Fatou's lemma

Fatou's lemma is a result in measure theory that provides an inequality relating the limit of a sequence of integrals to the integral of their pointwise limit. Specifically, it states that if you have a sequence of functions that are all non-negative, then the limit inferior of their integrals is less than or equal to the integral of the limit inferior of these functions. In simple terms, it helps us understand how the total "area under the curve" for a sequence of functions compares to the area under the limiting behavior of those functions, ensuring the limit isn't unexpectedly larger.