Let $(X,\mathcal{S},\mu)$ be a measure space, and let $E$ be a subset of $X$ of finite measure. If $f_{n}$ is a sequence of measurable functions converging to $f$ almost everywhere, then for each $\delta>0$ there exists a set $E_{\delta}$ such that $\mu(E_{\delta})<\delta$ and $f_{n}\rightarrow f$ uniformly on $E-E_{\delta}$.