Uniform Convergence - Almost Uniform Convergence

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.