. And that, in turn, will be useful in discussing the continuity of operations that combine a generalized query with a generalized premise.

Definition. We write 𝖯(𝑆) for the set of Borel probability measures on topological space 𝑆. (The Borel probability measures are those defined on the Borel sigma-algebra of 𝑆.)

We write ℙ for 𝖯(ℳ → 𝔹), the space of Borel probability measures on the space of truth assignments on the manifest symbols.

In this article we do the following:

• Define and characterize the weak topology on 𝖯 when 𝑆 is a Stone space, a class that includes the space of truth assignments (ℳ → 𝔹).

• Define and characterize what it means for a sequence of probability measures to converge weakly, and how this connects to the weak topology.

• Define the concept of a Kolmogorov quotient map from 𝑆 to 𝑇: essentially, a relabeling of the points of 𝑆 that preserves the topology while collapsing topologically indistinguishable points.

• Show that the mapping pm : 𝒫 → ℙ, associating a unique probability measure 𝜇 ∈ ℙ to every generalized premise 𝒳 ∈ 𝒫 as described in Theorem 3 in The Epistemic Representation Theorem, is a Kolmogorov quotient map.

• Use pm to relate converging sequences in 𝒫 and converging sequences in ℙ.

Throughout this article 𝑆 will always be understood to be a Stone space: a topological space that is compact, has a base of clopen sets, and is Hausdorff (for any 𝑥,𝑦 with 𝑥 ≠ 𝑦 there is an open set containing 𝑥 but not 𝑦). (ℳ → 𝔹), the space of truth assignments on the manifest symbols with the cylinder-set topology, is a Stone space.

Although we will only state and discuss them in the context of Stone spaces, many of the definitions and theorems we’ll discuss here apply to broader categories of topological spaces. The results presented here are mostly straightforward applications of well-known results in measure theory and functional analysis.

The weak topology and weak convergence

The weak topology and notion of weak convergence are defined in terms of Lebesgue integrals of bounded, continuous functions on the topological space 𝑆, so we’ll start by defining these.

Definition. 𝐶(𝑆) is the space of continuous, real-valued functions 𝑓 : 𝑆 → ℝ.

We don’t introduce a separate notation for bounded continuous functions because, for a Stone space 𝑆, which is compact, all the functions in 𝐶(𝑆) are bounded.

Now a few words about the Lebesgue integral:

Definition. Given probability measure 𝜇 ∈ 𝖯(𝑆) and measurable function 𝑓 : 𝑆 → ℝ, we write 𝜇(𝑓) for the Lebesgue integral of 𝑓 w.r.t. the measure 𝜇. Alternative notations used in the mathematics literature are ∫ 𝑓d𝜇 and ∫ 𝑓(𝑥)d𝜇(𝑥). The region of integration is omitted because it is the entire space 𝑆.

Rather than dive into Lebesgue integration theory, which unifies probability densities and probability mass functions, I’ll just say that 𝜇(𝑓) amounts to the expected value of 𝑓(𝑥) when 𝑥 is sampled from 𝜇. Apart from being concise, the motive for using this notation is that if 𝑓 is the indicator function for a set 𝐴, 𝑓(𝑥) = [𝑥 ∈ 𝐴], then the expected value of 𝑓 is just 𝜇(𝐴), the probability of 𝐴.

Definition. 𝜏w, the weak (a.k.a. initial) topology on 𝖯, is the smallest (coarsest) topology that makes every evaluation map 𝜇 ↦ 𝜇(𝑓) continuous, for 𝑓 ∈ 𝐶(𝑆).

We often abbreviate “evaluation map” as “ev-map,” which can also be thought of as “expected value map.”

Definition. A sequence (𝜇ᵢ) in 𝖯 converges weakly to 𝜇 ∈ 𝖯, written 𝜇ᵢ ⇒ 𝜇, if

Ev-maps are real-valued, so the above is convergence in ℝ.

We say “converges weakly” instead of just “converges” because there are several distinct notions of convergence of probability measures discussed in the academic literature: apart from the weak topology, there is for example the topology based on the total variation distance, as well as the setwise-convergence topology, which differs from the weak topology on ℙ and our canonical topology on 𝒫 in requiring convergence of probabilities for all Borel sets, not just for a restricted class of sets.

We have given the standard definition of 𝜏w in terms of a certain property it has, but it is often more useful to have a base or subbase for the topology.

Lemma 1. The weak topology 𝜏w has as subbase the collection of all sets of form

for some 𝑓 ∈ 𝐶(𝑆) and 𝑎 < 𝑏.

Proof. Consider the requirements for a topology 𝜏 that makes the ev-map for every 𝑓 ∈ 𝐶(𝑆) continuous. The open intervals (𝑎, 𝑏) form a base for the topology on ℝ. Therefore the ev-map 𝜇 ↦ 𝜇(𝑓) is continuous under 𝜏 iff 𝑈₀(𝑓, 𝑎, 𝑏) belongs to 𝜏 for every 𝑎 < 𝑏. A topology 𝜏 then makes all these ev-maps continuous iff 𝑈₀(𝑓, 𝑎, 𝑏) ∈ 𝜏 for every 𝑓 ∈ 𝐶(𝑆) and every 𝑎 < 𝑏. 𝜏w is the smallest such 𝜏, which means that it contains only these sets and such additional sets as are required by the definition of a topology; i.e., this collection of sets is a subbase for 𝜏w. ∎

The expected equivalence between “converges weakly” and “converges in the weak topology” holds:

Theorem 2. 𝜇ᵢ ⇒ 𝜇 if and only if (𝜇ᵢ) → 𝜇 in 𝜏w.

Proof. By Lemma 1, the collection of sets 𝑈₀(𝑓, 𝑎, 𝑏) for some 𝑓 ∈ 𝐶(𝑆) and 𝑎 < 𝑏 is a subbase of 𝜏w. Then

∎

An alternative characterization for Stone spaces

We want to relate the weak topology on ℙ to our canonical topology on 𝒫, but the former is defined in terms of expected values of continuous functions and the latter is defined in terms of probabilities of (what amount to) clopen sets. The definition of 𝜏w given above is valid for probability measures on a broad class of topological spaces1, but for Stone spaces in particular there is an alternative characterization that is more in line with our logical view of probability, one that will help us bridge this gap:

Theorem 3. 𝜏w is the smallest (coarsest) topology that makes every evaluation map 𝜇 ↦ 𝜇(𝐴) continuous where 𝐴 ⊆ 𝑆 is clopen.

Proof. Based on approximating continuous functions with step functions, which are finite linear combinations of indicator functions for clopen sets. See full version of article for details. ∎

The connection to our canonical topology on 𝒫 stems from the fact that the clopen sets of (ℳ → 𝔹) are exactly those that can be expressed as [𝜑]: the set of truth assignments satisfying some propositional formula / query 𝜑 ∈ 𝛷(ℳ).

An alternative subbase

For this alternative characterization of 𝜏w it is, again, useful to have a corresponding base or subbase for 𝜏w.

Theorem 4. 𝜏w has as a subbase the collection of all sets of the form

for some clopen set 𝐴 and rational numbers 𝑞 < 𝑟.

Proof. Consider the requirements for a topology 𝜏 that makes the ev-map 𝜇 ↦ 𝜇(𝐴) continuous whenever 𝐴 is clopen. The open intervals with rational endpoints (𝑞, 𝑟) form a base for the topology on ℝ. Therefore the map 𝜇 ↦ 𝜇(𝐴) is continuous under 𝜏 iff 𝑈(𝐴, 𝑞, 𝑟) belongs to 𝜏 for every rational 𝑞 < 𝑟. A topology 𝜏 then makes all these ev-maps continuous iff 𝑈(𝐴, 𝑞, 𝑟) ∈ 𝜏 for every clopen 𝐴 and every 𝑞 < 𝑟. 𝜏w is the smallest such 𝜏, which means that it contains only these sets and such additional sets as are required by the definition of a topology; i.e., the collection of sets 𝑈(𝐴, 𝑞, 𝑟) is a subbase for 𝜏w. ∎

Corollary. 𝜏w is second countable, hence first countable, hence sequential.

Proof. Theorem 4 gives a countable subbase for 𝜏w; therefore the base constructed from it is also countable, as it is formed by finite intersections of sets in the subbase. ∎

A sequential space is one for which continuity (defined in terms of open sets) and sequential continuity (defined in terms of sequential limits2) are the same. If two sequential topologies have the same converging sequences, they are the same topology; if one of them is not sequential, they may have the same converging sequences but differ on which sets are open. Many of the intuitions you may have built up about convergence and continuity by working with functions on ℝ rely on the fact that ℝ is a sequential space, so it is reassuring that ℙ with the weak topology 𝜏w is a sequential space.

And an alternative criterion for weak convergence

This alternative subbase for 𝜏w then gives us an alternative criterion for when a sequence of probability measures converges weakly:

Theorem 5. 𝜇ᵢ ⇒ 𝜇 (equivalently, (𝜇ᵢ) → 𝜇 in 𝜏w) iff (𝜇ᵢ(𝐴)) → 𝜇(𝐴) for all clopen 𝐴.

Proof. By Theorem 4,

∎

Kolmogorov quotient maps

You may have noticed that the subbase for 𝜏w given in Theorem 4 looks an awful lot like the subbase that defines 𝜏₁, our canonical topology on 𝒫 given in Generalized Premises: A Redo: all sets of the form

for some query 𝜑 and 𝑞 < 𝑟. This suggests that the space of probability measures ℙ with the weak topology 𝜏w is essentially the same as our space of generalized premises 𝒫 with its canonical topology, except that the latter may have multiple points that are functionally identical. We will show this in fact to be so, using the notion of a Kolmogorov quotient map.

Definitions

In an earlier article we mentioned the concept of topological indistinguishability. We write 𝑥 ∼ 𝑦 to mean that 𝑥 and 𝑦 are topologically indistinguishable: they belong to exactly the same open subsets of the space, and therefore no topological property can tell them apart. For the space of generalized premises 𝒫 we defined the open sets in terms of the probabilities a point 𝒳 ∈ 𝒫 yields when applied to queries, and 𝒳 ∼ 𝒴 if and only if 𝒳 and 𝒴 yield the same probabilities on all queries.

This notion plays an important role in what follows. Our goal is to relate the topology on 𝒫 to the weak topology on the space of probability measures ℙ. We want a way to move back and forth between spaces while preserving exactly the topological structure that matters for convergence and continuity—and no more.

That leads us to the following concept.

Definition. Let 𝑆 and 𝑇 be topological spaces. A function 𝑓 : 𝑆 → 𝑇 is called a Kolmogorov quotient map if it preserves the topology of 𝑆 up to topological indistinguishability. More precisely, 𝑓 is required to satisfy the following:

1. Onto: every point 𝑡 ∈ 𝑇 is the image of some point 𝑠 ∈ 𝑆 (i.e., 𝑡 = 𝑓(𝑠)).

2. Quasi one-to-one: for all 𝑥, 𝑦 ∈ 𝑆, 𝑓(𝑥) = 𝑓(𝑦) if and only if 𝑥 ∼ 𝑦. Thus 𝑓 distinguishes points of 𝑆 exactly when the topology of 𝑆 does.

3. Open sets are preserved both ways:

   1. If 𝑈 ⊆ 𝑆 is open then 𝑓[𝑈] ⊆ 𝑇 is open.

   2. If 𝑉 ⊆ 𝑇 is open then 𝑓⁻¹[𝑉] ⊆ 𝑆 is open.

Note that if 𝑓 : 𝑆 → 𝑇 is a Kolmogorov quotient map, then 𝑇 is necessarily a 𝑇₀-space: any two distinct points of 𝑇 are topologically distinguishable.

Intuition and motivation

A Kolmogorov quotient map behaves like a homeomorphism, except that it collapses points that the topology cannot distinguish, while retaining all topological distinctions.

The reason for introducing this notion is practical. Previously, in The Epistemic Representation Theorem, we mapped the space 𝒫 of generalized premises onto the space ℙ of probability measures on truth assignments. We want to use this mapping to import existing results from measure theory—especially results about convergence of probability measures and continuity of operations on probability measures—into our theory of epistemic probability, without having to re-prove them from scratch.

