Lebesgue Measure & Integration - Frank Burk : Summary
Lebesgue Integration is usually introduced in either one of the two ways , in most of the books. Either using measure theory OR skipping measure theory altogether and using approximating functions like step functions, monotone functions etc. The second approach is usually taken so that one can actually understand all the concepts of Lebesgue integral with out going through measure theory. However that approach would make it difficult for a reader to connect Lebesgue Integral and concepts relating to stochastic processes.