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Let $(X,M,\mu)$ be a finite or $\sigma$-finite measure space. Let $\{f_n\}$ be a sequence of finite a.e., measurable functions such that $f_n \to f$ [a.e.]. Then there is a partition of $X$ into a sequence $E_0,\,E_1,\,E_2,\,\dots$ of disjoint measurable sets such that $\mu(E_0) = 0$ and $f_n \to f$ uniformly on each $E_i$, $i \geqslant 1$.

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    By the way, welcome to math.stackexchange.2012-11-12

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