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

- a.e.
- .
- 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,

and thus

Consequently, . Now let .

**Applications**.

- 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
in .

This is because is bounded, a.e. and

.

- The fact that strongly in implies that . Therefore,

.

- As a consequence, one has the following result

.