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==Definition==
Let <math>(\Omega, \mathcal F, \mathfrak P)</math> be a [[probability space]], and let <math>T:\Omega\rightarrow E</math> be a [[random variable]], defined as a [[Borel measure|Borel-]][[measurable function]] from <math>\Omega</math> to its [[Probability_space#Random_variables|state space]] <math>(E, \mathcal E).</math> Then a '''regular conditional probability''' is defined as a function <math>\nu:E \times\mathcal F \rightarrow [0,1],</math> called a "transition probability", where <math>\nu(x,A)</math> is a valid probability measure (in its second argument) on <math>\mathcal F</math> for all <math>x\in E</math> and a measurable function in ''E'' (in its first argument) for all <math>A\in\mathcal F,</math> such that for all <math>A\in\mathcal F</math> and all <math>B\in\mathcal E</math><ref>D. Leao Jr. et al. ''Regular conditional probability, disintegration of probability and Radon spaces.'' Proyecciones. Vol. 23, No. 1, pp. 15–29, May 2004, Universidad Católica del Norte, Antofagasta, Chile [http://www.scielo.cl/pdf/proy/v23n1/art02.pdf PDF]</ref>
:<math>\mathfrak P\big(A\cap T^{-1}(B)\big) = \int_B \nu(x,A) \,d\mathfrak P\big(T^{-1}(x)\big).</math>
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