Brezis–Lieb 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

${\displaystyle \lim _{n\to \infty }\int _{X}{\Big |}|f|^{p}-|f_{n}|^{p}+|f-f_{n}|^{p}{\Big |}\,d\mu =0,}$

provided that the sequence f_n is uniformly bounded in L^p(X, μ).[2] A significant consequence, which sharpens Fatou's lemma as applied to the sequence |f_n|^p, is that

${\displaystyle \int _{X}|f|^{p}\,d\mu =\lim _{n\to \infty }\left(\int _{X}|f_{n}|^{p}\,d\mu -\int _{X}|f-f_{n}|^{p}\,d\mu \right),}$

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.[3]

The essence of the proof is in the inequalities

${\displaystyle {\begin{aligned}W_{n}\equiv {\Big |}|f_{n}|^{p}-|f|^{p}-|f-f_{n}|^{p}{\Big |}&\leq {\Big |}|f_{n}|^{p}-|f-f_{n}|^{p}{\Big |}+|f|^{p}\\&\leq \varepsilon |f-f_{n}|^{p}+C_{\varepsilon }|f|^{p}.\end{aligned}}}$

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

${\displaystyle W_{n}\leq \max {\Big (}0,W_{n}-\varepsilon |f-f_{n}|^{p}{\Big )}+\varepsilon |f-f_{n}|^{p},}$

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

${\displaystyle \limsup _{n\to \infty }\int _{X}W_{n}\,d\mu \leq \varepsilon \sup _{n}\int _{X}|f-f_{n}|^{p}\,d\mu .}$

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