Editing Uniform convergence (section)

==Almost uniform convergence==
If the domain of the functions is a [[measure space]] ''E'' then the related notion of '''almost uniform convergence''' can be defined. We say a sequence of functions <math>(f_n)</math> converges almost uniformly on ''E'' if for every <math>\delta > 0</math> there exists a measurable set <math>E_\delta</math> with measure less than <math>\delta</math> such that  the sequence of functions <math>(f_n)</math> converges uniformly on <math>E \setminus E_\delta</math>. In other words, almost uniform convergence means there are sets of arbitrarily small measure for which the sequence of functions converges uniformly on their complement.

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. However, [[Egorov's theorem]] does guarantee that on a finite measure space, a sequence of functions that converges [[Pointwise convergence#Almost everywhere convergence|almost everywhere]] also converges almost uniformly on the same set.

Almost uniform convergence implies [[almost everywhere convergence]] and [[convergence in measure]].