Your solution seems good (I only skimmed it though), just notice that the approach is very similar to when you try to calculate the lenght of a curve in some .

For the construction of the spaces there are several equivalent approaches (all of which require a knowledge of the Lebesgue integral): There's the measure theoretic approach, the metric space completion of certain spaces under the norm, etc. If you want an analytic approach Jöst's "Postmodern analysis" is a good option, if you're interested in measure theory Halmos' "Measure theory" and Bartle's "Elements of integration and Lebesgue measure" are classics.