Fatou's lemma

Fatou's lemma is the safety net you get when functions converge but might misbehave: even with no control on the sequence, area cannot be created out of nothing in the limit, only lost. The integral of the eventual lower envelope is at most the smallest the integrals settle down to. It is a one-sided inequality, and that one-sidedness is exactly right — mass can escape to infinity but cannot materialize.

Statement: for any sequence of non-negative measurable functions f_n, the integral of the liminf of the f_n is less than or equal to the liminf of the integrals of the f_n. No monotonicity, no domination, no convergence is required — only non-negativity. When the f_n actually converge pointwise to f, the left side is just the integral of f, giving the integral of f less than or equal to liminf of the integrals.

The inequality can be strict, and that is the whole point. For the moving bump f_n = indicator of [n, n+1], the liminf function is 0 (integral 0) while each integral is 1 (liminf 1): 0 is strictly less than 1, mass having marched off to infinity. Fatou is the lightweight tool from which the dominated convergence theorem is proved.