What is fatou lemma?

The Fatou lemma is named after French mathematician Pierre Fatou. It is a basic result in measure theory, which states that if a sequence of non-negative measurable functions $f_n$ converges almost everywhere to a measurable function $f$, then the integral of $f$ over the set on which it is defined is no larger than the lim inf of the integrals of the $f_n$'s over the same set.

More formally, if $(X,\mathcal{A},\mu)$ is a measure space, and $(f_n)$ is a sequence of non-negative measurable functions on $X$ that converges almost everywhere to a measurable function $f$, then:

$$\int_X f \ d\mu \leq \liminf_{n\to\infty} \int_X f_n \ d\mu.$$

In other words, the Fatou lemma states that the integral of the limit of a sequence of non-negative measurable functions is not greater than the limit of the integrals of the sequence of functions.

The Fatou lemma has many important applications in analysis, probability theory, and other areas of mathematics. For example, it is often used to prove other important theorems in measure theory, such as the dominated convergence theorem.