Regular conditional probability

Remarks

Clauses 2 and 3 say that $\nu(\cdot, A)$ is a version of the conditional probability $P(A \mid \mathcal{G}) = E[\mathbf{1}_A \mid \mathcal{G}]$ for each $A$. Then $E[Z \mid \mathcal{G}](\omega) = \int Z(\omega') \, \nu(\omega, d\omega')$ almost surely for every integrable $Z$. A regular conditional probability exists when $(\Omega, \mathcal{F})$ is a standard Borel space, for example a Polish space with its Borel σ-algebra. It is then unique up to a $P$-null set of $\omega$. It need not exist in general. A related notion is the regular conditional distribution of a random variable $Y$ given $X$. It is a Markov kernel $\kappa$ with $P(X \in A, Y \in B) = \int_A \kappa(x, B) \, P_X(dx)$. It exists whenever $Y$ takes values in a standard Borel space. Example: let $\Omega = [0, 1]^2$ with Lebesgue measure, and let $\mathcal{G}$ be generated by the first coordinate. Then $\nu((x, y), A) = \lambda(\set{y' : (x, y') \in A})$, with $\lambda$ the Lebesgue measure on $[0, 1]$, is a regular conditional probability; clause 3 is Fubini's theorem. Non-example: if $\mathcal{G} = \set{\emptyset, \Omega}$ and $0 < P(A) < 1$ for some $A$, then $\nu(\omega, A) = \mathbf{1}_A(\omega)$ satisfies clause 1 but fails clause 2, since $\mathbf{1}_A$ is not constant.