Applications of Lebesgue integration to elementary analysis problems
In elementary analysis one might have come across the problem of showing
for . It turns out (Real and Complex Analysis, Rudin) that there is a more advanced proof using Lebesgue integration with respect to a counting measure!
Specifically, we can consider the counting measure where is the power set of , so that is the cardinality of if is finite, and otherwise it is (e.g. while ). Indeed the domain is so that in fact all functions are measurable.
It turns out that integration of such a function with respect to the counting measure is
That is, integrals with respect to the counting measure are just sums (if you are interested, see this document for a proof of this). A consequence (Real and Complex Analysis, Rudin) of Lebesgue’s Monotone Convergence Theorem is: If (are measurable) and
From what was noted earlier the integrals can be replaced with sums:
Consequently simply choosing , which is measurable because as noted earlier all functions are measurable with respect to the counting measure, does the trick! This completes the proof.