conditional dominated convergence

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Josia Pietsch 2023-07-19 20:59:36 +02:00
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@ -113,11 +113,10 @@ We want to derive some properties of conditional expectation.
\begin{theorem}[Conditional dominated convergence theorem]
\label{ceprop7}
\label{cdct}
Let $X_n,X \in L^1(\Omega, \cF, \bP)$.
Suppose $|X_n(\omega)| < X(\omega)$ a.e.~
and $\int |X| \dif \bP < \infty$.
Then $X_n(\omega) \to X\left( \omega \right) \implies \bE[ X_n | \cG] \to \bE[X | \cG]$.
Let $X_n,Y \in L^1(\Omega, \cF, \bP)$.
Suppose that $\sup_n |X_n(\omega)| < Y(\omega)$ a.e.~
and that $X_n$ converges to a pointwise limit $X$.
Then $\bE[ X_n | \cG] \to \bE[X | \cG]$ a.e.
\end{theorem}
\begin{proof}
\notes