Theoretical Advantages of Lebesgue Integration

Question

This wiki article opens with the two sentences

In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the $L_1$ norm. Its power and utility are two of the primary theoretical advantages (my boldface) of Lebesgue integration over Riemann integration.

I thought the term "theoretical advantages" was interesting. I only know enough about Lebesgue integration to be able to understand terms like "Lebesgue measurable" when they crop up, and always thought of Lebesgue integration as a way of making a wider class of functions integrable than would be for the Riemann integral alone.

Is there a generally agreed upon set of "theoretical advantages" like this for Lebesgue integration?

Answer 1

One theoretical advantage comes from the fact that Lebesgue integration makes certain function spaces complete.

For example, consider the real vector space $\mathcal{R}$ of real-valued Riemann integrable functions on $[0,1]$, with addition and scalar multiplication defined in the usual way. One can define an inner product and associated norm on $\mathcal{R}$ in the standard way via $$\langle f,g\rangle = \int_{0}^1 f(x)g(x)\,dx.$$

As it turns out, the space $\mathcal{R}$ is not complete (which is the essential obstacle for it being a Hilbert space), because there are Cauchy sequences of functions $f_1,f_2,\ldots\in\mathcal{R}$ which converge to a function that is not bounded and thus not Riemann integrable.

The Lebesgue integral gives us a way to form a completion of $\mathcal{R}$, the Lebesgue class $L^2([0,1])$. The introduction of this function space was convenient in a way similar to the way we introduced the real numbers as a completion for the rationals.

Source: Stein and Shakarchi, Volume I: Fourier analysis.

Also, Volume III by the same authors should have some nice sources.