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: 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 X_n ≥ 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.