Integrals over Manifolds (part 2)
Well, paradoxically, we start by getting smaller. Specifically, I say that we can always find an orientable open cover of such that each set in the cover is contained within the image of a singular cube.
We start with any orientable atlas, which gives us a coordinate patch around any point we choose. Without loss of generality we can pick the coordinates such that . There must be some open ball around whose closure is completely contained within ; this closure is itself the image of a singular cube, and the ball obviously contained in its closure. Hitting everything with we get an open set — the inverse image of the ball — contained in the image of a singular cube, all of which contains . Since we can find such a set around any point we can throw them together to get an open cover of .
So, what does this buy us? If is any compactly-supported form on an -dimensional manifold , we can cover its support with some open subsets of , each of which is contained in the image of a singular -cube. In fact, since the support is compact, we only need a finite number of the open sets to do the job, and throw in however many others we need to cover the rest of .
We can then find a partition of unity subordinate to this cover of . We can decompose into a (finite) sum:
which is great because now we can define
But now we must be careful! What if this definition depends on our choice of a suitable partition of unity? Well, say that is another such partition. Then we can write
so we get the same answer no matter which partition we use.