Markov kernel: Difference between revisions

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== Formal definition ==

Let <math>(X,\mathcal A)</math> and <math>(Y,\mathcal B)</math> be [[measurable space]]s. A ''Markov kernel'' with source <math>(X,\mathcal A)</math> and target <math>(Y,\mathcal B)</math>, sometimes written as <math>\kappa:(X,\mathcal{A})\to(Y,\mathcal{B})</math>, is a mapfunction <math>\kappa : \mathcal B \times X \to [0,1]</math> with the following properties:

# For every (fixed) <math>B_0 \in \mathcal B</math>, the map <math> x \mapsto \kappa(B_0, x)</math> is <math>\mathcal A</math>-[[measurable function|measurable]]

# For every (fixed) <math> x_0 \in X</math>, the map <math> B \mapsto \kappa(B, x_0)</math> is a [[probability measure]] on <math>(Y, \mathcal B)</math>

In other words it associates to each point <math>x \in X</math> a [[probability measure]] <math>\kappa(dy|x): B \mapsto \kappa(B, x)</math> on <math>(Y,\mathcal B)</math> such that, for every measurable set <math>B\in\mathcal B</math>, the map <math>x\mapsto \kappa(B, x)</math> is measurable with respect to the [[Σ-algebra|<math>\sigma</math>-algebra <math>\mathcal A</math>]].<ref>{{cite book |last1=Klenke |first1=Achim |title=Probability Theory: A Comprehensive Course|series=Universitext |year=2014 |publisher=Springer|page=180|edition=2|doi=10.1007/978-1-4471-5361-0|isbn=978-1-4471-5360-3 }}</ref>

== Examples ==