Lebesgue integral vs Riemann integral

Advantages over Riemann:

Lebesgue integral of $L^{p}$ is complete but Riemann isn't.
youtu.be/PGPZ0P1PJfw?t=710 you are able to switch the order of integrals and limits of function sequences on non-uniform convergence. TODO why do we care? This is linked to the Fourier series of course, but concrete example?

Video 1. Riemann integral vs. Lebesgue integral by The Bright Side Of Mathematics (2018) Source.

youtube.com/watch?v=PGPZ0P1PJfw&t=808 shows how Lebesgue can be visualized as a partition of the function range instead of domain, and then you just have to be able to measure the size of pre-images.

One advantage of that is that the range is always one dimensional.

But the main advantage is that having infinitely many discontinuities does not matter.

Infinitely many discontinuities can make the Riemann partitioning diverge.

But in Lebesgue, you are instead measuring the size of preimage, and to fit infinitely many discontinuities in a finite domain, the size of this preimage is going to be zero.

So then the question becomes more of "how to define the measure of a subset of the domain".

Which is why we then fall into measure theory!

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Lebesgue integral