josia-notes/s23-probability-theory
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Josia Pietsch 2023-06-07 03:35:56 +02:00
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inputs/lecture_15.tex
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@ -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. If $X \ge 0$, then $\bE[X | \cG] \ge 0$ a.s.
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}
Let $W $ be a version of $\E[X | \cG]$. Let $W $ be a version of $\bE[X | \cG]$.
Suppose $\bP[ W < 0] > 0$. Suppose $\bP[ W < 0] > 0$.
Then $G \coloneqq \{W < -\frac{1}{n}\} \in \cG$ Then $G \coloneqq \{W < -\frac{1}{n}\} \in \cG$
For some $n \in \N$, we have $\bP[G] > 0$. 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 $G \in \cG$. Take some $G \in \cG$.
We know by (b) % TODO REF 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 The LHS increases to $\bE[Z \One_G]$ by the monotone
convergence theorem. convergence theorem.
Again by MCT, $\bE[X_n \One_G]$ increases to Again by MCT, $\bE[X_n \One_G]$ increases to