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Let $(X, \mathbb A, m)$ be a measurable space and let $\{f_n : X \to \mathbb R\}_{n \in \mathbb N}$ be a sequence of Borel measurable functions. If such sequence converges $m$-almost everywhere to some Borel measurable $f: X \to \mathbb R$, I have to prove that for any $\epsilon > 0$ there exists $A \in \mathbb A$ with $m(A) < \epsilon$ and such that $$\sup_{x \in X \setminus A} |f_n(x) - f(x) | \to 0$$ as $n \to + \infty$.

My question is simple: Isn't this just Egorov’s Theorem?

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Yes, it is and additionally you need a finite measure space. It is wrong for not-finite measures: For example take $f_n = 1_{[n,n+1]}$, then $f_n \rightarrow 0$ pointwise, but if $\lambda(A) < \varepsilon < 1$, then we must have $\lambda([n,n+1] \setminus A ) >0$. Thus $$\sup_{x \in \mathbb{R} \setminus A} |f(x) - 0| =1.$$

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