There's a general result which says any metric space can be embedded within a complete metric space(or in other words, the metric space can be completed).
If one completes the space of L1 Riemann integrable functions, how does this differ from the lebesgue integral?
Certainly the metric completion argument is very simple, skips measure theory altogether and gets nice results like MCT and DCT.
What are the reasons to prefer to measure-theoretic construction?