small fixes
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@ -50,7 +50,7 @@ We want to derive some properties of conditional expectation.
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If $X \ge 0$, then $\bE[X | \cG] \ge 0$ a.s.
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If $X \ge 0$, then $\bE[X | \cG] \ge 0$ a.s.
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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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Let $W $ be a version of $\E[X | \cG]$.
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Let $W $ be a version of $\bE[X | \cG]$.
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Suppose $\bP[ W < 0] > 0$.
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Suppose $\bP[ W < 0] > 0$.
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Then $G \coloneqq \{W < -\frac{1}{n}\} \in \cG$
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Then $G \coloneqq \{W < -\frac{1}{n}\} \in \cG$
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For some $n \in \N$, we have $\bP[G] > 0$.
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For some $n \in \N$, we have $\bP[G] > 0$.
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@ -88,7 +88,7 @@ We want to derive some properties of conditional expectation.
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Take some $G \in \cG$.
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Take some $G \in \cG$.
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We know by (b) % TODO REF
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We know by (b) % TODO REF
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that $\be[Z_n \One_G] = \bE[X_n \One_G]$.
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that $\bE[Z_n \One_G] = \bE[X_n \One_G]$.
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The LHS increases to $\bE[Z \One_G]$ by the monotone
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The LHS increases to $\bE[Z \One_G]$ by the monotone
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convergence theorem.
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convergence theorem.
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Again by MCT, $\bE[X_n \One_G]$ increases to
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Again by MCT, $\bE[X_n \One_G]$ increases to
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