Regular conditional distribution

Let $X:(\mathrm{\Omega },\mathcal{F},P)\to (T,\mathrm{T})$ be a measurable random element and $\mathcal{G}\subseteq \mathcal{F}$ a sub-sigma-algebra. A regular conditional distribution of X given $\mathcal{G}$ is a probability kernel $K:(\mathrm{\Omega },\mathcal{G})\to (T,\mathrm{T})$ satisfying ${\int }_{H}K(\omega ,A)P(d\omega )=P(H\cap \{X\in A\})(H\in \mathcal{G},A\in \mathrm{T})\mathrm{.}$

“Probability kernel” has the pointwise meaning of Measure kernel and probability kernel: every $K(\omega ,\cdot )$ is a probability measure, and every evaluation is $\mathcal{G}$-measurable. Its evaluations lie in $[0,1]$ and are integrable. Thus the testing identity says exactly that $K(\cdot ,A)$ is a conditional-expectation version of $1_{\{X\in A\}}$ in Conditional expectation given a sigma algebra. Conversely, a probability kernel with these version identities satisfies the displayed definition.

The kernel requirement holds at every sample point. Separate eventwise choices of conditional-expectation versions do not imply it and do not by themselves provide one exceptional set valid for all A. When A is empty the testing identity is zero on both sides, and when A=T it is P(H) on both sides. This defines a property of a supplied kernel and makes no existence or AC assertion.