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  "body": "\n## Definition\n\nLet $(X, \\mathcal{A})$ and $(Y, \\mathcal{B})$ be [[measurable spaces|N4KDT83V3S]].\nA *Markov kernel* from $X$ to $Y$ is a map $\\kappa \\colon X \\times \\mathcal{B} \\to [0, 1]$ such that:\n\n1. for each $B \\in \\mathcal{B}$, the map $x \\mapsto \\kappa(x, B)$ is [[measurable|YYNCJSR0PJ]] from $(X, \\mathcal{A})$ to $[0, 1]$;\n2. for each $x \\in X$, the map $B \\mapsto \\kappa(x, B)$ is a [[probability measure|059TJN0NZ1]] on $(Y, \\mathcal{B})$.\n\n## Remarks\n\nExample: 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 $1$.\nThen $\\kappa(i, B) = \\sum_{j \\in B} p_{ij}$ 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.\n\nNon-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 $[0, 1]$.\n\nAnother 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 $f$.\nA kernel acts on bounded measurable functions by $(\\kappa g)(x) = \\int_Y g(y) \\, \\kappa(x, dy)$ and on probability measures by $(\\mu \\kappa)(B) = \\int_X \\kappa(x, B) \\, \\mu(dx)$.\nMarkov kernels are also called stochastic kernels, transition kernels or probability kernels.\n",
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