Markov kernel: Difference between revisions

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

# For every (fixed) <math> x \in X</math>, the map <math> B \mapsto \kappa(B, x)</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 <math>\sigma</math>-algebra <math>\mathcal A.</math>.<ref>{{cite book |last1=Klenke |first1=Achim |title=Probability Theory: A Comprehensive Course|publisher=Springer|page=180|edition=2|doi=10.1007/978-1-4471-5361-0}}</ref>

== Examples ==