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Let $(X,B,\mu)$ be a measure space and let $E$ be a measurable set with $\mu(E)<\infty$.
Let $\{f_n\}$ be a sequence of measurable functions on $E$ such that each $f_n$ is finite almost everywhere in $E$ and $\{f_n\}$ converges almost everywhere in $E$ to a finite limit. Then for every $\epsilon>0$, there exists a subset $A$ of $E$ with $\mu(E-A)<\epsilon$ such that $\{f_n\}$ converges uniformly on $A$. |
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