When I was introduced to measure theory, the professor chose to use the Choquet integral to obtain the Lebesgue integral. An this uses the "good old" Riemann integral to integrate the pseudo-inverse of the cumulative distribution function (I think, it was this book).
As a student I enjoyed this approach because I really knew what the Riemann integral was about and also I had an understanding of the problems with it - but I was really confused by the way we had had the Lebesgue integral at the first place.
