# 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*

*then*

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.