A stochastic kernel on some set $X$ given another set $Y$ is a function $P(\cdot\mid \cdot)$ such that 1. $P(\cdot\mid y)$ is a probability measure on $X$ for each fixed $y\in Y$, and; 2. $P(B\mid \cdot)$ is a measurable function on $Y$ for each $B\in\mathcal{B}(X)$.

The set of all stochastic kernels on $X$ given $Y$ is denoted by $\mathcal{P}(X\mid Y)$.