Definition:Uniform Convergence Almost Everywhere
Definition
Given a measure space $(X, \Sigma, \mu)$ and a sequence of $\Sigma $-measurable functions $f_n:D\to\R$ for $D\in\Sigma$, $f_n$ is said to converge uniformly almost everywhere (or converge uniformly a.e.) on $D\ $ if for each $\epsilon > 0$, there is a measurable subset $E_\epsilon \subseteq D$ such that $\mu(E_\epsilon) < \epsilon\ $ and $f_n$ converges uniformly to $f$ on $D - E_\epsilon$.
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.
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