Examples of theorems from advanced calculus that are easier to prove with Lebesuge integral. I am looking for interesting examples of theorems in advanced calculus (at the level of Rudin's "Principles of Mathematical Analysis") that are easier to prove when using Lebesgue integration, or measure theory in general.
I want to see the power of Lebesgue integral other than being able to integrate weird functions.
I would be very grateful for any comment.
Thank you.
 A: [I hesitate to jump into this one, but it is a legitimate question and can be discussed ad nauseum.]
An analogy, which is actually not too far-fetched, is to consider what answer you would give
to a calculus student who wanted to develop the theory of infinite series but only in the context
of rational numbers (not including any "weird trancendentals").
You might say, Ok define $\sum_{k=1}^\infty a_k =r$  if all these numbers are rational and for
every integer $p$ there is an integer $q$ with
$$
n\geq q \implies 
\left|   \sum_{k=1}^n a_k -r\right|< \frac1p.$$ 
But you would quickly realize that the theory is pretty messy. Every theorem about convergent series has some lousy hypothesis added in about rational numbers.  

A series of positive terms $\sum_{k=1}^\infty a_k$ is convergent if
  and only if there is a rational number $r$ such that $\sum_{k=1}^n a_k
  < r$ for all $n$ and, moreover, for every  rational $s<r$, there
  exists an integer $q$ so that $\sum_{k=1}^q a_k > s$.

So a pretty pathetic theory.  No Cauchy criterion, comparison tests, ratio and root test, etc.
The situation is the same for the Riemann integral.  That theory is really quite terrible in spite of the fact that it dominated the late 19th century and still dominates modern freshman calculus classes.
Here is a typical theorem:

Let $f_n$ be a sequence of integrable functions on $[a,b]$ and  [add
  hypotheses here].  Then $$ \int_a^b \sum_{k=1}^\infty f_n(x)\,dx = 
  \sum_{k=1}^\infty \int_a^b f_n(x)\,dx .$$

Any legitimate theorem for the Riemann integral will have the extra hypothesis that $f(x)=\sum_{k=1}^\infty f_n(x)$ is Riemann integrable. [Just like the added hypothesis about rational numbers above.]
In the Lebesgue theory you can deduce integrability from the hypotheses.  Here it is an added cumbersome addition.
You might say "But I promise to integrate only nice functions--no weird ones".  That means, though, that at every step of some application you must pause to prove Riemann integrability.  But you don't normally know about the function $f$ here any more than that it is given by this sum.  Even if it is Riemann integrable you are going to have to find some way to prove it or you cannot use this theorem.  If your application has many limit steps, each one is bogged down with an extra 
obligation to check Riemann integrability.
But, of course, the "promise to integrate only nice functions--no weird ones" itself is rather naive.  Many completely natural processes (differentiation, integration, sequence limits, series, etc.) lead easily to functions that are not Riemann integrable.  That theory is completely crippled.
Lebesgue gave as his  motivation for extending the Riemann integral the problem of integrating derivatives.  Volterra had produced a "weird" derivative that was bounded but not Riemann integrable.  You might say "So what?"  The answer is that we have an entirely natural process of analysis (differentiation)
that leads outside the world of Riemann integration.  That alone is a serious indication that the theory is severely limited.  A further century of study shows that the Riemann theory is quite useless except, perhaps, as a teaching tool.
