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Let $(X,\mathcal M, \mu)$ be an arbitrary measure space and $1\le p<\infty$. I am curious whether the following statement holds:

Let $\{f_n:X\to\mathbb{R}:n\in\mathbb{N}\}_n$ be a sequence in $L^p=L^p(X,\mathcal M, \mu)$. If $f_n\to f\in L^p$ (with respect to the $p$-norm) and $f_n\le f$ almost everywhere, then $f=\sup_{n\in\mathbb{N}}f_n$ almost everywhere.

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