Fatou's lemma

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

$\int \liminf_{n\rightarrow\infty} f_n \leq \liminf_{n\rightarrow\infty} \int f_n.$

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 X_n ≥ Y (almost surely) for some Y such that E(|Y|) < ∞. Then by Fatou's lemma

$\operatorname{E}\left( \liminf_{n\rightarrow\infty} X_n \right) \leq \liminf_{n\rightarrow\infty} \operatorname{E}(X_n).$

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

Includes a link to a proof.

Categories: Real analysis