When I was introduced to measure theory, the professor chose to use the [Choquet integral][1] 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][2]).

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.


  [1]: http://en.wikipedia.org/wiki/Choquet_integral
  [2]: http://books.google.com/books?id=4m9jGLwntswC&printsec=frontcover&dq=non-additive+measure+and+integral&source=bl&ots=12EfwwxyvN&sig=i7J0LOBQRUijWdNEQehBAqpOgNE&hl=en&ei=pdo5Tar5F4fpOdyc4Z0L&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCkQ6AEwAg#v=onepage&q&f=false