Fatou's lemma

For nonnegative measurable functions, the integral of the liminf is bounded by the liminf of the integrals.

Fatou’s lemma

Fatou’s lemma: Let $(X,\Sigma,\mu)$ be a measure space and let $(f_n)_{n\ge 1}$ be a sequence of nonnegative measurable functions $f_n:X\to[0,\infty]$ . Then

\int_X \Bigl(\liminf_{n\to\infty} f_n\Bigr)\,d\mu \;\le\; \liminf_{n\to\infty}\int_X f_n\,d\mu,

where $\liminf_{n\to\infty} f_n$ is taken pointwise (and may take the value $+\infty$ ).

Fatou’s lemma expresses a fundamental lower-semicontinuity property of the Lebesgue integral and is closely related to the monotone convergence theorem . It is a standard ingredient in the dominated convergence theorem and many other limit arguments in measure theory.