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Suppose $1\leq p<\infty$. Let $E$ be a Banach space. Consider a filtration $F_n$ on some probability space $\Omega$. Let $X\in L^p(\Omega,E)$ where $L^p(\Omega,E)$ denote the Bochner space. In this case, we have

$$ \lim_{n\to \infty}\mathbb{E}(X|F_n)=\mathbb{E}(X|F_\infty) $$ in $L^{p}(\Omega,E)$ where $\mathbb{E}$ denote the conditional expectation operator.

I need a precise reference for this fact with a preference for a published book.

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up vote 1 down vote accepted

Vector measures, Joseph Diestel, John Jerry Uhl (Chapter V Martingales, Section 2 Convergence theorems).

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I took the liberty to add in a link. Verbatim the same answer (with modified parentheses) could go here – t.b. Nov 2 '11 at 23:19
@t.b. Thanks. $ $ – Did Nov 3 '11 at 8:38

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