Regular conditional distribution ($K(y,B)=P(X\in B\mid Y=y)$)

A regular conditional distribution of a random variable $X$ given $Y$ is a Markov kernel $K$ such that, for all measurable sets $A,B$,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.