Why do people stick with Riemann-Integration when dealing with differential geometry? I asked a question yesterday that is, "Is there an introductory differential geometry text using Lebesgue integration?"
Then, i got an answer that "since we are dealing with differential geometry we don't need to introduce Lebesgue integration".
However, let's look how one develops integration in differential geometry.
One first defines Riemann integration on a closed interval. Then, one defines Riemann integration on a closed cube in $\mathbb{R}^n$. Then, one expand the domain of integrable functions to a larger class using partition of unity.
Now, let's do the same thing starting from Lebesgue integration. The process of defining Riemann integration on a closed cube via iterated Riemann integration on a closed interval is merely Fubini's theorem. Hence, it is now a theorem in the view of Lebesgue integration. I believe other definitions(Riemann approach) above can be viewed as theorems here (Lebesgue approach).
The above illustrates why i think it would be much great to study differential geometry via Lebesgue integration. Am I not making sense? Is there really no differential geometry text using Lebesgue integration?
 A: My understanding is that differential geometry deals mainly with smooth manifold, which is regular enough for Riemann-integration to handle. So why use 'general relativity' when 'Newtonian mechanics' is suffice?
A: As other people have said, is usual in differential geometry to assume the maps in the manifold to be "as differentiable as required", which is a phrase I consistently see in textbooks.
That being said, if you insist in a book where every integration is in Lebesgue sense, I would strongly recommend "Analysis, Manifolds and Physics" by Choquet-Bruhat, DeWitt-Morette and Dillard-Bleick. Don't be discouraged by the "physics" in the title, is just that they take some examples from general relativity, gauge theory or PDEs, but the books is pure math. Chapter IV on integration in manifolds is specifically with Lebesgue integration. 
From what you've mentioned to have studied you should be well prepared. Only the last chapters on infinite dimensional manifolds required some functional analysis. By the way they do a brief summary of measure theory and point out that only in infinite dimensional manifolds one must worry with those things.
