Fatou lemma
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.
Let f1, f2, f3, . . . be a sequence of non-negative measurable functions on a measure space (S,Σ,μ). Define the function f : S → [0, ∞] by
Then f is measurable and
Note: The functions are allowed to attain the value +∞ and the integrals may also be infinite.
Fatou's lemma may be proved directly as in the first proof presented below, which is an elaboration on the one that can be found in Royden (see the references). The second proof is shorter but uses the monotone convergence theorem – which is usually proved using Fatou's lemma and thus could create a circular argument.
We will prove something a bit weaker here. Namely, we will allow fn to converge μ-almost everywhere on a subset E of S. We seek to show that
Let
Then μ(E-K)=0 and
Thus, replacing E by K we may assume that fn converge to f pointwise on E. Next, by the definition of the Lebesgue Integral, it is enough to show that if φ is any non-negative simple function less than or equal to f, then
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