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Dose $\lim_{k\to\infty}\langle f_k\rangle=\langle f\rangle$ imply $\lim_{k\to\infty}f_k=f_k$ almost everywhere?

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We know that for any $\langle f\rangle\in L^p(X,\mathscr{A},\mu)$ there is a sequence $\{\langle f_k\rangle\}$ of elements of $L^p(X,\mathscr{A},\mu)$, where each $f_k$ is a simple function that belongs to $\mathscr{L}^p(X,\mathscr{A},\mu)$, such that\begin{align*} \lim_{k\to\infty}\|\langle f_k\rangle-\langle f\rangle\|_p=\lim_{k\to\infty}d(\langle f_k\rangle,\langle f\rangle)=0;\end{align*}that is, $\lim_{k\to\infty}\langle f_k\rangle=\langle f\rangle$. Does this imply that for any $f\in\langle f\rangle$ and $f_k\in\langle f_k\rangle$ we have $\lim_{k\to\infty}f_k=f$?

I think intuitively it is true, because every element in $\langle h\rangle$ agrees almost everywhere. But I am having a difficult time producing a formal proof. Could someone please help me out? Thanks a lot in advance!


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