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As with conditional expectation, this can be further generalized to conditioning on a sigma algebra <math>\mathcal{F}</math>. In that case the conditional distribution is a function <math>\Omega \times \mathcal{B}(\mathbb{R}) \to [0, 1]</math>:
:<math> \kappa_{Y\mid\mathcal{F}}(\omega, A) = \operatorname E[1_{Y \in A} \mid \mathcal{F}](\omega)</math>
=== Regularity ===
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In other words <math>\kappa_{Y\mid X}</math> is a [[Markov kernel]].
The second condition holds trivially, but the proof of the first is more involved. It can be shown that if ''Y'' is a random element <math>\Omega \to S</math> in a [[Radon space]] ''S'', there exists a <math>\kappa_{Y\mid X}</math> that satisfies the first condition.<ref>{{cite book |last1=Klenke |first1=Achim |title=Probability theory : a comprehensive course |date=30 August 2013 |___location=London |isbn=978-1-4471-5361-0 |edition=Second}}</ref> It is possible to construct more general spaces where a regular conditional probability distribution does not exist.<ref>Faden, A.M., 1985. The existence of regular conditional probabilities: necessary and sufficient conditions. ''The Annals of Probability'', 13(1), pp. 288–298.</ref>
=== Relation to conditional expectation ===
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:: <math>P\big(A\cap T^{-1}(B)\big) = \int_B \nu(x,A) \,(P\circ T^{-1})(\mathrm{d}x)</math>
where <math>P\circ T^{-1}</math> is the [[pushforward measure]] <math>T_*P</math> of the distribution of the random element <math>T</math>,
<math>x\in\
Specifically, if we take <math>B=E</math>, then <math>A \cap T^{-1}(E) = A</math>, and so
:<math>P(A) = \int_E \nu(x,A) \, (P\circ T^{-1})(\mathrm{d}x),</math>
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