Fatou's lemma
/ fah-TOO /
Sometimes you cannot swap a limit and an expectation exactly, but you still want to say something. Fatou's lemma is the safety-net inequality for exactly that situation: even when the limit of expectations does not equal the expectation of the limit, Fatou guarantees the expectation of the limit can never overshoot — it can only be the same or smaller. It is the one-directional control you can always count on for non-negative variables, with no extra assumptions whatsoever.
Precisely, for any sequence of non-negative random variables X1, X2, X3, ... (no monotonicity, no domination required), E[liminf of Xn] <= liminf of E[Xn]. The liminf is the eventual floor of a sequence — the largest value the terms stay above in the long run — so the lemma says the expectation of the eventual-floor variable is at most the eventual floor of the expectations. The intuition is that probability mass can leak away to infinity in the limit (lowering the left side) but it can never be created out of nowhere, so the limiting expectation cannot exceed what the sequence's expectations were already heading toward.
Why keep a mere inequality around? Because it needs essentially no hypotheses, so it applies when the cleaner theorems do not, and it is the standard tool for proving lower bounds and establishing that limits exist and are finite. It is also the engine that proves the dominated convergence theorem: apply Fatou to the dominated sequence and to its reflection, and the two inequalities pinch the limit into an equality. A concrete picture of strict loss: let Xn be a tall spike of height n on an interval of width 1/n, so E[Xn] = 1 always, yet Xn -> 0 pointwise and E[lim Xn] = 0 < 1 — Fatou holds (0 <= 1), and the gap is exactly mass escaping to infinity.
Let Xn equal n on the interval (0, 1/n] and 0 elsewhere, under the uniform measure on (0,1]. Then E[Xn] = n times (1/n) = 1 for every n, but Xn -> 0 at every point, so E[lim Xn] = 0. Fatou's lemma is satisfied: 0 = E[liminf Xn] <= liminf E[Xn] = 1.
The limiting expectation can fall short but never overshoot — mass may escape, never appear from nowhere.
Fatou only goes one way (<=), and it requires non-negativity; for sequences that change sign or where mass could blow up, the inequality can reverse or fail, so you cannot use it to bound expectations from above.