## Saturday, May 18, 2019

### Regularity

Plausibly—though there are some set-theoretic worries that require some care if the language is rich enough—for a fixed language, there are only countably many situations we can describe. Consequently, we only need to do Bayesian epistemology for countably many events. But this solves the problem of regularity for uncountable sample spaces. For even if there are uncountably many events, only countably many are describable and hence matter, and they form a field (i.e., are closed under finite unions and complements) and:

Proposition: For any countable field F of subsets of a set Ω, there is a countably additive probability measure P on the power set of Ω such that every event in F has non-zero probability.

Proof: Let the non-empty members of F be u1, u2, .... Let a1, a2, ... be any sequence of positive numbers adding up to 1 (e.g., an = 2n). Choose one point xn ∈ un. Let P(A)=∑nanAn where An is 1 if xn ∈ A and 0 otherwise.

Note that this proof uses the countable Axiom of Choice, but almost nobody is worried about that.