Improper Regular Conditional Distributions

Abstract

Improper regular conditional distributions (rcd’s) given a $\sigma$-field $\mathscr{A}$ have the following anomalous property. For sets $A \in \mathscr{A}, \mathrm{Pr}(A|\mathscr{A})$ is not always equal to the indicator of $A$. Such a property makes the conditional probability puzzling as a representation of uncertainty. When rcd’s exist and the$\sigma$-field $\mathscr{A}$ is countably generated, then almost surely the rcd is proper. We give sufficient conditions for an rcd to be improper in a maximal sense, and show that these conditions apply to the tail $\sigma$-field and the $\sigma$-field of symmetric events.