Completeness of $L_1$ spaces over finitely additive probabilities

Representing Finitely Additive Invariant Probabilities

Learning Finitely Additive Probabilities: an Impossibility Theorem

If (X,X ) is a measure space and F◦ ⊂ X is a field generated either by a countable class of sets or by a Vapnik-Červonenkis class, then if μ is purely finitely additive, there exist uncountably many μ′ agreeing with μ on F◦ and having |μ(A) − μ′(A)| = 1 for uncountably many A. If μ is also non-atomic, then for any r ∈ (0, 1], |μ(Ar)− μ(Ar)| = r for uncountably many Ar. Al-Najjar’s [1] unlearnab...

Finitely additive representation of L spaces

Let λ̄ be any atomless and countably additive probability measure on the product space {0,1}N with the usual σ -algebra. Then there is a purely finitely additive probability measure λ on the power set of a countable subset T ⊂ T̄ such that Lp(λ̄) can be isometrically isomorphically embedded as a closed subspace of Lp(λ). The embedding is strict. It is also ‘canonical,’ in the sense that it maps si...

Finitely Additive Beliefs and Universal Type Spaces

In this paper we examine the existence of a universal (to be precise: terminal) type space when beliefs are described by finitely additive probability measures. We find that in the category of all type spaces that satisfy certain measurability conditions (κ-measurability, for some fixed regular cardinal κ), there is a universal type space (i.e. a terminal object, that is a type space to which e...

Completeness and interpolation of almost-everywhere quantification over finitely additive measures

We give an axiomatization of first-order logic enriched with the almosteverywhere quantifier over finitely additive measures. Using an adapted version of the consistency property adequate for dealing with this generalized quantifier, we show that such a logic is both strongly complete and enjoys Craig interpolation, relying on a (countable) model existence theorem. We also discuss possible exte...