= Markov kernel
{c}
{wiki}
= Transition kernel
{synonym}
A <Markov kernel> from a <measurable space> $(E,\mathcal E)$ to a <measurable space> $(F,\mathcal F)$ is a <function> $K:E\times\mathcal F\to[0,1]$ satisfying two conditions: for each $x\in E$, $A\mapsto K(x,A)$ is a <Probability measure>; for each $A\in\mathcal F$, $x\mapsto K(x,A)$ is a <measurable function>. The second condition makes <integrals> against a distribution of current states well defined.
If the current state has <probability distribution> $\mu$, the next-state distribution is
$$
(\mu K)(A)=\int_E K(x,A)\,\mu(dx).
$$
On a finite or countable <state space> with all subsets measurable, a <transition matrix> gives $K(i,A)=\sum_{j\in A}P_{ij}$. A deterministic measurable map $T:E\to F$ gives $K(x,A)=\mathbf1_A(T(x))$, so the kernel formalism includes deterministic dynamics as well as random transitions.
Successive kernels compose by $(KL)(x,A)=\int_F L(y,A)K(x,dy)$. The <tower property of conditional expectation>[tower property] of <conditional expectation> explains this rule: first condition on the intermediate state, then average over it. For a homogeneous <Markov chain>, the $n$-step transition kernel is $K^n$; its composition law is the <Chapman-Kolmogorov equation>.
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