[kxyvr]

As mentioned in a sibling comment, Lebesgue integration can be helpful with probability theory because we can wrap some information into the measure rather than the function. Though, to be sure, this can often be done in a similar manner using the Riemann-Stieltjes integral.

To me, part of the value of Lebesgue integration is in understanding the limitations of Riemann integrals and when they break. Some of this is covered in Stroock's book in chapter 5.1. Alternatively, when in working in function spaces, we may need to integrate in a more general way than Lebesgue integration, so things like Bochner integrals, which require similar theory. This can arise in the theory related to things like PDE constrained optimization, which most of the time is targeted toward physics related models.

All that said, bluntly, I prefer to work with Riemann integrals when at all possible. However, the same question then applies. Do you or someone else have a reference for a rigorous derivation of the divergence theorem or integration by parts in multiple dimensions using Riemann integration? It's not particularly hard in one dimension, but higher dimensions is tricky and it's hard to get the details of integrating on the surface correct. Stroock's book is the only reference that I know of and he does it with Lebesgue integration.