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Concept
Fatou's lemma
Summary
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.
Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem.
In what follows, denotes the -algebra of Borel sets on .
Fatou's lemma. Given a measure space and a set let be a sequence of -measurable non-negative functions . Define the function by setting for every .
Then is -measurable, and also , where the integrals may be infinite.
Fatou's lemma remains true if its assumptions hold -almost everywhere. In other words, it is enough that there is a null set such that the values are non-negative for every To see this, note that the integrals appearing in Fatou's lemma are unchanged if we change each function on .
Fatou's lemma does not require the monotone convergence theorem, but the latter can be used to provide a quick proof. A proof directly from the definitions of integrals is given further below.
In each case, the proof begins by analyzing the properties of . These satisfy:
the sequence is pointwise non-decreasing at any x and
.
Since , we immediately see that f is measurable.
Moreover,
By the Monotone Convergence Theorem and property (1), the limit and integral may be interchanged:
where the last step used property (2).
To demonstrate that the monotone convergence theorem is not "hidden", the proof below does not use any properties of Lebesgue integral except those established here.
Denote by the set of simple -measurable functions such that on .
If everywhere on then
If and then
If f is nonnegative and , where is a non-decreasing chain of -measurable sets, then
- Since we have By definition of Lebesgue integral and the properties of supremum,
- Let be the indicator function of the set It can be deduced from the definition of Lebesgue integral that if we notice that, for every outside of Combined with the previous property, the inequality implies
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