Markov kernel

Remarks

Example: let $X = Y = \set{1, \ldots, n}$ with all subsets measurable, and let $P = (p_{ij})$ be a matrix with nonnegative entries and row sums $1$. Then $\kappa(i, B) = \sum_{j \in B} p_{ij}$ is a Markov kernel: clause 1 holds because every function on a finite discrete space is measurable, and clause 2 because row $i$ is a probability vector.

Non-example: on $\R$ with its Borel sets, $\kappa(x, B) = 2 \, \mathbf{1}_B(x)$ fails clause 2, since $\kappa(x, \R) = 2$; it also takes values outside $[0, 1]$.

Another standard example: for a measurable map $f \colon X \to Y$, the kernel $\kappa(x, B) = \mathbf{1}_B(f(x))$ is a Markov kernel, the deterministic kernel of $f$. A kernel acts on bounded measurable functions by $(\kappa g)(x) = \int_Y g(y) \, \kappa(x, dy)$ and on probability measures by $(\mu \kappa)(B) = \int_X \kappa(x, B) \, \mu(dx)$. Markov kernels are also called stochastic kernels, transition kernels or probability kernels.