Today we prove Fatou’s Lemma, which is a precursor to the Fatou-Lebesgue theorem, and an important result in its own right.
If is a sequence of non-negative integrable functions then the function defined pointwise as
is also integrable, and we have the inequality
In fact, the lemma is often stated for a sequence of measurable functions and concludes that is measurable (along with the inequality), but we already know that the limit inferior of a sequence of measurable functions is measurable, and so the integrable case is the most interesting part for us.
So, we define the functions
so that each is integrable, each and the sequence is pointwise increasing. Monotonicity tells us that for each we have
and it follows that
We also know that
which means we can bring the monotone convergence theorem to bear. This tells us that
If it happened that were not integrable, then some of the would have to be only measurable — not integrable — themselves. And it couldn’t just be a finite number of them, or we could just drop them from the sequence. No, there would have to be an infinite subsequence of non-integrable , which would mean an infinite subsequence of their integrals would diverge to . Thus when we take the limit inferior of the integrals we get , as we do for the integral of itself, and the inequality still holds.