In the development of complex analysis you use Riemann integration and not Lebesgue integration to define line integrals. My questions are:
Are the theories developed the same? (i.e. does it not matter which integral you use in the development? Since all the functions usually involved are analytic or meromorphic can you use things such as analytic is equivalent to having a power series representation and uniform convergence within the radius of convergence to somehow show that the choice doesn't matter. I feel as if the function involved is analytic and has a finite radius of convergence this should be the case but I'm not so sure about what would happen if the function was meromorphic and/or has an infinite radius of convergence (Am I on the right track?))
If the theories developed are the same, does it become significantly easier to develop the theory with Riemann integration rather than Lebesgue integration.
If they are not the same, what are examples to show that show they are different?