The conditional expectation of a random variable $X$ given a nonnegligible event $A$ is usually defined as $\mathbb{E}(X\mathbb{1}_A)/\mathbb{P}(A)$. How does one derive the fact that $\mathbb{E}(X|A)=\int_{\Omega}X(\omega)\mathbb{P}(d\omega|A)$ where $\mathbb{P}(\cdot|A)$ denotes the regular conditional distribution?
2026-05-15 00:09:34.1778803774
Definition of conditional expectation given an event
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