Markov kernel

In probablity theory, a Markov kernel is a map that plays the role, in the general theory of Markov processes, that the transition matrix does in the theory of finite Markov chains.

Formal Definition

Let $(X,{\mathcal {A}})$, $(Y,{\mathcal {B}})$ be measurable spaces. A Markov kernel is a map K that associated to each point x of X a probability measure K(x) on $(Y,{\mathcal {B}})$ such that, for every measurable set $B\in {\mathcal {B}}$, the map $x\mapsto K(x)(B)$ is measurable with respect to the $\sigma$-algebra ${\mathcal {A}}$.