Who first realized that it is possible to define the integral of a function as the limit of the integrals of a sequence of step functions that converge uniformly to the given function?
This is normally how the Lebesgue integral is taught now (although the uniform convergence is avoided by Egorov's theorem), but Lebesgue defined the integral in an analogous way to how Cauchy, Riemann and Darboux did (although he sort of flipped everyting horizontally).