Improper Regular Conditional Distributions

Improper regular conditional distributions (rcd's) given a σ-field .A have the following anomalous property. For sets A ∈, Pr(A I 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 σ-field 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 σ-field and the σ-field of symmetric events