Kolmogorov quotient maps are exactly what makes this possible: they allow us to transport convergence and continuity results faithfully between spaces.

Fundamental properties of Kolmogorov quotient maps

The key point is that Kolmogorov quotient maps preserve all topological information relevant to convergence and continuity. In the following, 𝑆, 𝑇, and 𝑊 are topological spaces and 𝑓 : 𝑆 → 𝑇 is a Kolmogorov quotient map.

Proposition 6. (𝑠ᵢ) → 𝑠 in 𝑆 if and only if (𝑓(𝑠ᵢ)) → 𝑓(𝑠) in 𝑇.

Proof. See full version of article. ∎

Proposition 7. For any function 𝑔 : 𝑇 → 𝑊 and point 𝑠 ∈ 𝑆,

Proof. See full version of article. ∎

Application to 𝒫 and ℙ

Now let’s use this idea to relate the topologies of 𝒫 and ℙ.

Definition. For any 𝒳 ∈ 𝒫, the probability measure corresponding to 𝒳, denoted pm 𝒳, is the unique 𝜇 ∈ ℙ such that (𝒳, 𝑔) represents 𝜇, where 𝑔 : ℳ → 𝜎(𝒞) is the canonical mapping defined by 𝑔(𝑠) = [𝑠].

Remark. 𝑔 is chosen so that 𝑔⟦𝜑⟧ = [𝜑]; thus pm 𝒳 is the unique 𝜇 ∈ ℙ such that

for all 𝜑 ∈ 𝛷(ℳ); furthermore, pm 𝒳 = pm 𝒴 if and only if 𝒳 ∼ 𝑌. Theorem 3 of my earlier article The Epistemic Representation Theorem guarantees that pm 𝒳 exists and is unique, while Theorem 4, what I have been calling the Epistemic Representation Theorem itself, guarantees that pm is an onto function.

And now we come to the main result of this article:

Theorem 8. The function pm : 𝒫 → ℙ is a Kolmogorov quotient map.

Proof. Let’s consider each of the properties required of a Kolmogorov quotient map.

1. The Epistemic Representation Theorem tells us that pm is onto.

2. We previously proved that pm 𝒳 = pm 𝒴 if and only if 𝒳 ∼ 𝒴.

3. Open sets preserved both ways:

   1. If 𝑈 is an open set in 𝒫 then it is the union of sets of the form

      for some query 𝜑 and rational 𝑞 < 𝑟. Then pm[𝑈] is the union of sets of the form pm[ 𝐵(𝜑, 𝑞, 𝑟) ]; but

      which is a basic open set in ℙ, since [𝜑] is a clopen set for any query 𝜑.

   2. If 𝑉 is an open set in ℙ then it is the union of sets of the form 𝑈(𝐴, 𝑞, 𝑟) for some clopen 𝐴 and rational 𝑞 < 𝑟. Then pm⁻¹[𝑉] is the union of sets of the form pm⁻¹[ 𝑈(𝐴, 𝑞, 𝑟) ] for clopen 𝐴 and rational 𝑞 < 𝑟; but as shown in Theorem 1 of Generalized Queries, Part 2, 𝐴 = [𝜑] for some query 𝜑, and

      which is a basic open set in 𝒫.

∎

And here is why that is important:

Corollary. Let 𝜇ᵢ = pm 𝒳ᵢ and 𝜇 = pm 𝒳; then the following are equivalent:

1. (𝒳ᵢ) → 𝒳

2. (𝜇ᵢ) → 𝜇 in 𝜏w

3. 𝜇ₙ ⇒ 𝜇.

Proof. Theorem 2 says that (2) and (3) are equivalent. Theorems 5 and 8 and Proposition 6 together imply that (1) is equivalent to (2). ∎

Up next

Having shown that pm is a Kolmogorov quotient map from 𝒫 to ℙ, we next need to determine what is the appropriate topology on generalized queries, one that is consistent with the Jaynesian requirement that infinities arise only as the well-defined and well-behaved limits of systems of finite objects. The obvious topology isn’t quite suitable, as it is not sequential. We will fix that by applying a well-known process known as sequentialization.

Once we’ve done that we’ll verify that all of our operations involving generalized queries and/or generalized premises are continuous w.r.t. the appropriate topologies.

(Extended version with all proofs.)

We continue our discussion of how to generalize queries, the propositional formula 𝜑 in 𝘗𝘳(𝜑 | 𝒳), beyond those expressible with simple propositional formulas. In Part 1 we discussed the Borel hierarchy, which is a decomposition of the measurable sets into an uncountable number of levels containing a bewildering zoo of exotic sets, and concluded that we didn’t need all that. In this article we argue that the second level of the hierarchy contains exactly the class of sets we want to express. In keeping with our Jaynesian philosophy, we want generalized queries to be well-defined and well-behaved limits of finite queries. It turns out that the 𝚫⁰₂, the intersection of 𝜮⁰₂ and 𝚷⁰₂, is exactly the class of sets that can be expressed as the limit of a sequence of sets specified by propositional formulas, for not just one but two different notions of “limit.”

We start by reviewing the first level of the Borel hierarchy and then moving on to the second. Throughout we shall blur the distinction between truth assignments on ℳ and infinite binary sequences, as any ordering of the manifest symbols in ℳ yields for each truth assignment an equivalent infinite binary sequence.

Understanding 𝜮⁰₁ and 𝚷⁰₁

Let’s review some facts about (ℳ → 𝔹), the space of truth assignments on the set of manifest symbols ℳ, and its topology:

• The cylinder sets are a base for the standard topology on this space.

• The cylinder sets are exactly those sets that can be expressed as [𝜑] (the set of truth assignments satisfying 𝜑) for some product term 𝜑. A product term is a propositional formula of the form 𝑙₁ ∧ ⋯ ∧ 𝑙ₙ, where each 𝑙ᵢ is a literal, either 𝑠 or ¬ 𝑠 for some propositional symbol 𝑠.

• The finite unions of cylinder sets are exactly those sets that can be expressed as [𝜑] for some propositional formula 𝜑, since every such 𝜑 is logically equivalent to some disjunctive normal form formula 𝜓₁ ∨ ⋯ ∨ 𝜓ₙ, where each 𝜓ᵢ is a product term.

• For any propositional formula 𝜑 the set [𝜑] is clopen: both closed and open. It is open because it is a union of cylinder sets. It is closed because its complement is [¬𝜑], which is also a union of cylinder sets and hence open.

Recall that 𝜮⁰₁ is the class of open sets for a topological space.

• Therefore 𝜮⁰₁ (for the space of truth assignments) is exactly those sets that are the countable unions of sets [𝜑] for some formula 𝜑. (Countable unions because there are only countably many propositional formulas.)

• So suppose we have a set 𝐴 ∈ 𝜮⁰₁ with

  Defining

  we find that

  and furthermore that [𝜑′ₙ] ⊆ [𝜑′ₙ₊₁]: 𝐴 is the limit of an ascending sequence of sets corresponding to finite queries.

Likewise, 𝚷⁰₁ is the class of closed sets.

• The sets in 𝚷⁰₁ are the complements of the sets in 𝜮⁰₁, so have the form

  and this is exactly those sets that can be expressed as the countable intersection of sets [𝜑] for some formula 𝜑.

• Similarly, if you have a set 𝐴 ∈ 𝚷⁰₁ with

  then defining

  we find that

  and furthermore that [𝜑′ₙ₊₁] ⊆ [𝜑′ₙ]: 𝐴 is the limit of a descending sequence of sets corresponding to finite queries.

Example. 𝜮⁰₁. Consider the set

𝐴ₙ is the set of all infinite binary sequences that start with 𝑛 0’s followed by a 1. This is a cylinder set, and hence clopen. 𝐴 is the set of all binary sequences that start with 0 but have a 1 somewhere. It is the union of clopen (hence open) sets, and so is itself an open set, a member of 𝜮⁰₁.

Example. 𝚷⁰₁. Consider the set 𝐵 = { 1^𝜔 } , the set whose sole element is all 1’s. This set can be expressed as

𝐵ₙ is the set of binary sequences that start with 𝑛 1’s. This is a cylinder set, and hence clopen. 𝐵 is the intersection of clopen (hence closed) sets, and so itself is a closed set, a member of 𝚷⁰₁.

Recall that 𝚫⁰₁ = 𝜮⁰₁ ∩ 𝚷⁰₁: the class of all sets that are in both 𝜮⁰₁ and 𝚷⁰₁. This is exactly the class of clopen sets, and that set has a simple characterization for the space of truth assignments:

Theorem 1. The clopen sets of (ℳ → 𝔹) are exactly those sets expressible as [𝜑] for some finite query 𝜑 ∈ 𝛷(ℳ).

Proof. See the extended version of this post. ∎

Understanding 𝚫⁰₂

Now let’s turn our attention to 𝜮⁰₂, 𝚷⁰₂, and 𝚫⁰₂ = 𝜮⁰₂ ∩ 𝚷⁰₂.

Proposition. 𝜮⁰₂ is the collection of countable unions of closed sets.

Proof. Recall that CU(𝒜) is the collection of all countable unions of sets from 𝒜. We have

∎

Proposition. 𝚷⁰₂ is the collection of countable intersections of open sets.

Proof. We have

∎

Let’s take a look at some examples. (In the following, since Substack doesn’t support inline LaTeX and there is no unicode superscript-𝑐, I will sometimes write ∼𝐴 for the complement of set 𝐴.)

Example. 𝚫⁰₂ but neither 𝜮⁰₁ nor 𝚷⁰₁. Consider the set 𝐷 = 𝐴 ∪ 𝐵, with 𝐴 and 𝐵 defined as in the previous examples:

𝐷 is not open. 𝐷 contains the point 1^𝜔, and every open neighborhood of 1^𝜔 is a superset of 1ⁿ 𝔹^𝜔 for some 𝑛; but 1ⁿ 𝔹^𝜔 is not a subset of 𝐷.

Nor is 𝐷 closed. A closed set contains all of its limit points, but 0ⁿ 1 0^𝜔 ∈ 𝐷 for all 𝑛 and

But 𝐷 is the countable union of closed sets: each set 0ⁿ1𝔹^𝜔 is a cylinder set, hence closed, and 𝐵 is also closed. Hence 𝐷 ∈ 𝜮⁰₂.

In addition, ∼𝐷 = ∼𝐴 ∩ ∼𝐵 is the countable union of closed sets. ∼𝐴 is the set of all sequences that either start with 1 or are all 0’s, and ∼𝐵 is the set of all sequences that have a 0 somewhere, so ∼𝐷 is the union of { 0^𝜔 } and the set of all sequences that start with a 1 but have a 0 somewhere:

The set { 0^𝜔 } is closed for the same reason that 𝐵 = { 1^𝜔 } is closed, and each set 1ⁿ 0 𝔹^𝜔 is a cylinder set, hence closed, so ∼𝐷 is also the countable union of closed sets. Hence ∼𝐷 ∈ 𝜮⁰₂ and therefore 𝐷 ∈ 𝚷⁰₂.

Combining these facts, we have 𝐷 ∈ 𝚫⁰₂.

Example. Not in 𝚫⁰₂. Some measurable sets that lie beyond 𝚫⁰₂:

The set 𝐴 is just a little bit beyond 𝚫⁰₂, as it still belongs to the second level of the Borel hierarchy: 𝐴 ∈ 𝚷⁰₂ but 𝐴 ∉ 𝜮⁰₂. The set 𝐵 can be shown to belong to 𝚷⁰₃ but not 𝜮⁰₃ nor any earlier level of the hierarchy.

These sets are not even finitely observable under any circumstances: if you imagine an algorithm scanning a binary sequence 𝑥 starting at index 0, returning true once it determines that 𝑥 ∈ 𝐴 (or 𝑥 ∈ 𝐵), or false once it determines that 𝑥 ∉ 𝐴 (or 𝑥 ∉ 𝐵), the algorithm will never terminate, regardless of which 𝑥 is scanned. No matter how far you go into the sequence, the yet-unseen remainder of the sequence could still put 𝑥 in the set or outside of the set. There is no possibility of ever resolving the membership question with a finite number of observations for any 𝑥.

