conditional dominated convergence

inputs/lecture_15.tex

@@ -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