a regular conditional distribution

Conditioning event by event, P(A given G) = E[1_A given G], gives each event its own conditional probability — but each is defined only 'almost surely', with its own exceptional null set. We would dearly like to fix a single outcome omega and read off a whole genuine probability distribution P( · given G)(omega), the way physical intuition demands: 'given everything G tells me, here is the actual distribution of X now'. A regular conditional distribution is exactly a coherent way to do that.

Formally, a regular conditional distribution of X given G is a map mu(omega, B) such that, for each fixed outcome omega, B -> mu(omega, B) is an honest probability measure (countably additive, total mass 1), and, for each fixed measurable set B, omega -> mu(omega, B) is a version of P(X in B given G). The difficulty it overcomes is real: the individual conditional probabilities are each ambiguous up to a null set, and uncountably many such ambiguities could clash. A theorem (using that the values of X live in a 'nice' space — a standard Borel space such as the real line) shows the versions can be chosen consistently so that for almost every omega you really do get a bona fide measure. Once you have it, you can compute E[ f(X) given G ](omega) as an ordinary integral of f against mu(omega, ·).

This is the rigorous foundation under the everyday phrase 'the conditional distribution of X given Y = y' and behind constructions like Markov transition kernels. The honest caveats are two. First, existence is not automatic: on pathological spaces a regular conditional distribution can fail to exist; the standard-Borel hypothesis is what saves it (and it covers essentially every space in practice). Second, even when it exists it is unique only up to a null set of omega, so it is a representative, not a canonical object — but a representative good enough to integrate against, which is all the theory needs.