From cc6f7fd3812c80593a76bc9744e3167c0b72face Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Wed, 7 Jun 2023 03:35:56 +0200 Subject: [PATCH] small fixes --- inputs/lecture_15.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex index 0eaf742..5bd845d 100644 --- a/inputs/lecture_15.tex +++ b/inputs/lecture_15.tex @@ -50,7 +50,7 @@ We want to derive some properties of conditional expectation. If $X \ge 0$, then $\bE[X | \cG] \ge 0$ a.s. \end{theorem} \begin{proof} - Let $W $ be a version of $\E[X | \cG]$. + Let $W $ be a version of $\bE[X | \cG]$. Suppose $\bP[ W < 0] > 0$. Then $G \coloneqq \{W < -\frac{1}{n}\} \in \cG$ For some $n \in \N$, we have $\bP[G] > 0$. @@ -88,7 +88,7 @@ We want to derive some properties of conditional expectation. Take some $G \in \cG$. We know by (b) % TODO REF - that $\be[Z_n \One_G] = \bE[X_n \One_G]$. + that $\bE[Z_n \One_G] = \bE[X_n \One_G]$. The LHS increases to $\bE[Z \One_G]$ by the monotone convergence theorem. Again by MCT, $\bE[X_n \One_G]$ increases to