𝚫⁰₂ sets as pointwise limits of clopen sets

Definition. A sequence of sets (𝐴ᵢ) converges pointwise to 𝐴 if, for every 𝑥, the sequence of binary values ([𝑥 ∈ 𝐴ᵢ]) converges to [𝑥 ∈ 𝐴]. The sequence converges pointwise if it converges pointwise to some 𝐴.

Remark. An equivalent criterion is that for all 𝑥 either

• 𝑥 ∈ 𝐴 and there exists 𝑛 such that 𝑥 ∈ 𝐴ₘ for all 𝑚 ≥ 𝑛, or

• 𝑥 ∉ 𝐴 and there exists 𝑛 such that 𝑥 ∉ 𝐴ₘ for all 𝑚 ≥ 𝑛.

Pointwise convergence fits in with the Jaynesian program of only allowing infinities that arise as well-defined and well-behaved limits of finite objects. We will find that, for the space (ℳ → 𝔹), 𝚫⁰₂ is exactly the class of sets that are pointwise limits of some sequence of sets corresponding to finite queries, which recommends it as a candidate for the class of sets to which our generalized queries will correspond. We actually obtain a more general result: for any Stone space (a particular broad and important class of topological spaces), 𝚫⁰₂ is exactly the class of sets that are pointwise limits of some sequence of clopen sets.

First, we show that 𝚫⁰₂ is large enough for our purposes: it contains all sets obtainable as the pointwise limit of a sequence of clopen sets.

Theorem 2. For any topological space 𝛺, if (𝐴ᵢ) is a sequence of clopen subsets of 𝛺 that converges pointwise to set 𝐴, then 𝐴 ∈ 𝚫⁰₂.

Proof. See the extended version of this post. ∎

Now we need to consider the other direction: do we need all of 𝚫⁰₂? I.e., is every set in 𝚫⁰₂ the pointwise limit of a sequence of clopen sets? Answering this question takes a bit more work. We’ll show that the answer is yes whenever we are dealing with a (quasi-)Stone space, which includes the space (ℳ → 𝔹) of truth assignments on the manifest symbols.

Definition. A quasi-Stone space is a topological space that is

• compact (every open cover has a finite subcover), and

• has a base consisting of clopen sets.

Remark. This is non-standard terminology. It is standard to define a Stone space to be a topological space that has the above two properties, and is also Hausdorff, but we won’t need that third property.

Proposition. (ℳ → 𝔹) is a quasi-Stone space.

Proof. The cylinder sets form a basis for this space, and they are clopen. The discrete topology on 𝔹 is trivially compact (it is finite); the cylinder-set topology on (ℳ → 𝔹) is what’s known as a product topology; and a standard result known as Tychonoff’s Theorem says that a product topology of any collection of compact spaces is also compact.

Theorem 3. For any quasi-Stone space, if 𝐴 ∈ 𝚫⁰₂ then there is a sequence of clopen sets (𝐴ᵢ) that converges pointwise to 𝐴.

Proof. See the extended version of this post. ∎

Theorem 4. For any quasi-Stone space 𝛺, 𝚫⁰₂ is exactly the collection of sets 𝐴 ⊆ 𝛺 that are the pointwise limit of some sequence (𝐴ᵢ) of clopen sets.

Proof. Follows directly from Theorems 2 and 3. ∎

So, in particular, the 𝚫⁰₂ sets for (ℳ → 𝔹) are exactly those that can be expressed as the pointwise limit of some sequence of finite queries.

Generalized queries

Given the above, it should be clear what we’re going to do next.

Definition. A generalized query is a sequence ℰ = (𝐸ᵢ) of queries such that the sequence ([𝐸ᵢ]) converges pointwise. We write 𝒬 for the set of generalized queries. We extend to generalized queries the definitions of several operations on queries, as follows:

In the above,

• Recall that 𝛼⟦𝐸⟧ for a finite formula 𝐸 is the truth value that 𝐸 evaluates to under the truth assignment 𝛼.

• [ℰ] is the set of truth assignments corresponding to ℰ.

• 𝒟 = (𝐷ᵢ) is another generalized query.

• 𝒳 is a generalized premise.

It is straightforward to verify that ¬ℰ and 𝒟 ∨ ℰ as defined converge pointwise, and hence are valid generalized queries, and that [¬ℰ] = ∼[ℰ] and [𝒟 ∨ ℰ] = [𝒟] ∪ [ℰ] as expected. The other logical operators can be defined in terms of ¬ and ∨ , e.g.

If you’re paying attention, you’ll notice a possible problem with the definition of 𝘗𝘳(ℰ | 𝒳): how do we know that the required limit actually exists?

We’ll talk about that in Part 3, where, among other things, we’ll see that you have pointwise convergence if and only if you have convergence of probabilities for all generalized premises.

The notion of a compact topological space captures a certain sort of finiteness of that space. To explain what that means, let’s start with the notion of an open cover.

Definition. A collection 𝒜 of subsets of a topological space 𝑆 is said to cover 𝑆, or to be a cover of 𝑆, if ⋃𝒜 = 𝑆 (or, equivalently, every 𝑥 ∈ 𝑆 belongs to some set in 𝒜). It is called an open cover of 𝑆 if its elements are open subsets of 𝑆. Any subcollection of 𝒜 that also covers 𝑆 is called a subcover of 𝒜.

As an example, one open cover of the real line ℝ is the collection of sets of the form

for any integer 𝑛. Every real number belongs to one of these open sets. However, we don’t need all of these to cover ℝ; the subcollection of all those 𝐴ₙ where 𝑛 is even suffices.

Definition. A topological space 𝑆 is said to be compact if every open cover of 𝑆 contains a finite subcover. A subset 𝑅 ⊆ 𝑆 is said to be compact if 𝑅, viewed as a topological space in its own right with the subspace topology derived from 𝑆, is a compact space.

So, a topological space is compact if, whenever a collection of open sets covers the space, you only need a finite number of them to cover the space. The rest are, in a sense, superfluous. So clearly ℝ is not compact: the open cover { 𝐴ₙ : 𝑛 ∈ ℤ } does have subcovers, but it has no finite subcover.

However the subset [0, 1] is indeed compact. (The Heine-Borel Theorem says that any closed and bounded subset of ℝ is compact.) The open sets of [0, 1] (as a topological space) are all sets of the form 𝐴 ∩ [0, 1], where 𝐴 is an open set of ℝ; in particular, they are all unions of intervals of one of these three types:

• an open interval (𝑎, 𝑏) with 0 < 𝑎 < 𝑏 < 1;

• an interval [0, 𝑏) with 0 < 𝑏 < 1;

• an interval (𝑎, 1] with 0 < 𝑎 < 1.

And no matter what collection of intervals of one of the above three types you come up with, if it covers [0, 1], then a finite subcollection will suffice to do the same. For example, pick some arbitrary 𝜀 ∈ (0, 1) and consider the collection 𝒜 defined as

This is an infinite collection that is an open cover of [0, 1]. Note that we had to include that set [0, 𝜀) to completely cover [0, 1], as the sets (1/𝑛, 1] omit the point 0. But inclusion of [0, 𝜀) means that we don’t need to include in the cover any of the intervals (1/𝑛, 1] where 1/𝑛 < 𝜀; in particular, the finite subcollection

also covers [0, 1].

Of relevance to our explorations of Epistemic Probability, the space 𝔹^𝜔 of infinite binary sequences is compact, and so of course is the space (ℳ → 𝔹) of truth assignments on the manifest symbols. Furthermore, any cylinder set, or finite union of cylinder sets, is also compact.

The problem

Up to this point the queries 𝐴 in a probability expression 𝘗𝘳(𝐴 | 𝒳) have all been propositional formulas, each involving at most a finite number of atomic propositions (as propositional symbols) from a countable set of the same. Let’s see what we can and cannot do with these finite queries.

Suppose that we have propositional symbols1 (𝚡 < 𝑎), (𝚡 > 𝑎), (𝚢 < 𝑎), and (𝚢 > 𝑎) for every rational number 𝑎, with the obvious intended meanings; and suppose that generalized premise 𝒳 contains information about the logical relations between these symbols, axiomatizing their intended meanings, such as the implication

This is the same sort of thing we did in the previous post Turning Concrete Facts into a Probability Distribution.

The queries that we can express are then effectively unions of rectangles on the plane whose borders are at rational coordinates, such as this rectangular approximation to the unit disk:

We cannot express any of the following as queries, even though they seem quite reasonable:

1. 𝑥 < 𝜋;

2. 𝑥 ≥ 0 ∧ 𝑦 ≥ 0 ∧ 𝑥 + 𝑦 ≤ 1;

3. 𝑥² + 𝑦² ≤ 1.

(1) is inexpressible because it involves comparison with an irrational number; (2) is inexpressible because it is a triangular region of the plane, having one diagonal boundary; and (3) is inexpressible because it is the unit disc, a region of the plane that has a curved boundary.

We can identify any point in the plane with the truth assignment 𝛼 that assigns a value of 1 (true) to every symbol whose intended meaning is true for that point, and 0 (false) to the rest. For example, for the point (2, 3) we would have

and so on. Likewise, we can identify any region of the plane, such as those specified by (1), (2), and (3), with the set of truth assignments corresponding to the points in that region. These particular sets cannot be expressed as the set of truth assignments satisfying some finite query, because sets of that form correspond to “pixelated” regions of the plane, with all boundaries being vertical or horizontal lines. But they can be expressed as the countable union of such sets—sort of an infinite OR of propositional formulas. This countably union can be constructed by repeatedly adding to the union the largest possible rectangle that does not overlap any existing one, and lies entirely within the desired region.

A process similar to the above discussion will hold for any sample space of interest to us. Determine its topology and identify a countable sub-base for that topology. Corresponding to every element of the sub-base is a proposition (like 𝑥 < 3/2) to which we associate some manifest propositional symbol (which might be chosen to have a suggestive name such as “𝚡 < 𝟹/𝟸”.) Then any region of the sample space of interest to us can be identified with a set of truth assignments.

But what sorts of sets should a generalized query correspond to? Should it be the entire 𝜎-algebra generated by the sets corresponding to finite queries? I am going to argue no; when we analyze the structure of the 𝜎-algebra we find it ludicrously extravagant, containing a bewildering zoo of exotic sets that are of no interest to us. A much smaller and more manageable collection of sets will suffice, all of which can be expressed as the point-wise limit of a sequence of finite queries.

Borel sets

Let’s get some terminology out of the way.

The notion of a Borel 𝜎-algebra connects the topology 𝜏 on a space 𝛺 to a notion of measurable subsets of 𝛺. The Borel 𝜎-algebra on 𝛺 (with topology 𝜏) is just 𝜎(𝜏): the smallest 𝜎-algebra containing every open set, or equivalently, the 𝜎-algebra generated by the open sets. The Borel 𝜎-algebra is also generated by any countable base or sub-base for the topology. A Borel set is any member of the Borel 𝜎-algebra.

Recall that a truth assignment on the set of manifest symbols ℳ can be viewed as an infinite binary sequence if we choose some arbitrary ordering of the symbols of ℳ. Writing (ℳ → 𝔹) for the set of truth assignments (functions taking any value from ℳ as input and producing a value from 𝔹 as output), we can define the (single-coordinate) cylinder sets of (ℳ → 𝔹) analogously to the (singe-coordinate) cylinder sets for 𝔹^𝜔. A cylinder set of (ℳ → 𝔹) is any set of truth values defined by fixing the values for a finite set of manifest symbols:

for some finite set of manifest symbols 𝑆 ⊆ ℳ and choice of values for those symbols 𝑎 : 𝑆 → 𝔹. A single-coordinate cylinder set of (ℳ → 𝔹) is any set of truth values of the form

for some 𝑠 ∈ ℳ and 𝑎 ∈ {0, 1} . The standard product topology on (ℳ → 𝔹) has the cylinder sets as a base and the single-coordinate cylinder sets as a sub-base. The Borel 𝜎-algebra on (ℳ → 𝔹) is generated by the collection of cylinder sets of (ℳ → 𝔹), and is also generated by the collection of single-coordinate cylinder sets.

