If the domain of the functions is a measure space then the related notion of almost uniform convergence can be defined. We say a sequence of functions converges almost uniformly on E if there is a measurable subset F of E with arbitrarily small measure such that the sequence converges uniformly on the complement E \ F.
Note that almost uniform convergence of a sequence does not mean that the sequence converges uniformly almost everywhere as might be inferred from the name.
Egorov's theorem guarantees that on a finite measure space, a sequence of functions that converges almost everywhere also converges almost uniformly on the same set.
Almost uniform convergence implies almost everywhere convergence and convergence in measure.
Read more about this topic: Uniform Convergence
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