Many fields in mathematics start from the "dirty" approach. In calculus we do all sort of $\epsilon$-$\delta$ stuffs, until topology gives an elegant formulation using open sets. A first course in linear algebra usually starts with defining matrices, their multiplications, determinants, etc, then the whole theory is built upon that. But if we start from vector spaces and linear operators on them, everything seems to fit in much more cleanly.
What about integrals? There are at least two approaches to define Riemann integrals: upper and lower integrals; or Riemann sums of partitions. None of them looks elegant to me. So, is there a "clean" way to develop the theory of integrals?