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Ira Leviton (talk | contribs) m Fixed a typo found with Wikipedia:Typo_Team/moss. |
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<math>\mu(\cdot, \omega)</math> is a probability measure on <math>\mathcal{R}^1</math> for each <math>\omega \in \Omega</math> (i.e., it is [[Regular conditional probability|'''regular''']]) and <math>\mu(H, \cdot) = P(X^{-1}( H )\mid \mathcal{G})</math> (almost surely) for every <math>H \in \mathcal{R}^1</math>.
For any <math>\omega \in \Omega</math>, the function <math>\mu(\cdot, \omega) : \mathcal{R}^1 \to \mathbb{R}</math> is called a '''[[Conditional expectation#Definition of conditional probability|conditional probability]] distribution''' of <math>X</math> given <math>\mathcal{G}</math>. In this case,<math>E[X \mid \mathcal{G}] = \int_{-\infty}^\infty x \, \mu(d x, \cdot)</math> almost surely.
== Relation to conditional expectation ==
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