Integral of differential forms in Lee's Smooth Manifolds

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?

• Can you isolate the question from the set up? Mar 24, 2020 at 13:01
• @PraphullaKoushik You're right, the question is not very readable. I tried to clean it up a bit. Mar 24, 2020 at 13:06

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.