A nice application of Fatou’s Lemma
Let me show you an exciting technique to prove some convergence statements using exclusively functional inequalities and Fatou’s Lemma. The following are two classic problems solved this way. Enjoy!
Exercise 1 Let be a measurable space and suppose is a sequence of measurable functions in that converge almost everywhere to a function and such that the sequence of norms converges to . Prove that the sequence of integrals converges to the integral for every measurable set .
Proof: Note first that
Set then (which are non-negative functions) and apply Fatou’s Lemma to that sequence. We have then
It must then be . But this proves the statement, since
Exercise 2 Let be a finite measure space and let . Suppose that is a sequence of measurable functions in whose norms are uniformly bounded in and which converge almost everywhere to a function . Prove that the sequence converges to for all where is the conjugate exponent of .
Proof: The proof is very similar to the previous problem. We start by noticing that under the hypotheses of the problem,
If we prove that , we are done.
We will achieve this by using the convexity of , since in that case it is
Set then (which are non-negative functions) and apply Fatou’s Lemma as before.