Well, one avenue you can take is that the Riemann integral exists. If a function is Riemann integrable, it is Lebesgue integrable.
I am trying to prove that exists as a Lebesgue integral. I've tried evaluating the integral a couple ways, and if I kept pursuing these routes I might find an answer, but these methods are getting a little tortuous and I suspect this is not the best or intended answer. I have all of Levi's Theorems and the Lebesgue Dominated Convergence Theorem, and I sort of suspect that I'm supposed to derive the result from these or something like them, but I'm having a hard time seeing how that would go.