Along a similar line, the only book that I've found that has goes through a careful derivation of the divergence theorem using Lebesgue integration is, "A Concise Introduction to the Theory of Integration," Second Edition by Daniel W. Stroock. I prefer this one to his later book, "Essentials of Integration Theory for Analysis." Does anyone else have a good reference or book on the topic?
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 integr…
The more practical advantage measure theory provides for probability is you can simultaneously handle continuous and discrete distributions. Most of the time what works for one works for the other but you can get some weird mistakes (Shannon’s differential entropy has a few issues as a measure of information not found in the discrete case because he got lazy and just replaced the sum with an integral).
A good chunk of the time I come across measure theoretic probability papers I feel like they’re making the paper a lot more complicated and messier than it needs to be, but it does serve a purpose.