Also when you talk about the divergence theorem using Lebesuge integration, you're tending into the territory of Geometric Measure Theory which is very deep and im not sure why anyone would do this unless they were a working mathematician and had a practical need for it, like say they were studying geometric analysis (which only a small group of people in the world do).
This is an amazing book and I'm so suprised to see someone else knows about it. However
, who cares about the Lebesgue integral? The only thing its good for is integrating pathological functions like the rationals, the indicator of the Cantor set, and fractals. Riemann integration is just fine and I'm not sure what all the fuss is about the Lebesgue Integral. Sure expectation of a random variable is a Lebesgue integral, but most of the time you have a density anyways and you can use the Riemann Integral.