In mathematics, Fatou’s lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions.
If \(X_{n}\geq 0\) then \[ \liminf_{n \to \infty}\mathbb{E}[X_{n}] \geq \mathbb{E}\left[\liminf_{n \to \infty}X_{n}\right]. \]