Let be a measure space and let be a sequence of complex valued measurable functions which are uniformly bounded in for some . Suppose that pointwise almost everywhere (a. e.). What can be said about ?
The simplest tool for estimating is Fatou’s lemma, which yields
The purpose of this note is to point out that much more can be said, namely
More generally, if is a continuous function such that , then, when a.e. and
it follows that
under suitable conditions on and/or .
Statement. The case (): Suppose a.e. and for all and for some . Then the following limit exists and the equality holds
(i) By Fatou’s lemma, .
(ii) In case , and if we assume that , then we do not need the hypothesis that is uniformly bounded. [This follows from the inequality
and the dominated convergence theorem.] However, when , the hypothesis that is uniformly bounded is really necessary (even if we assume that ) as a simple counterexample shows.
(iii) When , the hypotheses imply that weakly in . [By the Banach-Alaoglu theorem, for some subsequence,, converges weakly to some ; but since a.e.] However, weak convergence in is insufficient to conclude that
holds, except in the case . When it is easy to construct counterexamples, that is
under the assumption only of weak convergence. When the proof of
is trivial under the assumption of weak convergence. Indeed,
Since in , then (note that the dual space of is itself). Thus