On Existence and Non-Existence of Proper, Regular, Conditional Distributions

Abstract

If $\mathscr{A}$ is the tail, invariant, or symmetric field for discrete-time processes, or a field of the form $\mathscr{F}_{t+}$ for continuous-time processes, then no countably additive, regular, conditional distribution given $\mathscr{A}$ is proper. A notion of normal conditional distributions is given, and there always exist countably additive normal conditional distributions if $\mathscr{A}$ is a countably generated sub $\sigma$-field of a standard space. The study incidentally shows that the Borel-measurable axiom of choice is false. Classically interesting subfields $\mathscr{A}$ of $\mathscr{B}$ possess certain desirable properties which are the defining properties for $\mathscr{A}$ to be "regular" in $\mathscr{B}$.