Brezis–Lieb lemma

The lemma and its proof

Statement of the lemma

Let (X, μ) be a measure space and let f_n be a sequence of measurable complex-valued functions on X which converge almost everywhere to a function f. The limiting function f is automatically measurable. The Brezis–Lieb lemma asserts that if p is a positive number, then

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\int }_X||f{|}^p-|f_n{|}^p+|f-f_n{|}^p|d\mu =0,}$

provided that the sequence f_n is uniformly bounded in L^p(X, μ). A significant consequence, which sharpens Fatou's lemma as applied to the sequence |f_n|^p, is that

${\displaystyle {\int }_X|f{|}^{p}d\mu =\underset{n\to \mathrm{\infty }}{lim}\left({\int }_X|f_n{|}^{p}d\mu -{\int }_X|f-f_n{|}^{p}d\mu \right),}$

which follows by the triangle inequality. This consequence is often taken as the statement of the lemma, although it does not have a more direct proof.

Proof

The essence of the proof is in the inequalities

${\displaystyle \begin{array}{cc}{W}_n\equiv ||f_n{|}^p-|f{|}^p-|f-f_n{|}^p|& \le ||f_n{|}^p-|f-f_n{|}^p|+|f{|}^p\\ & \le \epsilon |f-f_n{|}^p+C_{\epsilon }|f{|}^p.\end{array}}$

The consequence is that W_n − ε|f − f_n|^p, which converges almost everywhere to zero, is bounded above by an integrable function, independently of n. The observation that

${\displaystyle W_n\le max(0,W_n-\epsilon |f-f_n{|}^p)+\epsilon |f-f_n{|}^p,}$

and the application of the dominated convergence theorem to the first term on the right-hand side shows that

${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}{\int }_X{W}_{n}d\mu \le \epsilon \underset{n}{sup}{\int }_X|f-f_n{|}^{p}d\mu .}$

The finiteness of the supremum on the right-hand side, with the arbitrariness of ε, shows that the left-hand side must be zero.