TITLE: How do people apply the Lebesgue integration theory?
QUESTION [68 upvotes]: This question has puzzled me for a long time. It may be too vague to ask here. I hope I can narrow down the question well so that one can offer some ideas. 
In a lot of calculus textbooks, there is usually a chapter about "applications" after the one for Riemann integral. Students can do a lot of calculations and appreciate the power of Riemann integration --- they solve many problems in physics and geometry. 
While learning the Lebesgue integral, or more generally, integration on measure space, I cannot appreciate the power of this kind of integration util I learn some modern PDE. One the other hand, I found that there are much much more inequalities when doing Lebesgue integration than equations  when applying Riemann integral. Instead of calculating something, people do estimation with the convergence theorems.
Here are my questions:

How do people apply Lebesgue integration theory? If putting the methods into categories, can I say that it primarily deals with the problems related to convergence?
What's the fundamental difference between applying these two different integration techniques? 
Is there an example such that people solve some problem which may be very hard (but still can be solved) when using Riemann integral but relatively easy with Lebesgue integral?

REPLY [3 votes]: Lebesgue Integral is used for the Birkhoff Ergodic Theorem.  From these notes

Let $(X, \mathcal{B}, \mu)$ be a probablity space and let $T: X \to X$ be an ergodic measure-preserving transformation.  Let $f \in L^1 (X, \mathcal{B}, \mu)$ then
  $$ \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^{n-1} (f \circ T^k)(x) = \int f \, d\mu $$
  For $x$ almost everywhere in $X$.  

That's Lebesgue not Riemann.  That convergence is not guaranteed to occur at any particular point.