Fatou's lemma
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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 states that if f1, f2, ... is a sequence of non-negative (measurable) functions, then
- <math>\int \liminf_{n\rightarrow\infty} f_n \leq \liminf_{n\rightarrow\infty} \int f_n.<math>
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
- <math>\operatorname{E}\left( \liminf_{n\rightarrow\infty} X_n \right) \leq \liminf_{n\rightarrow\infty} \operatorname{E}(X_n).<math>
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
- Template:Planetmath reference Includes a link to a proof.