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Let <math>(S,Y)</math> be a [[Borel set#Standard Borel spaces and Kuratowski theorems|Borel space]], <math>X</math> a <math>(S,Y)</math>-valued random variable on the measure space <math>(\Omega, \mathcal{F}, P)</math> and <math>\mathcal G \subseteq \mathcal F</math> a sub-<math>\sigma</math>-algebra. Then there exists a Markov kernel <math>\kappa</math> from <math>(\Omega, \mathcal G)</math> to <math>(S,Y)</math>, such that <math>\kappa(\cdot,B)</math> is a version of the [[conditional expectation]] <math>\mathbb{E}[\mathbf 1_{\{X \in B\}} \mid \mathcal G]</math> for every <math>B \in Y</math>, i.e.
:<math>P(X \in B\mid\mathcal G)=\mathbb{E} \left [\mathbf 1_{\{X \in B\}}\mid\mathcal G \right ] = \kappa(\
It is called regular conditional distribution of <math>X</math> given <math>\mathcal G</math> and is not uniquely defined.
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