So we see a close parallel between the spaces 𝔹^𝜔 and (ℳ → 𝔹), and we shall take advantage of this to write 𝒞 and 𝒞₁ (respectively) for the cylinder sets and single-coordinate cylinder sets of (ℳ → 𝔹) or 𝔹^𝜔, trusting to the context to disambiguate which we’re talking about.

One feature of the topological spaces 𝔹^𝜔 and (ℳ → 𝔹) that you should keep in mind, and that is very different from what we see for the real numbers ℝ, is that every cylinder set is clopen—both closed and open. For ℝ the only clopen sets are the empty set and ℝ itself. Note that, since an open set is the complement of a closed set, and a closed set is the complement of an open set, the complement of any clopen set is also clopen.

Now let’s take a look at the structure of the Borel 𝜎-algebra for a broad class of topological spaces that includes the space of truth assignments (ℳ → 𝔹).

The Borel hierarchy

Naïvely, one might think that one could obtain the Borel 𝜎-algebra by just starting with a base for the topology and adding in all countable unions and their complements. The problem is that, having added additional sets to your collection of sets, there are now additional countable unions to consider, along with their complements. So we might have to repeat the process; and then repeat it again; and so on.

Since the cylinder sets are a countable base for the product topology on (ℳ → 𝔹), taking all countable unions of cylinder sets gives us all the open sets. Taking their complements then gives us all the closed sets. We will write

This is the first level of the Borel hierarchy, which is defined for arbitrary topological spaces, not just (ℳ → 𝔹). There are additional levels 𝜮⁰_𝛼 and 𝚷⁰_𝛼 for other indices 𝛼, and in general we define

(Note that here and in much of what follows, we use 𝛼 to indicate an index of some sort, rather than a truth assignment. This is customary notation for the sort of indices we’ll be using.) Since 𝜮⁰₁ is the collection of open sets and 𝚷⁰₁ is the collection of closed sets, 𝚫⁰₁ is the collection of clopen sets.

Unfortunately, 𝚷⁰₁ is not closed under arbitrary countable unions; we shall have to extend the hierarchy further. First we define

Then for 𝛼 > 1 we define

That is, first we form a collection of sets 𝒜 containing all the sets from any 𝚷⁰_𝛽 for any earlier level 𝛽 < 𝛼; then 𝜮⁰_𝛼 is CU(𝒜), the collection of all sets we can form by taking the union ⋃ᵢ𝐴ᵢ of any sequence of sets 𝐴₀, 𝐴₁ ,𝐴₂, … where each 𝐴ᵢ belongs to 𝒜. Then we construct 𝚷⁰_𝛼 by taking the complements of all the sets in 𝜮⁰_𝛼.

This gives us 𝜮⁰₂ and 𝚷⁰₂, then 𝜮⁰₃ and 𝚷⁰₃, and so on, for arbitrarily large 𝛼 ∈ ℕ. In general it can be shown that 𝜮⁰_𝛼 ∪ 𝚷⁰_𝛼 ⊆ 𝚫⁰_{𝛼+1}; once a set is in one level of the hierarchy it is at all higher levels. Furthermore, the inclusion is proper: the higher levels include new sets that are not in the lower levels. (This is not necessarily true of the Borel hierarchy for an arbitrary topology, but it is true for any Polish space, a class of topological spaces that includes 𝔹^𝜔, (ℳ → 𝔹), ℝⁿ for arbitrary 𝑛, and in fact most spaces of interest.)

So if we take every set that occurs in 𝜮⁰_𝛼 or 𝚷⁰_𝛼 for any 𝛼 ∈ ℕ, that should get us all of the Borel sets, right?

Wrong.

To infinity and beyond: the ordinal numbers

(I know, I promised we wouldn’t delve into infinite ordinals and transfinite induction, but in order to fully appreciate just how ludicrously extravagant the full 𝜎-algebra on (ℳ → 𝔹) is, we’re going to have to dip our toe into the topic.)

We’re going to need more numbers in order to index the remaining levels of the Borel hierarchy; the natural numbers ℕ alone won’t suffice. Since the purpose of these numbers is to indicate positions in a total ordering (rather than the size of something), they’re called the ordinal numbers. It is traditional to use Greek letters like 𝛼 and 𝛽 for variables that are ordinal numbers.

The first three properties of ordinal numbers are familiar properties of the natural numbers:

• 0 is an ordinal number.

• Every ordinal number 𝛼 has a successor, 𝛼+1. This is the least ordinal number greater than 𝛼.

• The ordinal numbers are well-ordered: they are totally ordered (either 𝛼 < 𝛽, 𝛼 = 𝛽, or 𝛼 > 𝛽 for any two ordinals 𝛼 and 𝛽) and every nonempty set of ordinal numbers has a least element.

To the above we add an additional way to construct ordinal numbers:

• Every set of ordinal numbers has a least upper bound: the smallest ordinal 𝛼 that is ≥ every ordinal in the set.

The first ordinal beyond the natural numbers—their least upper bound—is 𝜔, the first infinite ordinal. Equations (1) and (2) apply also to infinite ordinals, so level 𝜔 of the Borel hierarchy is defined as

But 𝜮⁰_𝜔 and 𝚷⁰_𝜔 aren’t the end of the story either, as the latter still isn’t closed under countable unions. So we repeat the process again to get level 𝜔+1:

And again to produce level 𝜔+2, and again and again, producing level 𝜔+𝑘 for every 𝑘 ∈ ℕ. Of course there’s a least ordinal greater than all of those, so that gives us level 𝜔⋅2 of the hierarchy.

Then level 𝜔⋅2 + 1, level 𝜔⋅2 + 2, and so on, and eventually we get to level 𝜔⋅3.

In like fashion we eventually arrive at levels 𝜔⋅4, 𝜔⋅5, …, and so on, then to the least level greater than all those, level 𝜔⋅𝜔, written 𝜔².

Continuing onwards produces a level 𝜔²⋅𝑎 + 𝜔⋅𝑏 + 𝑐 for every 𝑎, 𝑏, 𝑐 ∈ ℕ, and then the least level greater than all of those is level 𝜔³.

In like fashion we eventually arrive at level 𝜔^𝑘 for any 𝑘 ∈ ℕ; in fact, there is a level

for every 𝑘 ∈ ℕ and every 𝑎₀, 𝑎₁, …, 𝑎ₖ ∈ ℕ.

Just above all of those levels is level 𝜔^𝜔.

And we’re still just getting started! Ahead of us still lies level

and level

which are examples of tetration, a.k.a. power towers, which we can write using Knuth’s notation as 𝜔↑↑2 and 𝜔↑↑3, so of course there are levels 𝜔↑↑𝑘 for every 𝑘 ∈ ℕ, and of course there’s a least level above all of those, 𝜔↑↑𝜔 (an infinite power tower of 𝜔’s) and…

is it just me or is the air getting thin up here?…

we’ve still barely scratched the dizzying heights of the Borel hierarchy.

If the set of ordinals less than 𝛼 is countable then we say that 𝛼 is a countable ordinal. All of the ordinals we have discussed so far are countable. The set of countable ordinals itself has a least upper bound, denoted 𝜔₁: this is the first uncountable ordinal. (Hence there are uncountably many countable ordinals.)

And here, finally, we come to the end of the Borel hierarchy (although not the end of the ordinal numbers):

There are uncountably many levels to the Borel hierarchy. There are so many levels that it’s not possible, even in principle, to give names (nor mathematical expressions) for them all!2

Not far to the finish line

Let us now carefully back away from the Cliffs of Insanity and consider what we really need. That will be the topic of the next article, but for now I’ll say this: we don’t need anything past level 2 of the Borel hierarchy. It turns out that the sets of 𝚫⁰₂ are exactly the sets that may be obtained as the point-wise limit of a sequence of of sets, each of which corresponds to some finite query.

(The preceding article, A Brief Intro to Measure Theory, is a prerequisite for this one. Even if you’re already familiar with measure theory and its use in probability theory, that article introduces some non-standard notation we use here, specifically 𝑔⟦𝐴⟧ and and alg(𝒢).)

The question

A natural question to ask about the epistemic approach to probability theory is this: do we lose any expressive power by working with generalized premises instead of the probability measures used in the standard approach to probability theory?

There is the issue of the sorts of questions we can ask—up to this point we have only defined 𝘗𝘳(𝐴 | 𝒳) for 𝐴 a propositional formula. Propositional formulas form (up to equivalence) only a set algebra, not a 𝜎-algebra, as we cannot express (countably) infinite disjunctions ⋁ᵢ𝐴ᵢ or (countably) infinite conjunctions ⋀ᵢ𝐴ᵢ. That is a limitation we shall address and remove at a later point, when we define generalized queries.

In this article we are focused on the the right-hand side of the probability expression 𝘗𝘳(𝐴 | 𝒳)—can we construct, for every probability measure 𝜇, a corresponding generalized premise that “implements” or represents 𝜇? Let’s say exactly what that means.

Definition. We say that (𝒳, 𝑔) represents probability measure 𝜇 on measurable space (𝛺, 𝒜) if

• 𝒳 ∈ 𝒫 is a generalized premise and 𝑔 :⊆ ℳ → 𝒜;

• 𝒜 = 𝜎(rng(𝑔)); and

• For all 𝐴 ∈ 𝛷(dom(𝑔)), 𝜇(𝑔⟦𝐴⟧ ) = 𝘗𝘳(𝐴 | 𝒳).

We say that 𝜇 is 𝑔-representable if (𝒳, 𝑔) represents 𝜇, for some 𝒳 ∈ 𝒫.

We say that 𝜇 is representable if it is 𝑔-representable for some 𝑔 :⊆ ℳ → 𝒜.

So 𝑔 provides the link between a generalized premise 𝒳 and a measure 𝜇 by telling us how to interpret a query 𝐴—as the event1 𝑔⟦𝐴⟧. It does so by assigning an event in 𝒜 to each manifest symbol in its domain, such that this collection of events—the range of 𝑔—generates the full 𝜎-algebra 𝒜.

The above only gives the value of 𝜇 on the set algebra generated by 𝒢 = rng(𝑔), not the full 𝜎-algebra, but this is sufficient, as the values of 𝜇 on alg(𝒢) determine its values on 𝜎(𝒢):

Proposition. If (𝒳,𝑔) represents both probability measure 𝜇 and 𝜈 on the same measurable space (𝛺, 𝒜), then 𝜇 = 𝜈.

Proof. Follows from the fact that every set in alg(rng(𝑔)) can be expressed as 𝑔⟦𝐴⟧ for some 𝐴 ∈ 𝛷(dom(𝑔)), combined with Carathéodory’s extension theorem. ∎

We can now rephrase our question as this: is every probability measure 𝜇 representable? No; only probability measures on countably-generated 𝜎-algebras could possibly be representable, as there are only countably many manifest symbols. But that is the only limitation; we will see that the answer is yes for any probability measure on a countably-generated 𝜎-algebra, and these are the only ones of any practical importance.

Countably generated 𝜎-algebras are the only ones that matter in any setting where the events in our 𝜎-algebra are meant to be computable, or specifiable. If a 𝜎-algebra has no countable generating set, then there is no hope of treating its members as objects we can systematically name, manipulate, or computably approximate, as there are only countably many formulas or expressions we can write using a countable set of symbols.

In contrast, when our 𝜎-algebra 𝒜 is countably generated, any measure on 𝒜 is fully defined by its values on a countable set of events; and every event in 𝒜 is approximable to arbitrary precision by the members of a countable set of events. This is what allows probabilities of events to be computed.

It is no surprise, then, that the standard 𝜎-algebras on the spaces that dominate real applications—such as ℝⁿ, separable metric spaces, Polish spaces, and the product spaces used for stochastic processes—are all countably generated.

The 𝜎-algebra of infinite binary sequences

We start by considering the set of infinite binary sequences 𝔹^𝜔. It is of interest because, if we order the manifest symbols ℳ, then 𝔹^𝜔 can be viewed as the set of truth assignments on ℳ. We will see that, for any given one-to-one correspondence 𝑔 : ℳ → 𝒞₁⁼¹, each generalized premise represents a probability measure on the standard 𝜎-algebra for 𝔹^𝜔. As usual, that 𝜎-algebra is defined by specifying a generating set.

