The Monotone Convergence Theorem
We want to prove a strengthening of the dominated convergence theorem. If is an a.e. increasing sequence of extended real-valued, non-negative, measurable functions, and if converges to pointwise a.e., then
If is integrable, then dominates the sequence , and so the dominated convergence theorem itself gives us the result we assert. What we have to show is that if , then the limit diverges to infinity. Or, contrapositively, if the limit doesn’t diverge then must be integrable.
But this is the limit of a sequence of real numbers, and so if it converges then it’s Cauchy. That is, we can conclude that
Our assumption that is a.e. increasing tells us that for any fixed and , the difference is either a.e. non-negative or a.e. non-positive. That is,
And thus the sequence is mean Cauchy, and thus mean convergent to some integrable function , which must be equal to almost everywhere.
One nice use of this is when talking about series of functions. If is a sequence of integrable functions so that
then I say that the series
converges a.e. to an integrable function , and further that
To see this, we let be the partial sum
which gives us a pointwise increasing sequence of non-negative measurable functions. The monotone convergence theorem tells us that these partial sums converge pointwise to some and that
But this is exactly the sum we assumed to converge before. Thus the function is integrable and the series of the is absolutely convergent. That is, since must be a.e. finite, the series
is absolutely convergent for almost all , and so it must be convergent pointwise almost everywhere. Since dominates the partial sums
the bounded convergence theorem tells us that limits commute with integrations here, and thus that