Fatou Lemma

1 Fatou Lemma

In mathematics, the Fatou 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 \[ \liminf_{n \to \infty}\mathbb{E}[X_{n}] \geq \mathbb{E}\left[\liminf_{n \to \infty}X_{n}\right]. \]

Proof: Define \[ Y_k(\omega) := \inf_{n\ge k} X_n(\omega), \qquad k \ge 1. \] Then each \(Y_k\) is measurable and nonnegative, and \[ Y_k \uparrow \liminf_{n\to\infty} X_n \quad \text{pointwise as } k\to\infty. \] By the Monotone Convergence Theorem, \[ \mathbb E\!\left[\liminf_{n\to\infty} X_n\right] = \mathbb E\!\left[\lim_{k\to\infty} Y_k\right] = \lim_{k\to\infty} \mathbb E[Y_k]. \] For each \(k\), since \(Y_k \le X_n\) for all \(n \ge k\), \[ \mathbb E[Y_k] \le \inf_{n\ge k} \mathbb E[X_n]. \] Taking limits in \(k\), \[ \lim_{k\to\infty} \mathbb E[Y_k] \le \lim_{k\to\infty} \inf_{n\ge k} \mathbb E[X_n] = \liminf_{n\to\infty} \mathbb E[X_n]. \] This proves the result. \(\square\)