Direct hyperfinite representations of finitely additive probabilities

Abstract

Fix a standard measurable space (X,X) and an internal, hyperfinite H⊂∗X that contains each x∈X. All finitely additive probabilities on (X,X) can be represented by setting p(B) equal to the standard part of Q(∗B∩H) for some internal probability Q supported on H. From this starting point, we have the following: a decomposition of two ways in which a probability can fail to be countably additive, a nonstandard characterization of countable additivity for probabilities on complete separable metic spaces, and results on the multiplicity of hyperfinitely supported probabilities that represent a given finitely additive p, from which we have set-valued integrals with respect to products of finitely additive probabilities that respect statistical independence, and for subfields or sub-σ-fields of X, a proper ∗X-based disintegration of finitely additive probabilities as a countably additive integral over the set of finitely additive probabilities. We end with several applications for which the alternative Stone space approach to representing finitely additive probabilities is an impediment to the analyses.