How do people apply the Lebesgue integration theory?

For the second question, you have that whenever the Riemann integral is defined, so is the Lebesgue integral. The only thing I'm aware of that you have for the Riemann integral that you don't for the Lebesgue integral is the fundamental theorem of calculus. This is more because the functions that Riemann-integbrale functions are nice, and possess more features than $L^1$ functions.

In some sense, what Riemann integration lacks is not the inability to integrate functions that you care about (because most of the functions that people are concerned with are at least piece-wise continuous), but rather that it is easy to take the limit of continuous functions and get discontinuities, and so you lack a lot of nice formal properties becasue you cannot work by going from nice functions and passing to limits. Therefore, if something is true for a function, you must be able to show it directly. For Lebesgue integration, the fact that you can take limits means you get completeness of the $L^p$ spaces, and thus that you can bring Hilbert/Banach space techniques to bear.

Here are a few applications:

• The use of function spaces ($L^p$ spaces, sobelev spaces, etc). As you mentioned, weak derivatives allow you to solve PDE's in a two step process: show that a weak solution exists, and then prove some sort of regularity result that the weak solution is actually a real function. However, there are many many more uses of these function spaces, and they allow you to do some pretty neat things.

• Fourier transforms. There are a number of results on convergence of Fourier transforms, and function spaces are necessary to prove them. It has been said that putting Fourier transforms on solid footing was the first major triumph of Lebesgue integration

• Probability. The modern point of view of probability is based on measure theory: a probability space is just a measure space of measure $1$, and an "event" is just an element of a sigma algebra. Even for a simple problem like flipping a coin an infinite number of times, this allows you to make sense of the fact that some "events" just aren't talked about, because they aren't part of the sigma algebra, which lets you escape seeming paradoxes related to infinity. Once you have recast the definitions or probability, you can then phrase many results in terms of integration, where their proofs become rather easy, at least compared to the traditional proofs. It also allows for discussing more complicated stochastic processes, like stochastic differential equations, and I honestly don't even know if there are fully satisfactory ways to talk about these things without the measure theoretic point of view.

The Riemann integral depends heavily on the structure of the line, where the general process of integration does not. Because of the way the Riemann integral is constructed, continuity of integrands is very important. We now have two abstractions "siamese twinned": integration and topology. The Lebesgue integral is an excellent abstraction that gives a minimal structure in which to do integration. This is cleaner and better.

Here's an example which shows the power of Lebesgue integration theory.

The following theorem is sometimes useful in calculus(only continuous functions are involved). It is an easy special case of Lebesgue dominated convergence theorem. However, it's difficult to prove within Riemann integration theory.

Let $M$ be a real number such that $0 \le M < \infty$. Let $(f_n)$ be a sequence of continuous functions on a finite interval $[a, b]$ such that $|f_n| \le M$ for every $n$. Let $f(x)$ be a continuous function on $[a, b]$. Suppose $\lim_{n\rightarrow\infty} f_n(x) = f(x)$ for every $x \in [a, b]$.

Then $\lim_{n\rightarrow\infty} \int_{a}^{b} f_n(x) dx = \int_{a}^{b} f(x) dx$.