Time-homogeneous Markov chain with transition kernel

Assume the axiom of choice. Let $K$ be a probability kernel on $(E,\mathcal{E})$, let $(\mathrm{\Omega },\mathcal{F},\mathbb{P},({\mathcal{F}}_n{)}_{n\ge 0})$ be a filtered probability space, and let $X=(X_n{)}_{n\ge 0}$ be an adapted $E$-valued process. We call $X$ a time-homogeneous Markov chain with transition kernel $K$ relative to $({\mathcal{F}}_n)$ if, for every $n\ge 0$ and $A\in \mathcal{E}$, $\mathbb{P}(X_{n+1}\in A\mid {\mathcal{F}}_n)=K(X_n,A)\text{a.s.}$ Here the left side is an almost-everywhere class of conditional-probability versions. The right side is ${\mathcal{F}}_n$-measurable because $x\mapsto K(x,A)$ is $\mathcal{E}$-measurable and $X_n$ is ${\mathcal{F}}_n\mathrm{/}\mathcal{E}$-measurable.

If ${\mathcal{F}}_n=\sigma (X_0,\dots ,X_n)$, we say simply that $X$ is a Markov chain relative to its natural filtration. A chain relative to a larger filtration is therefore a stronger assertion, not merely a change of notation.

Choice is used only through the library construction that supplies conditional expectations, hence conditional probabilities, simultaneously as almost-everywhere classes; this definition asserts no pointwise regular conditional distribution.