I'd like to know if there are examples of problems that don't "outwardly" seem to require the powerful tools of measure theory, but whose solutions nonetheless require (or, say, are very greatly simplified by) proper use of measure theory, and in particular Lebesgue integration.
An analogy might be the use of complex numbers in mathematics; we all know problems whose statements refer only to real numbers, but for which the easiest solution is to use complex numbers.
I realise that Lebesgue integration simplifies and/or enables and/or unifies many a theory; as such, the soundness of some solution methods relies on Lebesgue theory. But eschewing the grander underlying theory that establishes the soundness of the method, what about the solution itself? I haven't heard of "natural" situations where, in the midst of a computation, one crucially needs the Lebesgue integral at hand in order to carry forth. This is what I would be interested in seeing, and which is more similar in spirit to the complex number analogy above.
PS: I am aware of this post (How can using a different definition for the integral be useful?), but the main answer focuses on the theory-building prowess of Lebesgue integration, while I am interested in the problem-solving process instead.
