Different integrals (Riemann, Lebesgue) – are they equivalent? If yes, what benefits are there to studying them? If not, where do they differ, how? (And what in the complex realm)

Question

inb4 “not eli5-able”

As the rules say, the replies should be aimed at laymen, not five year olds. I let you assume the laymen reading this have at least had a calculus course (so you don’t have to explain how integral is are under the curve), maybe even heard of some basic real analysis like the epsilon-delta definition, but whose knowledge in any of this is rudimentary.

Roughly these questions which I think are mostly connected:

1. Why do we have these different definitions of integrals in the
first place

2. When are they not equivalent?

3. Why would anyone care about them, study them?

4. Assuming these answers are mostly concerned with R, in the complex world does anything change?

In: 4

Answer 1

Lesbegue integral can be seen as an extension of the Reimann integral. Whenever you can use Reimann integral you can also use the Lesbegue integral, and the results would be the same. We use Reimann in these cases because it’s simpler. However, the Lesbegue integral can also be used in cases where the Reimann integral can’t be applied.