Regular conditional probability

Summary

A regular conditional probability is a version of conditional probability given a σ-algebra that is a genuine probability measure for each outcome. Conditional probabilities $P(A \mid \mathcal{G})$ are defined one event at a time and only up to null sets. So in general they need not fit together into a measure. A regular version is a Markov kernel that does fit them together, and it lets one compute conditional expectations as integrals. It exists when the underlying space is nice enough, for example a Polish space with its Borel σ-algebra.