conditional dominated convergence
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@ -113,11 +113,10 @@ We want to derive some properties of conditional expectation.
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\begin{theorem}[Conditional dominated convergence theorem]
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\begin{theorem}[Conditional dominated convergence theorem]
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\label{ceprop7}
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\label{ceprop7}
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\label{cdct}
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\label{cdct}
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Let $X_n,X \in L^1(\Omega, \cF, \bP)$.
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Let $X_n,Y \in L^1(\Omega, \cF, \bP)$.
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Suppose $|X_n(\omega)| < X(\omega)$ a.e.~
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Suppose that $\sup_n |X_n(\omega)| < Y(\omega)$ a.e.~
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and $\int |X| \dif \bP < \infty$.
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and that $X_n$ converges to a pointwise limit $X$.
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Then $X_n(\omega) \to X\left( \omega \right) \implies \bE[ X_n | \cG] \to \bE[X | \cG]$.
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Then $\bE[ X_n | \cG] \to \bE[X | \cG]$ a.e.
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\end{theorem}
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\end{theorem}
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\begin{proof}
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\begin{proof}
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\notes
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\notes
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