[verylongname]

But the only obstruction to developing convergence theorems for Riemann integrals is that the limit of a sequence of Riemann integrable functions need not be Riemann integrable. This, of course, follows from the standard convergence theorems for Lebesgue integrals and the fact that the two notions coincide where both are defined. So it really does come down to the class of functions which are integrable.

[WallWextra]

Of course. I'm saying the important part is not that the class is bigger, the important part is that it's a Banach space.