Main Page | See live article | Alphabetical index Fatou's lemma establishes an inequality relating the integral (in the sense of Lebesgue) of the limit inferior of a sequence of functions to the limit inferior of the sequence of integrals of the functions. It is named after the French mathematician Pierre Fatou (1878 - 1929). Fatou's lemma: If f1, f2, ... is a sequence of non-negative (measurable) functions, then Fatou's lemma is proved using the monotone convergence theorem. Applications Fatou's lemma is particularly useful in probability theory, in establishing results about the convergence of the expectations of the elements of a sequence of random variables. Suppose that the sequence of functions is a sequence of random variables, X1, X2, ..., with Xn ≥ Y (almost surely) for some Y such that E(|Y|) < ∞. Then by Fatou's lemma It is often useful to assume that Y is a constant. For example, taking Y = 0 it becomes clear that Fatou's lemma can be applied to any sequence of non-negative random variables. External links Fatou's lemma at PlanetMath, with a link to a proof. This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.