Integrating on Compact Manifolds

[AnalysisQuest]

Homework Statement

This problem is in Analysis on Manifolds by Munkres in section 25. R means the reals
Suppose M \subset R^m and N \subset R^n be compact manifolds and let f: M \rightarrow R and g: N \rightarrow R be continuous functions.

Show that \int_{M \times N} fg = [\int_M f] [ \int_N g ]

The hint states to consider the case where supports of f and g are contained in coordinate patches.

Homework Equations

I know the general formulation of integration over a compact manifold when the support of the function is contained in a single coordinate patch. Also, I know how to integrate functions on manifolds using partitions of unity.

The Attempt at a Solution

I know the general idea. First I must prove that this holds when the support of f and g are in a single coordinate patch. Then using partitions of unity, I want to notice the integral is just the sum of integrals over finitely many coordinate patches. What I can't get is the part where the support of f and g are in a single coordinate patch:

Suppose supp(f) \subset V with \phi : U \rightarrow V \subset M and supp(g) \subset V' with \psi : U' \rightarrow V' \subset N.

Then \int_{M\times N} fg = \int_{U \times U'} (fg) \circ (\phi\times\psi) V_k(\phi \times \psi) but I don't know how to deal with this...

 

[Dick]

I think you just want to use Fubini's theorem. The integral of |fg| exists because f and g are continuous on compact sets, so they are uniformly continuous and bounded. Doesn't that work?

 

[AnalysisQuest]

The problem is that I don't know how to split up the k-volume term and I don't know how to deal with phi x psi. Unless what I've done so far is completely wrong. I've been stuck for days now and I'm just frustrated and missing something obvious, I think.

 

[Dick]

The problem is that I don't know how to split up the k-volume term and I don't know how to deal with phi x psi. Unless what I've done so far is completely wrong. I've been stuck for days now and I'm just frustrated and missing something obvious, I think.

I think you are over complicating it. If you are working on a single coordinate patch and you've applied a partition of unity then the functions you are integrating are continuous and the domains over which you are integrating have compact support. The Jacobean is differentiable. Everything is bounded. I think you can just say it's Fubini's theorem on R^m x R^n. I don't think you need to do anything complicated.