Fatou's lemma

Fatou's lemma is the inequality companion to the convergence theorems of Lebesgue integration, establishing that the integral of the limit inferior of a sequence of non-negative measurable functions is less than or equal to the limit inferior of the integrals. Unlike the monotone convergence theorem or the dominated convergence theorem, Fatou's lemma does not guarantee equality; it provides a one-sided bound that is often sufficient for proving finiteness, non-negativity, and lower semicontinuity of integral functionals. The lemma is deceptively simple — its proof requires only the definition of the Lebesgue integral and the preservation of inequalities under limits — yet it appears in virtually every existence proof in the calculus of variations, optimal control, and partial differential equations.

The asymmetry of Fatou's lemma — inequality rather than equality — is not a weakness but a feature. In many physical and economic settings, what matters is not the exact value of a limit but the guarantee that it does not vanish or diverge. Fatou's lemma provides this guarantee. It is the tool by which analysts prove that energy functionals have minima, that variational problems have solutions, and that physical systems do not spontaneously cease to exist. The lemma is named after Pierre Fatou, who proved it in 1906, but its true significance was not recognized until the development of the direct method in the calculus of variations in the mid-twentieth century.

Fatou's lemma is the mathematical expression of a pessimistic principle: things can only get worse in the limit. The integral of the limit is never greater than the limit of the integrals. This is not cynicism; it is topology. Lower semicontinuity — the preservation of inequalities under limits — is a topological property, and Fatou's lemma is its measure-theoretic manifestation. In optimization, in physics, in any domain where one seeks guarantees rather than exact values, Fatou's lemma is often the only tool one needs. Equality is overrated. Inequality is where the action is.