Almost Uniform Convergence
From the conclusion of Egoroff’s Theorem we draw a new kind of convergence. We say that a sequence of a.e. finite-valued measurable functions converges to the measurable function “almost uniformly” if for every there is a measurable set with so that the sequence converges uniformly to on .
We have to caution ourselves that this is not almost everywhere uniform convergence, which would be a sequence that converges uniformly once we cut out a subset of measure zero. Our example sequence yesterday converges almost uniformly to the constant zero function, but it doesn’t converge almost everywhere uniformly. Maybe “nearly uniform” would be better, but “almost uniformly” has become standard.
Now we can restate Egoroff’s Theorem to say that on a set of finite measure, a.e. convergence implies almost uniform convergence. Conversely, if is a sequence of functions (on any measure space) that converges to almost uniformly, then it converges pointwise almost everywhere.
We start by picking a set for every positive integer so that , and so that converges uniformly to on . We set
and conclude that , since for all . If , then there must be some for which . Since converges to uniformly on , we conclude that converges to . Thus converges pointwise to except on the set of measure zero.