= Egorov's theorem
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= Egorov theorem
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On a finite-measure measurable <set>, a <sequence> of <measurable functions> converging pointwise almost everywhere converges uniformly outside a <set> of arbitrarily small measure. For tolerance $1/k$, the <sets> on which every index after $N$ has error at most $1/k$ increase to full measure as $N\to\infty$. Choose $N_k$ with exceptional measure at most $\varepsilon2^{-k-1}$ and intersect the good <sets>. <Countable subadditivity> bounds the discarded measure, and the resulting tail estimates prove <uniform convergence>. Finite measure is essential: $\mathbf1_{[n,\infty)}$ on $[0,\infty)$ is a counterexample.
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