What we did in this topic was just an version of the Brezis-Lieb lemma. In this topic, we will discuss the generalization of this lemma.
Roughly speaking, what we are going to prove is the following: If is a continuous function such that , then, when a.e. and
we claim that
under suitable conditions on and/or .
To be exact, in addition let satisfy the following hypothesis:
For every sufficiently small , there exists two continuous, nonnegative functions and such that
for all .
Theorem. Let satisfy the above hypothesis and let be a sequence of measurable functions from to such that
- for some constant , independent of and .
- for all .
Then, as ,
Proof. Fix and let
where . As , a.e. On the other hand,
Therefore, . By the Lebesgue Dominated Convergence theorem, as . However,
Consequently, . Now let .
- The simplest example is when we choose where . In this situation, one has
- We now assume in . As a consequence and up to a subsequence, in for every and a.e. Therefore, for a fixed , the fact that in implies, by the Brezis-Lieb lemma, that
This is because is bounded, a.e. and
- The fact that strongly in implies that . Therefore,
- As a consequence, one has the following result