Definition. We write 𝒞 for the collection of all cylinder sets, which are subsets of 𝔹^𝜔 of the form

for some finite set of indices 𝐼 ⊆ ℕ and choice of values for these indices 𝑎 : 𝐼 → 𝔹. As an example, if 𝑔 maps a to 3 and b to 12, then

is a cylinder set.

We write 𝒞₁ for the collection of all single-coordinate cylinder sets, which are cylinder sets where 𝐼 is a singleton for some 𝑖 ∈ ℕ. These have the general form

for some 𝑖 ∈ ℕ and 𝑎 ∈ { 0, 1 } .

We write 𝒞₁⁼¹ for the collection of single-coordinate cylinder sets having the specific form

for some 𝑖 ∈ ℕ.

Remark. Note that, since 𝒞₁ ⊆ 𝒞 ⊆ alg(𝒞₁⁼¹), and 𝒞₁⁼¹ ⊆ 𝒞₁ ⊆ 𝒞, we have

and

𝜎(𝒞), or equivalently, 𝜎(𝒞₁⁼¹), is the standard 𝜎-algebra one associates with the space 𝔹^𝜔.

Lemma 1. For every bijection 𝑔 : ℳ → 𝒞₁⁼¹ there is a corresponding bijection ℎ : (ℳ → 𝔹) → 𝔹^𝜔, mapping truth assignments to infinite binary sequences, such that

for all 𝐴 ∈ 𝛷(ℳ).

Proof. Since 𝑔 is a bijection from ℳ to 𝒞₁⁼¹ there exists a bijection 𝑛 : ℳ → ℕ such that

For every truth assignment 𝛼 : ℳ → 𝔹, define ℎ(𝛼) to be the infinite binary sequence 𝑥 ∈ 𝔹^𝜔 such that

Intuitively, ℎ orders the manifest symbols in an infinite sequence according to 𝑛 and then replaces each symbol 𝑠 with 𝛼(𝑠). The function ℎ has an inverse, with ℎ⁻¹(𝑥) being the truth assignment 𝛼 such that α ( s ) = x n ( s ) (obtained by substituting 𝑛(𝑠) for 𝑖 in equation (1).)

A straightforward induction on the structure of 𝐴 then shows that that 𝑔⟦𝐴⟧ = ℎ([𝐴]) for every 𝐴 ∈ 𝛷(ℳ). (Details in PDF.) ∎

Definition. We call the function ℎ of Lemma 1 the serializer of 𝑔.

Corollary 2. Let 𝑔 : ℳ → 𝒞₁⁼¹ be a bijection. Then

Proof. Uses the serializer of 𝑔 to relate 𝑔⟦𝜑⟧ and [𝜑]. (Details in PDF.)

Theorem 3. Let 𝑔 : ℳ → 𝒞₁⁼¹ be a bijection. Then for every generalized premise 𝒳 ∈ 𝒫 there exists a unique probability measure 𝜇 on 𝜎(𝒞) such that (𝒳, 𝑔) represents 𝜇.

Proof. Define the pre-measure 𝜇₀ on alg(𝒞) via

The remaining proof steps (see PDF) are:

1. Show that 𝜇₀ is well-defined.

2. Show that 𝜇₀ is a pre-measure (satisfies the Kolmogorov axioms).

3. Invoke Carathéodory’s Extension Theorem to uniquely extend 𝜇₀ to a probability measure on 𝜎(𝒞).

∎

Definition. Let us write 𝒳 ∼ 𝒴 to mean that generalized premises 𝒳 and 𝒴 are topologically equivalent, and hence 𝘗𝘳(𝐴 | 𝒳) = 𝘗𝘳(𝐴 | 𝒴) for every query 𝐴.

Proposition. For any given bijection 𝑔 : ℳ → 𝒞₁⁼¹, generalized premises 𝒳, 𝒴 ∈ 𝒫 represent the same measure 𝜇 on 𝜎(𝒞) if and only if 𝒳 ∼ 𝒴.

Proof. Follows directly from Theorem 3, Carathéodory’s Extension Theorem, and the definition of “(𝒳, 𝑔) represents 𝜇.” ∎

𝜎(𝒞) is special in this regard: Theorem 3 does not generalize to arbitrary countably-generated 𝜎-algebras. Consider, for example, the case where 𝛺 = ℝ and 𝒜 = 𝜎(𝒢), where 𝒢 is the countable collection of all sets of the form (-∞, 𝑎) or (𝑎, ∞) for rational 𝑎. For any bijection 𝑔 : ℳ → 𝒢 there are symbols 𝑠₁, 𝑠₂ ∈ ℳ such that 𝑔 maps 𝑠₁ to (-∞, 0) and 𝑠₂ to (1, ∞), and even the simple premise 𝑠₁ ∧ 𝑠₂ does not represent any measure on 𝜎(𝒢) with respect to 𝑔, because 𝑔⟦𝑠₁ ∧ 𝑠₂⟧ = ∅.

Main theorem

Now let’s state and prove the main result. The proof makes use of mixtures: recall that in a previous article we defined a function

that constructs a new generalized premise from two existing ones 𝒳 and 𝒴 and a mixture probability 𝜃, and has the property that

Theorem 4. Let 𝑔 :⊆ ℳ → 𝒜 and 𝒜 = 𝜎(rng(𝑔)). Then every probability measure 𝜇 on 𝒜 is 𝑔-representable.

Proof. Here is an outline (see PDF for details; step numbering differs):

1. Enumerate (without duplicates) the elements of the countable set dom(𝑔) as 𝑠ₙ, 0 ≤ 𝑛 < |dom(𝑔)|.

2. For 𝑢 ∈ 𝔹* define 𝜑(𝑢) to be the product term corresponding to 𝑢, i.e.

   1. 𝜑(01) = ¬ 𝑠₀ ∧ 𝑠₁;

   2. 𝜑(101) = 𝑠₀ ∧ ¬ 𝑠₁ ∧ 𝑠₂; and so on.

3. Define 𝑝(𝑎 | 𝑢) to be the probability 𝜇 assigns to 𝑔⟦𝜑(𝑢𝑎)⟧ conditional on 𝑔⟦𝜑(𝑢)⟧ .

4. For 𝑢 ∈ 𝔹* of length at most 𝑛, recursively define

5. Show that for 𝑣 ∈ 𝔹ⁿ,

6. Re-express any query 𝐴 using at most the first 𝑛 manifest symbols in disjunctive normal form, as the OR of 𝜑(𝑣) for 𝑣 ranging over some subset of 𝔹ⁿ. Use this to show that

7. Therefore, for any query 𝐴, the sequence of probabilities (𝘗𝘳(𝐴 | 𝒴ᵢ(𝜀))) converges to 𝜇(𝑔⟦𝐴⟧). This implies that the sequence of generalized premises (𝒴ᵢ(𝜀)) is convergent, and we may define

(𝒳, 𝑔) then represents 𝜇. ∎

Corollary. Every probability measure 𝜇 on a countably-generated 𝜎-algebra 𝒜 is representable.

Proof. Choose any countable generating set 𝒢 for 𝒜 and any 𝑔 :⊆ ℳ → 𝒜 such that rng(𝑔) = 𝒢. Then by Theorem 4, 𝜇 is 𝑔-representable. ∎

Introduction

We are about to make some connections between the epistemic approach to probability theory and the standard approach based on measure theory. In preparation for that, this article will review the basics of measure theory as they apply to probability measures.

A central concept of measure theory is that not all sets are measurable. Some subsets of ℝ are too weird to assign them any total length without violating the properties we expect of lengths; some subsets of ℝ² are too weird to assign them any total area without violating the properties we expect of areas; and some subsets of ℝ³ are too weird to assign them any total volume without violating the properties we expect of volumes. So we define a restricted collection of “measurable” sets that includes the empty set and is closed under the operations of complementation and countable union.

The standard approach to probability theory talks about probabilities of events, which are subsets of some sample space 𝛺. As with length, area, and volume, requiring that every arbitrary subset of 𝛺 must have an associated probability can be problematic when 𝛺 is uncountable. For example, we find that we cannot define a uniform probability measure on the unit interval [0, 1] without violating Kolmogorov’s axioms for probability. So again, we must restrict the class of sets to which we can assign probabilities.

Before continuing, let’s get some notation out of the way.

Definition. We define the following:

• 𝑃(𝑆) is the powerset of a set 𝑆, the collection of all subsets of 𝑆.

• dom(𝑓) is the domain 𝑅 of function 𝑓 : 𝑅 → 𝑆 (the set of values on which it is defined).

• rng(𝑓) is the range { 𝑓(𝑥) : 𝑥 ∈ 𝑅 } of a function 𝑓 : 𝑅 → 𝑆; it is a subset of 𝑆.

• We write 𝑓 :⊆ 𝑅 → 𝑆 to indicate that 𝑓 is a partial function on 𝑅, that is, 𝑓 : 𝑅′ → 𝑆 for some set 𝑅′ ⊆ 𝑅.

Set algebras: syntactic approach

We’ve already seen one connection between propositional formulas and sets of values, when we defined [𝐴] to be the set of truth assignments on ℳ (the manifest symbols) satisfying 𝐴. To generalize that notation we need to associate a set of values with each manifest symbol used in 𝐴. If we think of 𝛺 as the set of all possible states of some system we are discussing, then associating some 𝐹 ⊆ 𝛺 to a manifest symbol 𝑠 is a a way of assigning 𝑠 the meaning “the system state is one of the values in 𝐹.” For example, if 𝛺 = [0, 1], with the state 𝑝 meaning “my car’s gas tank is 100𝑝 % full,” then we might assign the set [0, 0.57] to the symbol f57 to give it the meaning “my car’s gas tank is 57% full.”

Recall that when 𝛼 is a truth assignment we defined 𝛼⟦⋅⟧ to be its extension to propositional formulas, i.e. 𝛼⟦𝐴⟧ is the truth value you get by substituting 𝛼(𝑠) for every occurrence of a propositional symbol 𝑠 in 𝐴 and interpreting the symbols ∧ , ∨ , and ¬ as the expected logical operators. If 𝑔 assigns subsets of 𝛺 to the propositional symbols used in 𝐴, we can analogously define 𝑔⟦𝐴⟧ to be the expression you get by substituting 𝑔(𝑠) for every occurrence of a propositional symbol 𝑠 in 𝐴 and replacing ∧ with ∩, ∨ with ∪, and ¬ with set complement.

This notion will be of central importance when we discuss the Epistemic Representation Theorem. Here is the precise definition.

Definition. Let 𝑔 :⊆ ℳ → 𝑃(𝑆) be a partial function that assigns to each manifest symbol in its domain a subset of set 𝑆. Let 𝐴 ∈ 𝛷(dom(𝑔)) be a propositional formula that contains propositional symbols only from the domain of 𝑔. Then 𝑔⟦𝐴⟧ is defined recursively via

Remark. Treating 𝐴 ∧ 𝐵 as an abbreviation for ¬(¬ 𝐴 ∨ ¬ 𝐵) and similarly for the other logical operators, it follows that 𝑔⟦𝐴 ∧ 𝐵⟧ = 𝑔⟦𝐴⟧ ∩ 𝑔⟦𝐵⟧ and so on.

Example. If we define 𝑔 : ℳ → 𝑃(ℳ → 𝔹) such that 𝑔(𝑠) is the set of truth values 𝛼 for which 𝛼(𝑠) = 1, then 𝑔⟦𝐴⟧ = [𝐴], the set of truth values that satisfy 𝐴.

A set algebra on a set 𝛺 can be thought of as the collection of sets that can be expressed in the form 𝑔⟦𝐴⟧ , where 𝑔 :⊆ ℳ → 𝑃(𝛺) is fixed and 𝐴 ranges over 𝛷(dom(𝑔)), the set of all propositional formulas that can be constructed using only the symbols in the domain of 𝑔.

Definition. If 𝑔 :⊆ ℳ → 𝑃(𝛺) then

One might then define a (countably-generated) set algebra to be alg₀(𝑔) for some 𝑔 :⊆ ℳ → 𝛺. This is not the standard definition, although we shall see that it is closely related.

Set algebras: standard definition

The standard definition of a set algebra is rather more abstract than the construction we have outlined.

