In the mathematical field of analysis, the BrezisâLieb lemma is a basic result in measure theory. It is named for HaĂŻm BrĂ©zis and Elliott Lieb, who discovered it in 1983. The lemma can be viewed as an improvement, in certain settings, of Fatou's lemma to an equality. As such, it has been useful for the study of many variational problems. [1]
Let (X, ÎŒ) be a measure space and let fn 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 BrezisâLieb lemma asserts that if p is a positive number, then
provided that the sequence fn is uniformly bounded in Lp(X, Ό). [2] A significant consequence, which sharpens Fatou's lemma as applied to the sequence |fn|p, is that
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
The consequence is that Wn â Δ|f â fn|p, which converges almost everywhere to zero, is bounded above by an integrable function, independently of n. The observation that
and the application of the dominated convergence theorem to the first term on the right-hand side shows that
The finiteness of the supremum on the right-hand side, with the arbitrariness of Δ, shows that the left-hand side must be zero.
Footnotes
Sources
In the mathematical field of analysis, the BrezisâLieb lemma is a basic result in measure theory. It is named for HaĂŻm BrĂ©zis and Elliott Lieb, who discovered it in 1983. The lemma can be viewed as an improvement, in certain settings, of Fatou's lemma to an equality. As such, it has been useful for the study of many variational problems. [1]
Let (X, ÎŒ) be a measure space and let fn 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 BrezisâLieb lemma asserts that if p is a positive number, then
provided that the sequence fn is uniformly bounded in Lp(X, Ό). [2] A significant consequence, which sharpens Fatou's lemma as applied to the sequence |fn|p, is that
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
The consequence is that Wn â Δ|f â fn|p, which converges almost everywhere to zero, is bounded above by an integrable function, independently of n. The observation that
and the application of the dominated convergence theorem to the first term on the right-hand side shows that
The finiteness of the supremum on the right-hand side, with the arbitrariness of Δ, shows that the left-hand side must be zero.
Footnotes
Sources