Definition:Almost Uniform Convergence

Definition
Let $\left({X, \Sigma, \mu}\right)$ be a measure space, and let $D \in \Sigma$.

Let $\left({f_n}\right)_{n \in \N}, f_n: D \to \R$ be a sequence of $\Sigma$-measurable functions.

Then $\left({f_n}\right)_{n \in \N}$ is said to converge uniformly almost everywhere (or converge uniformly a.e.) on $D$ iff:


 * For all $\epsilon > 0$, there is a measurable subset $E_\epsilon \subseteq D$ of $D$ such that:


 * $(1): \quad \mu \left({E_\epsilon}\right) < \epsilon$;
 * $(2): \quad \left({f_n}\right)_{n \in \N}$ converges uniformly to $f$ on $D \setminus E_\epsilon$.

Relations to Other Modes of Convergence
Uniform convergence a.e. is weaker than uniform convergence.

Uniform convergence a.e. implies convergence a.e. (proof here). A partial converse to this result is given by Egorov's Theorem.

Uniform convergence a.e. also implies convergence in measure.