Definition. A set algebra (or field) ℱ on a nonempty set 𝛺 is a collection of subsets of 𝛺 with the following properties:

1. ∅ ∈ ℱ.

2. Closed under complementation: 𝛺 ∖ 𝐹 ∈ ℱ whenever 𝐹 ∈ ℱ.

3. Closed under union: 𝐹₁ ∪ 𝐹₂ ∈ ℱ whenever 𝐹₁, 𝐹₂ ∈ ℱ.

Remark. By mathematical induction and DeMorgan’s Laws, property (3) above may be replaced by any of the following equivalent properties:

• Closed under intersection: 𝐹₁ ∩ 𝐹₂ ∈ ℱ whenever 𝐹₁, 𝐹₂ ∈ ℱ.

• Closed under finite unions: 𝐹₁ ∪ ⋯ ∪ 𝐹ₙ ∈ ℱ whenever 𝐹₁, …, 𝐹ₙ ∈ ℱ, for any 𝑛 ∈ ℕ.

• Closed under finite intersections: 𝐹₁ ∩ ⋯ ∩ 𝐹ₙ ∈ ℱ whenever 𝐹₁, …, 𝐹ₙ ∈ ℱ, for any 𝑛 ∈ ℕ.

Here are some examples of set algebras:

• For any nonempty set 𝛺, 𝑃(𝛺) is trivially a set algebra—the requirements of a set algebra specify subsets of 𝛺 that the collection must contain, and 𝑃(𝛺) contains all subsets of 𝛺.

• For any nonempty set 𝛺, { ∅, 𝛺 } is a set algebra.

• The collection of all sets of the form [𝐴], where 𝐴 ∈ 𝛷(ℳ), is a set algebra on the set of truth assignments on ℳ. Let’s call this the truth-assignment set algebra.

Given a collection of subsets of 𝛺, we can minimally extend it to a set algebra.

Definition. Let 𝒢 ⊆ 𝑃(𝛺) for some nonempty set 𝛺. Then alg(𝒢), the set algebra generated by 𝒢, is the smallest (under the subset ordering) collection of sets ℱ ⊆ 𝑃(𝛺) such that 𝒢 ⊆ ℱ and ℱ is a set algebra on 𝛺.

Remark. We know that a smallest such collection exists because 𝑃(𝛺) itself is a set algebra containing 𝒢, and the intersection of any collection of set algebras on 𝛺, each containing 𝒢, is itself a set algebra on 𝛺 containing 𝒢.

Example. If we define

then alg(𝒢) is the truth-assignment set algebra.

We shall only be interested in set algebras that can be generated by a countable collection 𝒢. Such set algebras are said to be countably generated.

Relating alg(𝒢) and alg₀(𝑔)

A disadvantage of the standard definition of a set algebra is that it is non-constructive: it only defines alg(𝒢) indirectly, but doesn’t tell you how to construct its elements. But alg₀(𝑔) is defined constructively; from the definition you can construct any member of the set in a finite number of steps. We relate alg(𝒢) and alg₀(𝑔) by choosing a 𝑔 whose range is 𝒢. This is always possible if 𝒢 is countable.

Theorem. alg₀(𝑔) = alg(rng(𝑔)) for any 𝑔 :⊆ ℳ → 𝑃(𝛺).

Proof. Let 𝒢 = rng(𝑔). It is clear that alg₀(𝑔) is a set algebra that contains 𝒢, hence alg(𝒢) ⊆ alg₀(𝑔). In the other direction we show that alg₀(𝑔) ⊆ alg(𝒢), and hence alg₀(𝑔) = alg(𝒢), by showing that 𝑔⟦𝐴⟧ ∈ alg(𝒢) for every 𝐴 ∈ 𝛷(dom(𝑔)). This we prove by induction on the structure of 𝐴.

Base case: 𝐴 is some symbol 𝑠 in dom(𝑔). Then 𝑔⟦𝐴⟧ = 𝑔(𝑠) ∈ 𝒢 and 𝒢 ⊆ alg(𝒢).

Induction step: Suppose that 𝑔⟦𝐴⟧, 𝑔⟦𝐵⟧ ∈ alg(𝒢). Then

and

and the other logical operators are all defined in terms of ¬ and ∨. ∎

Remark. There are a few other properties of alg(𝒢) worth mentioning:

• If 𝒢₁ ⊆ 𝒢₂ then alg(𝒢₁) ⊆ alg(𝒢₂). Adding additional members to the generating set can only make the resulting set algebra larger.

• If 𝒢₁ ⊆ alg(𝒢₂) then alg(𝒢₁) ⊆ alg(𝒢₂).

• alg(ℱ) = ℱ if ℱ is a set algebra.

Probability pre-measures

A probability pre-measure assigns probabilities to the members of a set algebra. We use the term pre-measure because a measure has a larger, richer domains than just a set algebra.

Definition. Let ℱ be a set algebra on some nonempty set 𝛺. A probability pre-measure on ℱ is a function 𝜇₀ : ℱ → ℝ that satisfies the Kolmogorov axioms:

1. 𝜇₀(𝐹) ≥ 0 for all 𝐹 ∈ ℱ.

2. 𝜇₀(𝛺) = 1.

3. If 𝐹ᵢ ∩ 𝐹ⱼ = ∅ for 𝑖 ≠ 𝑗 then

In the countable additivity requirement (3) some of the 𝐹ᵢ may be ∅, so it applies to both finite and countably infinite collections of disjoint sets in ℱ. As we have previously seen for the truth-assignment set algebra, it may happen for a set algebra that (3) reduces to finite additivity because there is no countably infinite collection of disjoint, nonempty sets in ℱ.

Note that if 𝐹ᵢ = ∅ for all 𝑖 ∈ ℕ then countable additivity implies that

and so 𝜇₀(∅) = 0, as expected. Applying requirement (3) with 𝐹₀ = 𝐹, 𝐹₁ = 𝛺 ∖ 𝐹, and 𝐹ᵢ = ∅ for 𝑖 > 1, and using 𝜇₀(𝐹₀ ∪ 𝐹₁) = 1 due to (2), we also get the expected identity 𝜇₀(𝛺 ∖ 𝐹) = 1 - 𝜇₀(𝐹).

Measure theory applies to more than just probability; for example, it is also the theory of area and volume. In these other applications the requirement 𝜇₀(𝛺) = 1 is not appropriate; in fact we may have 𝜇₀(𝐹) = ∞ for some sets 𝐹, which invalidates the proof that countable additivity implies 𝜇₀(∅) = 0. Thus the general definition of a pre-measure (no “probability” prefix) allows 𝜇₀(𝐹) to be ∞ and replaces the requirement 𝜇₀(𝛺) = 1 with the requirement that 𝜇₀(∅) = 0.

Sigma algebras

Consider the case where 𝛺 = ℝ², the real plane, and our set algebra is generated by the half-planes, sets of the form

or

for some 𝑎 ∈ ℝ. It’s clear that the set algebra includes rectangles such as

and finite unions of such rectangles, but what if we want to talk about the probability of a set such as the interior of a circle? That set does not belong to our set algebra, but note that it can be expressed as the union of a countably infinite collection of sets in our set algebra. Specifically, assign to every point within the interior of the circle, having rational coordinates, a rectangle small enough that the entire rectangle fits within the circle. This is a countably infinite collection of rectangles, and their union is the interior of the circle.

This sort of thing is the motivation for defining a 𝜎-algebra, which is defined identically to a set algebra except that we require closure under all countable unions, not just finite unions.

Definition. A 𝜎-algebra 𝒜 on a nonempty set 𝛺 is a collection of subsets of 𝛺 with the following properties:

1. ∅ ∈ 𝒜.

2. Closed under complementation: 𝛺 ∖ 𝐴 ∈ 𝒜 whenever 𝐴 ∈ 𝒜.

3. Closed under countable unions: ⋃ᵢ𝐴ᵢ ∈ 𝒜 whenever 𝐴ᵢ ∈ 𝒜 for all 𝑖 ∈ ℕ.

A measurable space is a pair (𝛺, 𝒜), where 𝛺 is a nonempty set and 𝒜 is a 𝜎-algebra on 𝒜.

Remark. Every 𝜎-algebra is also a set algebra, as we have only strengthened the requirements. By DeMorgan’s Laws, a 𝜎-algebra is also closed under countable intersections.

There is an analog to alg(𝒢) for 𝜎-algebras.

Definition. Let 𝒢 ⊆ 𝑃(𝛺) for some nonempty set 𝛺. Then 𝜎(𝒢), the 𝜎-algebra generated by 𝒢, is the smallest (under the subset ordering) collection of sets 𝒜 ⊆ 𝑃(𝛺) such that 𝒢 ⊆ 𝒜 and 𝒜 is a 𝜎-algebra on 𝛺.

Remark. Some properties of 𝜎(𝒢):

• The definition of 𝜎(𝒢) is identical to the definition of alg(𝒢) except that we have replaced “set algebra” with “𝜎-algebra.” Since every 𝜎-algebra is also a set algebra, this means that alg(𝒢) ⊆ 𝜎(𝒢).

• If 𝒢₁ ⊆ 𝒢₂ then 𝜎(𝒢₁) ⊆ 𝜎(𝒢₂).

• 𝜎(𝒜) = 𝒜 if 𝒜 is a 𝜎-algebra.

• If 𝒢₁ ⊆ 𝜎(𝒢₂) then 𝜎(𝒢₁) ⊆ 𝜎(𝒢₂).

• 𝜎(alg(𝒢)) = 𝜎(𝒢).

A constructive definition?

As with alg(𝒢), the definition of 𝜎(𝒢) is indirect; it does not tell you how to construct all of the members of the collection. Unlike alg(𝒢), there is no nice constructive alternative. The closest we can come to something sort-of, kind-of like a direct construction of the members of 𝜎(𝒢) involves infinite ordinal numbers and transfinite induction, and that would take us far afield into the hinterlands of set theory. We’re not going there.

Probability measures

Now we have all the pieces in place needed to define a probability measure.

Definition. Let 𝒜 be a 𝜎-algebra on some nonempty set 𝛺. A probability measure on 𝒜 is a function 𝜇 : 𝒜 → ℝ that satisfies the Kolmogorov axioms (as previously listed for a probability pre-measure).

If 𝜇₀ is a probability pre-measure on a set algebra ℱ that generates 𝒜, and 𝜇(𝐹) = 𝜇₀(𝐹) for all 𝐹 ∈ ℱ, we say that 𝜇 is an extension of 𝜇₀ (to a probability measure on 𝒜.)

So a probability measure is the same as a probability pre-measure, except that its domain must be a 𝜎-algebra, not just a set algebra.

A standard result in measure theory, known as Carathéodory’s Extension Theorem, is that to specify a probability measure on a 𝜎-algebra 𝒜 it suffices to specify its values on any set algebra ℱ that generates 𝒜:

Theorem. Let ℱ be a set algebra on nonempty set 𝛺 and 𝜇₀ a probability pre-measure on ℱ. There is a unique extension of 𝜇₀ to a probability measure 𝜇 on 𝜎(ℱ).

This then completes the measure theory background we’ll need. In the next article we’ll prove an important result: that any probability measure on a countably-generated σ -algebra has an epistemic representation as a generalized premise.

(𝑅 ➝ 𝑆) is the set of all functions 𝑓: 𝑅 ➝ 𝑆, assigning a value from 𝑆 to each value in its domain 𝑅.

𝛷(𝑆)

𝛷(𝑆) is the set of propositional formulas constructed using only the propositional symbols in 𝑆. A formula in 𝛷(𝑆) is either

• a symbol 𝑠 ∈ S,

• ¬𝐴 (“not 𝐴”) for some 𝐴 ∈ 𝛷(𝑆), or

• 𝐴 ∨ 𝐵 (“𝐴 or 𝐵”) for some 𝐴,𝐵 ∈ 𝛷(𝑆).

The remaining logical operators can be defined in terms of ¬ and ∨:

• 𝐴 ∧ 𝐵 (“𝐴 and 𝐵”) is just ¬(¬𝐴 ∨ ¬𝐵).

• 𝐴 → 𝐵 (“𝐴 implies 𝐵”) is just ¬𝐴 ∨ 𝐵.

