On a finite-measure set, almost-everywhere convergence becomes uniform after discarding arbitrarily little measure.Suggest a slogan
Let \(m\) be Lebesgue measure on \(\RR^n\), let \(E\subseteq \RR^n\) be Lebesgue measurable with \(m(E) < \infty\), and let \(f_n\colon E\to\RR\) for \(n\geq1\) and \(f\colon E\to\RR\) be measurable with \(f_n \to f\) almost everywhere on \(E\). Then for every \(\varepsilon > 0\) there exists a closed set \(F\subseteq E\) such that \(m(E\setminus F)<\varepsilon\) and \(f_n\to f\) uniformly on \(F\).
State Egorov's theorem.