As I understand it, in "Introduction to smooth manifolds" Lee only defines the integral of a differential form for those who have compact support. Is this because he fundamentally uses the Riemann-integral to integrate the underlying coordinate function? He constructs the integral in the following way:
He first defines the integral of a form over a "domain of integration" in $\mathbb{R^n}$ which is a bounded subset of which the boundary is a set of Lebesgue measure zero. I guess this again is due to using the $n$-dimensional Riemann integral. Then by using charts he defines integration of compactly supported forms on a manifold.
Couldn't you just define the integral over an arbitrary measurable subset of $\mathbb{R^n}$ and then "pull the $\sigma$-algebra up to the manifold" and proceed likewise to define the integral over a measurable subset of a manifold? This then would have the benefit of not only being able to integrate compactly supported differential forms. Or is there a different reason the "domain of integration" is needed?
Best Answer
Yes, you certainly can define a natural $\sigma$-algebra on a manifold and then define integrals of differential forms over measurable subsets. In fact, this generalization is essential in many applications of geometric analysis. But I wanted my book to be accessible to first-year graduate students, and many universities don't cover Lebesgue integration in undergraduate courses.