• 𝐴 ↔︎ 𝐵 (“𝐴 if and only if 𝐵”) is just (𝐴 → 𝐵) ∧ (𝐵 → 𝐴).

𝛷⁺(𝑆)

𝛷⁺(𝑆) is the set of formulas in 𝛷(𝑆) that are satisfiable.

𝔹

The set of binary values, { 0, 1 }.

bijection

A bijection 𝑓: 𝑅 ➝ 𝑆 is a function from 𝑅 to 𝑆 that is both one-to-one and onto. It establishes a correspondence between the members of 𝑅 and 𝑆. A bijection is invertible: there exists a function 𝑓⁻¹: 𝑆 ➝ 𝑅 with the properties that

• 𝑓⁻¹(𝑓(𝑥)) = 𝑥 for all 𝑥 ∈ 𝑅, and

• 𝑓(𝑓⁻¹(𝑦)) = 𝑦 for all 𝑦 ∈ 𝑆.

countable

A set 𝑆 is countable if its elements can be enumerated, possibly with duplications. More rigorously, there is a function 𝑓: ℕ ➝ 𝑆 (which may be thought of as an infinite sequence) whose range is 𝑆 (every element of 𝑆 appears at least once in the sequence.)

dom

dom(𝑓) is the domain of a function 𝑓.

domain

The domain of a function 𝑓: 𝑅 ➝ 𝑆 is the set 𝑅 of values on which it is defined.

generalized premise

An infinite sequence of premises 𝒳 = (𝑋ᵢ) with the property that, for every query 𝐴, the sequence (Pr(𝐴 | 𝑋ᵢ)) converges, thus allowing us to define

invertible

A function 𝑓: 𝑅 ➝ 𝑆 is invertible if it is a bijection, and hence has an inverse 𝑓⁻¹: 𝑆 ➝ 𝑅.

ℒ

The countably infinite set of latent propositional symbols.

latent symbol

A propositional symbol that is not meant to appear in any query, as it is unobserved, much as the latent variables of a probabilistic model are unobserved.

ℳ

The countably infinite set of manifest propositional symbols.

manifest symbol

A propositional symbol that can appear in a query; not a latent symbol.

ℕ

The set of natural numbers: all nonnegative integers, including 0.

one-to-one

A function 𝑓: 𝑅 ➝ 𝑆 is one-to-one if, for all 𝑥, 𝑦 ∈ 𝑅, if 𝑥 ≠ 𝑦 then 𝑓(𝑥) ≠ 𝑓(𝑦). Distinct points in 𝑅 map to distinct points in 𝑆.

onto

A function 𝑓: 𝑅 ➝ 𝑆 is onto if for all 𝑦 ∈ 𝑆 there is some 𝑥 ∈ 𝑅 with 𝑓(𝑥) = 𝑦. That is, 𝑓 is onto if its range is 𝑆.

premise

1. The propositional formula 𝑋 in a probability expression Pr(𝐴 | 𝑋), which means “the probability of 𝐴 given 𝑋.” Here 𝑋 expresses what is known, the totality of the information from which we derive the probability 𝐴.

2. A satisfiable propositional formula.

proposition

A statement that is either true or false, although we may not know which.

propositional formula

See 𝛷(𝑆)

ℚ

The set of all rational numbers, expressible as a ratio of two integers 𝑎 / 𝑏, with 𝑏 > 0.

query

1. The propositional formula 𝐴 in a probability expression Pr(𝐴 | 𝑋), which means “the probability of 𝐴 given 𝑋.”

2. A propositional formula containing no latent symbol, only manifest symbols.

ℝ

The set of all real numbers.

range

The range of a function 𝑓: 𝑅 ➝ 𝑆 is that subset of 𝑆 mapped to by some element of 𝑅:
{ 𝑓(𝑥): 𝑥 ∈ 𝑅 }.

rng

rng(𝑓) is the range of a function 𝑓.

𝛴

The countably infinite set of all propositional symbols, partitioned into the manifest symbols ℳ and the latent symbols ℒ.

satisfiable

A propositional formula 𝑋 is satisfiable if some truth assignment 𝛼 satisfies 𝑋, meaning that 𝑋 evaluates to true when we replace each propositional symbol occurring in it with the value 𝛼 assigns the symbol.

truth assignment

A truth assignment on a set of propositional symbols 𝑆 is a function 𝛼: 𝑆 ➝ 𝔹 that assigns a truth value to each member of 𝑆.

ℤ

The set of all integers.

While I’m finishing up my article(s) on the Representation Theorem, here’s a fun little problem I ran across, and my solution.

The problem

Some years ago I read historian Richard Carrier’s book, On the Historicity of Jesus: Why We Might Have Reason for Doubt, in which he applies Bayesian reasoning to the question of whether there ever was historic Jesus. This of course requires assessing a prior probability for the hypothesis of historicity; Carrier proposes to do so by using the class of Rank-Raglan heroes (to which Jesus belongs) as a reference class. He identifies 15 members of this class, later expanded to 18,1 for a total of 𝑛 = 17 examples other than Jesus himself. The idea is that if 𝑘 of the 𝑛 examples we have of Rank-Raglan heroes are historic figures who actually existed, then by Laplace’s rule of succession we get a prior probability of (𝑘+1)/(𝑛+2) for any additional member of the class (e.g. Jesus) to be historic.

The snag is that there isn’t full agreement on which members of this reference class are historic. Carrier handles this by setting upper and lower bounds on what one may reasonably argue 𝑘 to be, and computing the posterior probability for each of these. There is, however, a more principled way of approaching the problem, although it is more work.

Model assumptions

Let 𝑥ᵢ be 1 if member 𝑖 of the Rank-Raglan hero class is historic, and 0 if they are mythical. Let 𝑑ᵢ be the evidence relevant to member 𝑖’s historicity; assume the 𝑑ᵢ are independent of each other conditional on 𝑥₁, …, 𝑥ₙ, and let

be the likelihood ratio in favor of historicity for member 𝑖. Finally, let 𝜃 be the long-run proportion of Rank-Raglan heroes who are historic, i.e., we treat the 𝑛+1 known members as randomly chosen examples of a much larger set, a fraction 𝜃 of whom are historic. We give 𝜃 a uniform prior. In equations:

Note that we are indulging in the usual abuse of notation where we often don’t distinguish variables from hypothesized values for those variables, e.g. writing 𝑝(𝒅 | 𝒙) in place of 𝘗𝘳(𝒅 = 𝒅′ | 𝒙 = 𝒙′).

Derivation of posterior distribution and its mean

Defining

we have

This gives the joint probability

and the marginal likelihood

Combining this with the uniform prior over 𝜃 we get

where

and 𝑁(𝒙) is just the number of 1’s in the binary vector 𝒙.

Since the density function for the beta distribution on the interval [0,1] is

where B(𝛼, 𝛽) is the beta function, we then have

where

Using the formula for the mean of a beta distribution, the prior probability of 𝑥ₙ₊₁ is then

Algorithm

The algorithm below is a fairly direct translation of the above equations, with the only complications being the need to vectorize things for implementation in R, and the need to work with the logs of probabilities to avoid numeric underflow. The function logsumexp takes as input a vector of values 𝒚 and computes

in a manner that avoids numeric underflow or overflow.

Equation (1) for the values 𝑣ₖ is impractical to directly compute when 𝑛 is large, as it requires computing 2ⁿ separate terms; but in this case 𝑛 = 17 and the algorithm, as implemented in R, runs nearly instantaneously on a 2019 laptop computer.

1. For 𝑗 ∈ { 1, …, 𝑛 } , let 𝑙ⱼ = 𝗅𝗈𝗀𝜆ⱼ. Let 𝒍 be the 𝑛-vector of values 𝑙ⱼ.

2. Let 𝐴 be a 2ⁿ × 𝑛 matrix whose rows are all possible length-𝑛 binary vectors.

3. Let 𝛋 = 𝐴𝟏 be a length-2ⁿ vector such that 𝜅ᵢ is the number of 1’s in row 𝑖 of 𝐴.

4. Let 𝒖 = (𝐴 - 1/2) 𝒍. Note that

5. For 𝑘 ∈ { 0, …, 𝑛 } , let 𝑣ₖ = 𝗅𝗈𝗀(𝑤′ₖ), computed as

6. Let 𝑧 = logsumexp(𝒗).

7. Return the vector 𝒘, where 𝑤ₖ = exp(𝑣ₖ - 𝑧).
   

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Laplace Ros With Uncertain Data

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laplace-uncertain.R : Code to compute a variant of Laplace’s rule of succession (posterior mean of a proportion 𝜃 given a uniform prior) when set membership is not known with certainty, but we have likelihood ratios 𝜆ᵢ for the evidence.

Intro Measure Theory

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A Brief Intro to Measure Theory

Rep Thm

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The Epistemic Representation Theorem.

Generalized Queries Part 1

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Generalized Queries Part 1: Borel Bedlam

Generalized Queries Part 2

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Generalized Queries Part 2: All You Need Is 𝚫⁰₂

Weak Topology

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The Weak Topology on Probability Measures

Sequential

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Sequential Spaces

Laws Of Probability

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Have we defined 𝘗𝘳(𝐴 | 𝒳) for generalized premises 𝒳 so as to continue satisfying the usual laws of probability, as finite premises do? Yes, due to continuity and the properties of limits, although there are some subtleties that are worth noting.

• We need to define 𝒳 ⊧ 𝐴 when 𝒳 is a generalized premise, and note that, unlike the case for finite premises, it is not identical to 𝘗𝘳(𝐴 | 𝒳) > 0.

• There is the issue of finite additivity vs. countable additivity, even though the finite queries we have worked with so far cannot express infinite disjunctions.

We’ll look at both the Jaynesian laws, based on conditional probabilities, and Kolmogorov’s laws, based on unconditional probabilities.

The Jaynesian laws

In the following, 𝐴 and 𝐵 are queries, and 𝑋 is a “state of information.” The Jaynesian formulation of the laws of probability makes all probabilities explicitly conditional, and can be expressed as follows:

1. Probability range: 0 ≤ 𝘗𝘳(𝐴 | 𝑋) ≤ 1.

2. Consistency with propositional logic:

   1. 𝘗𝘳(𝐴 | 𝑋) = 1 if 𝑋 logically implies the truth of 𝐴.

   2. 𝘗𝘳(𝐴 | 𝑋) = 𝘗𝘳(𝐵 | 𝑋) if 𝑋 logically implies the truth of (𝐴 ↔ 𝐵).

3. The sum rule: 𝘗𝘳(𝐴 | 𝑋) + 𝘗𝘳(¬ 𝐴 | 𝑋) = 1.

4. The product rule: 𝘗𝘳(𝐴 ∧ 𝐵 | 𝑋) = 𝘗𝘳(𝐴 | 𝑋) ⋅ 𝘗𝘳(𝐵 | 𝐴 ∧ 𝑋).

To express the equivalent laws for a generalized premise 𝒳 = (𝑋ᵢ), we need some additional definitions:

• 𝒳 ⊧ 𝐴 (“𝒳 logically entails 𝐴”) is defined to mean that 𝒳ₙ ⊧ 𝐴 for all but finitely many 𝑛. In other words, as 𝑛 → ∞, eventually 𝑋ₙ ⊧ 𝐴 and this continues to hold true. For finite premises 𝑋 ⊧ 𝐴 iff 𝘗𝘳(𝐴 | 𝑋) = 1, but for generalized premises they are not always the same; if 𝘗𝘳(¬ 𝐴 | 𝑋ₙ) > 0 for all 𝑛, but (𝘗𝘳(¬ 𝐴 | 𝑋ᵢ)) → 0, then we have 𝘗𝘳(𝐴 | 𝒳) = 1 but not 𝒳 ⊧ 𝐴.

• 𝐴 ∧ 𝒳 was defined in the article on conditionalization as the sequence (𝐴 ∧ 𝑋ᵢ), and shown to be a valid generalized premise whenever 𝘗𝘳(𝐴 | 𝒳) > 0.

