Improper regular conditional distributions 1 by

Abstract

Improper regular conditional distributions (red's) given a σ-field Script A Sign have the following anomalous property. For sets A ∈ Script A Sign, Pr(A | Script A Sign) is not always equal to the indicator of A. Such a property makes the conditional probability puzzling as a representation of uncertainty. When red's exist and the σ-field Script A Sign is countably generated, then almost surely the red is proper. We give sufficient conditions for an red 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.