So I've been trying to generalize a theorem that I had seen a while ago. I'm having a bit of trouble proving the more exotic cases. The generalization, as it's phrased now, is:

"Let {f_n: E -> R} be a sequence of functions which converges uniformly to f on E. If each f_n is integrable on E, then f is integrable on E, and
int_E f = lim_{n->infty} int_E f_n."

I know there's probably a measure-theory theorem that looks exactly like this, but I know around 0 measure theory. Obviously, this generalization isn't hard to prove when E is compact. The cases I'm having trouble with are when E is not Jordan-measurable (the boundary has nonzero measure), when the volume of E is infinite, and when E is not compact. Obviously, all of these cases are related, and (at least to me) they all seem non-arbitrary. Any suggestions?

Also, it could turn out that this generalization is too general. However, I don't think that's the case.

[Edit]: Looking back, I think that E having finite volume is the only other requirement. And, since it's not clear, I'm talking solely about Riemann integration.