Now let’s consider the equivalents laws for generalized premises:

1. 0 ≤ 𝘗𝘳(𝐴 | 𝒳) ≤ 1. Proof: By the fact that a converging sequence in a closed interval [𝑎,𝑏] converges to a point within that same interval. We have 𝘗𝘳(𝐴 | 𝑋ₙ) ∈ [0,1] for each 𝑛 ∈ ℕ, hence

2. Consistency with propositional logic:

   1. 𝘗𝘳(𝐴 | 𝒳) = 1 if 𝒳 ⊧ 𝐴. Proof: 𝘗𝘳(𝐴 | 𝑋ₙ) = 1 for all but finitely many 𝑛, so the limit is 1.

   2. 𝘗𝘳(𝐴 | 𝒳) = 𝘗𝘳(𝐵 | 𝒳) if 𝒳 ⊧ (𝐴 ↔ 𝐵). Proof: By continuity of subtraction. For all but finitely many 𝑛 we have

      and so

3. 𝘗𝘳(𝐴 | 𝒳) + 𝘗𝘳(¬ 𝐴 | 𝒳) = 1. Proof: By continuity of addition. We have

   for each 𝑛 ∈ ℕ, hence

4. 𝘗𝘳(𝐴 ∧ 𝐵 | 𝒳) = 𝘗𝘳(𝐴 | 𝒳) ⋅ 𝘗𝘳(𝐵 | 𝐴 ∧ 𝒳). Proof: shown in the article on conditionalization.

Law 2b above can in fact be strengthened to the following:

Theorem. 𝘗𝘳(𝐴 | 𝒳) = 𝘗𝘳(𝐵 | 𝒳) if 𝘗𝘳(𝐴 ↔ 𝐵 | 𝒳) = 1.

Proof. See extended version of this article. ∎

The Kolmogorov Laws

Kolmogorov’s laws for probability, which form the basis for the measure theoretic approach to probability, take unconditional probabilities as the fundamental concept, and assume probabilities are assigned to certain “measurable” subsets of some sample space 𝛺:

1. 𝘗𝘳(𝐴) ≥ 0 for any measurable 𝐴 ⊆ 𝛺.

2. 𝘗𝘳(𝛺) = 1.

3. If each 𝐴ᵢ ⊆ 𝛺 is measurable and the 𝐴ᵢ are mutually disjoint (𝐴ᵢ ∩ 𝐴ⱼ = ∅ when 𝑖 ≠ 𝑗) then

Item 3 is called countable additivity. A weaker version, where there are only a finite number of sets 𝐴ᵢ involved in the union and sum, is called finite additivity:

if each 𝐴ᵢ ⊆ 𝛺 is measurable and the 𝐴ᵢ are mutually disjoint.

The Kolmogorov laws are in terms of measurable sets, and we’re working in terms of propositions; to connect the two we’ll introduce the following notation:

Definition. If 𝜑 is a query then [𝜑] is the set of all truth assignments on ℳ (the set of manifest propositional symbols) that satisfy 𝜑.

Remark. If we impose some arbitrary ordering on the elements of ℳ, then we can treat these truth assignments as infinite sequences of binary values, and [𝜑] as a subset of 𝔹^𝜔, where 𝔹 ≜ { 0, 1 } . So, for example, if ℳ = { 𝑠₁, 𝑠₂, … } then

the set of all infinite binary sequences with 1 in the first and 0 in the third position. We have the familiar correspondence between logical operations and set operations:

Now let’s consider the equivalents of the Kolmogorov laws for generalized premises. In this case 𝔹^𝜔 plays the role of the sample space 𝛺. The first two laws are straightforward:

1. 𝘗𝘳(𝐴 | 𝒳) ≥ 0. Proven above.

2. 𝘗𝘳(𝐴 | 𝒳) = 1 whenever 𝒳 ⊧ 𝐴. Proven above.

For 3 we run into the issue that there is no way to express an infinite disjunction

as a propositional formula, so we express countable additivity as the following:

3. If [𝐵] = ⋃ᵢ [𝐴ᵢ] (𝑖 ranging from 1 to ∞) and ⊧ ¬(𝐴ᵢ ∧ 𝐴ⱼ) for all 𝑖 ≠ 𝑗, then

We prove this in two steps.

• Show that finite additivity holds.

• Show that if [𝐵] = ⋃ᵢ [𝐴ᵢ] and the 𝐴ᵢ are mutually exclusive, then all but a finite number of the 𝐴ᵢ are contradictions, [𝐴ᵢ] = ∅. Thus countable additivity reduces to finite additivity.

At a later time I’ll introduce a notion of generalized queries, for which countable additivity does not reduce to finite additivity, and we’ll have to revisit this issue.

Proof of countable additivity

Theorem. 𝘗𝘳(𝐴 ∨ 𝐵 | 𝒳) = 𝘗𝘳(𝐴 | 𝒳) + 𝘗𝘳(𝐵 | 𝒳) if 𝘗𝘳(𝐴 ∧ 𝐵 | 𝒳) = 0.

Proof. By the disjunction rule for finite premises, plus continuity of addition and subtraction. Let 𝒳 = (𝑋ᵢ). Then

∎

Corollary. If 𝘗𝘳(𝐴ᵢ ∧ 𝐴ⱼ | 𝒳) = 0 for all 𝑖 ≠ 𝑗 then

Proof. By induction using the preceding theorem. ∎

Theorem. If [𝐵] = ⋃ᵢ [𝐴ᵢ] and ⊧ ¬(𝐴ᵢ ∧ 𝐴ⱼ) for queries 𝐵 and 𝐴ᵢ, 𝑖 ≥ 1, then [𝐴ᵢ] = ∅ for all but a finite number of indices 𝑖.

Proof. See extended version of this article. ∎

Combining the results of the two steps gives us our final result:

Theorem. If [𝐵] = ⋃ᵢ [𝐴ᵢ] (𝑖 ranging from 1 to ∞) and ⊧ ¬(𝐴ᵢ ∧ 𝐴ⱼ) for all 𝑖 ≠ 𝑗 then

Proof. There is some finite 𝑛 such that [𝐴ᵢ] = ∅ for all 𝑖 > 𝑛, hence

and so 𝐵 ≡ 𝐴₁ ∨ ⋯ ∨ 𝐴ₙ, and so

since 𝘗𝘳(𝐴ᵢ | 𝒳) = 0 for 𝑖 > 𝑛. ∎

Conditionalization

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Introduction

Let’s now look at the equivalent of a conditional probability measure for our theory of epistemic probability based on the EPL Theorem. As one would expect, the product rule holds for arbitrary finite premises 𝑋 and queries 𝐴 and 𝐵:

A small bit of algebra then gives the familiar identity for conditional probabilities,

when 𝘗𝘳(𝐴 | 𝑋) > 0.

In our theory the premise 𝑋 plays the same role as a probability measure 𝜇 does in the standard theory of probability, and the conjunction 𝐴 ∧ 𝑋 plays the same role as the conditional probability measure 𝜇( ⋅ | [𝐴]), where [𝐴] is the set of truth assignments on ℳ satisfying 𝐴; therefore it is natural to define “𝑋 conditional on 𝐴” to be the premise 𝐴 ∧ 𝑋, with the requirement that 𝘗𝘳(𝐴 | 𝑋) > 0 so that 𝐴 ∧ 𝑋 is satisfiable.

To extend this to generalized premises 𝒳 = (𝑋ᵢ) it seems obvious that one should define “𝒳 conditional on 𝐴” to be the the sequence (𝐴 ∧ 𝑋ᵢ), but there are some complications. First, it may be that 𝘗𝘳(𝐴 | 𝑋ₙ) = 0 for some indices 𝑛, in which case 𝐴 ∧ 𝒳 is not a valid generalized premise. If this occurs for only a finite number of premises 𝑋ₙ, we may solve the problem by simply discarding a sufficiently large (but finite) initial prefix of the sequence, since

for any query 𝐵 and finite 𝑁 ∈ ℕ.

Second, even if 𝘗𝘳(𝐴 | 𝑋ₙ) > 0 for all 𝑛, and the sequence (𝘗𝘳(𝐵 | 𝐴 ∧ 𝑋ᵢ)) converges for all queries 𝐵, so that 𝐴 ∧ 𝒳 is in fact a valid generalized premise, there is still a problem if 𝘗𝘳(𝐴 | 𝒳) = 0 (which occurs if the sequence (𝘗𝘳(𝐴 | 𝑋ᵢ)) converges to 0.) In this case we can always construct an infinite number of generalized premises 𝒴 that are each topologically equivalent to 𝒳 (yielding the same probabilities) and yet for which 𝐴 ∧ 𝒴 is topologically distinct from 𝐴 ∧ 𝒳:

(The extended version of this article shows how to construct these premises 𝒴.)

Since we need our operations on generalized premises to preserve topological equivalence (and in fact be continuous), we therefore must restrict the domain of (𝐴 ∧ ⋅) to those 𝒳 ∈ 𝒫 for which 𝘗𝘳(𝐴 | 𝒳) > 0.

Definitions

Now for the formal definitions.

Definition. 𝒫⟨𝐴⟩ is the set of generalized premises for which query 𝐴 has nonzero probability:

Remark. The set 𝒫⟨ 𝐴⟩ is open because, for any 𝜀 > 0, it is equal to the basic open set

Definition. If 𝒳 = (𝑋ᵢ) is a generalized premise and 𝑁 ∈ ℕ, then 𝒳[𝑁:] is the sequence (𝑋ᵢ) with the first 𝑁 elements removed.

Remark. For any generalized premise 𝒳 and 𝑁 ∈ ℕ, 𝒳[𝑁:] is a generalized premise. Furthermore, 𝘗𝘳(𝐴 | 𝒳[𝑁:]) = 𝘗𝘳(𝐴 | 𝒳) for every query 𝐴, and therefore 𝒳 and 𝒳[𝑁:] are topologically equivalent. These follow from the fact that removing any finite number of elements from a convergent sequence leaves its convergence and its limits unchanged.

Definition. For any query 𝐴 and generalized premise 𝒳 = (𝑋ᵢ) ∈ 𝒫⟨𝐴⟩,

That is, we replace each 𝑋ₙ with 𝐴 ∧ 𝑋ₙ and drop initial elements of the resulting sequence until all that remain are satisfiable. Since 𝒳 ∈ 𝒫⟨𝐴⟩ only a finite number of elements will be discarded.

For symmetry we likewise define 𝒳 ∧ 𝐴 ≜ 𝐴 ∧ 𝒳.

Results

First we prove that 𝒳 ∈ 𝒫⟨𝐴⟩ is a sufficient condition for 𝐴 ∧ 𝒳 to be a generalized premise, and to ensure that the familiar formula for conditional probabilities still holds for generalized premises.

Theorem. 𝐴 ∧ 𝒳 ∈ 𝒫 for any query 𝐴 and generalized premise 𝒳 ∈ 𝒫⟨𝐴⟩; furthermore, for every query 𝐵 we have

Proof. Let 𝒳 = (𝑋ᵢ) and let 𝑁 be the smallest nonnegative integer such that 𝘗𝘳(𝐴 | 𝑋ₙ) > 0 for all 𝑛 ≥ 𝑁. Since 𝘗𝘳(𝐴 | 𝒳) > 0 and division is a continuous function, we have for any query 𝐵 that

and since we get this convergence for any query 𝐵, the sequence (𝐴 ∧ 𝒳) = (𝐴 ∧ 𝑋ᵢ)[𝑁:] defines a generalized premise, with

∎

The other important property is that (𝐴 ∧ ⋅) is continuous on its domain (and hence preserves topological equivalence).

Theorem. For any query 𝐴, the function (𝐴 ∧ ⋅) : 𝒫⟨𝐴⟩ → 𝒫 is continuous.

Proof. Let (𝒳ᵢ) be a sequence in 𝒫⟨𝐴⟩ converging to 𝒴 ∈ 𝒫⟨𝐴⟩ . Then for any query 𝐵 we have

with the second line following from the continuity of 𝘗𝘳(𝐴 ∧ 𝐵 | ⋅) and 𝘗𝘳(𝐴 | ⋅) and the fact that 𝘗𝘳(𝐴 | 𝒴) > 0. Therefore

and thus (𝐴 ∧ ⋅) is sequentially continuous, and hence (topologically) continuous. ∎