= Regular conditional distribution
{title2=$K(y,B)=P(X\in B\mid Y=y)$}
A regular conditional distribution of a <random variable> $X$ given $Y$ is a <Markov kernel> $K$ such that, for all measurable sets $A,B$,
$$
P(X\in B,Y\in A)=\int_A K(y,B)\,P_Y(dy).
$$
For each $y$, $K(y,\cdot)$ is a <probability measure>; for each $B$, $K(\cdot,B)$ is measurable. Thus it assigns an actual probability distribution to a specified observation even when $P(Y=y)=0$. Such kernels exist for standard Borel state spaces, including real-valued variables, and their uniqueness is only up to $P_Y$-null sets. With a joint density and positive marginal density, the ratio of the two densities defines the usual conditional density kernel. Values outside the marginal support or on a null set are not uniquely determined by the joint law.
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