Regular conditional probability

Let $(\mathrm{\Omega },\mathcal{F},P)$ be a probability space whose sample measurable space $(\mathrm{\Omega },\mathcal{F})$ is standard Borel in Standard Borel spaces, and let $\mathcal{G}\subseteq \mathcal{F}$ be a sub-sigma-algebra. A regular conditional probability given $\mathcal{G}$ is a regular conditional distribution, in Regular conditional distribution, of the identity random element ${\mathrm{id}}_{\mathrm{\Omega }}:(\mathrm{\Omega },\mathcal{F})\to (\mathrm{\Omega },\mathcal{F})$.

The identity is measurable because its inverse image of A is A. Written explicitly, this is a probability kernel $K:(\mathrm{\Omega },\mathcal{G})\to (\mathrm{\Omega },\mathcal{F})$ with ${\int }_{H}K(\omega ,A)P(d\omega )=P(H\cap A)(H\in \mathcal{G},A\in \mathcal{F})\mathrm{.}$

Its target sigma-algebra is the full sample sigma-algebra $\mathcal{F}$, while its source sigma-algebra is $\mathcal{G}$. Empty and whole target events give zero and one evaluations, respectively. The term here is used under the displayed standard-Borel hypothesis. This definition makes no existence claim on an arbitrary sample measurable space and uses no choice axiom merely to specify the property.