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Let $(X,\mathcal{F},\mu)$ be a measure space, and let $f,f_1,f_2,\ldots$ be $[+\infty,-\infty]$-valued $\mathcal{F}$-measurable functions on $X$. Show that if $\{f_n\}$ converges to $f$ almost everywhere, then there are $\mathcal{F}$-measurable functions $g_1,g_2,\ldots$ that are equal to $f_1,f_2,\ldots$ almost everywhere and satisfy $f=\lim_{n}{g_n}$ everywhere.

Hi everyone, I have the above problem, How I can prove that there exits masurable functions $g_1,g_2,\ldots$ that are equal to $f_1,f_2,\ldots$ almost everywhere?? Can give me a Hint!! Thanks!

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Assume all is defined in the measurable space $(X, \mathscr{F})$. There is a set $N$ of measure zero such that outside of it the sequence $f_n$ converges to $f.$ Define $g_n = f_n \chi_{X - N} + f\chi_N.$ ($\chi _{X - N}$ is the characteristic function of $X - N$). So, $g_n$ converges everywhere to $f.$

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    $\begingroup$ +1 You meant "$f_n$ converges to $f$". Also, shouldn't you add the "correct" values of $f$ to the $g_n$? i.e. $g_n = f_n \chi_{X - N} + f\chi_N$ $\endgroup$ – Ant Apr 5 '17 at 21:35

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