An application of partitions of unity: integrating over open sets.

Solution 1:

This issue is also discussed in this post.

I think that the key is at the begining of the proof of item ($1$) in Theorem $3$-$12$, where Spivak says "Since $\varphi\cdot f=0$ except on some compact set $C$..." This compact set seems to depend on $\varphi$ therefore the statement suggests that Spivak may have had in mind a slightly different version of item ($4$) in Theorem $3$-$11$.

It seems then that the word "closed" in item ($4$) should be changed to "compact" and hence the sentence in the proof "If $f:U\rightarrow [0,1]$ is a $C^{\infty}$ function which is $1$ on $A$ and $0$ outside of some closed set in $U$,..." (p.64) should have the word "closed" changed to "compact".

Solution 2:

I may well be misinterpreting something here (your notation is a bit unclear on a few points, and I don't have Spivak's book available), but provided the cover consists of bounded, Jordan measurable sets, and you already know how to integrate over such sets, the rest should be easy: if $J = \{ A_i \}_{i \in I}$, and the partition of unity consists of continuous functions $\varphi_i: A \to \mathbb{R}$ with $\varphi_i$ supported in $A_i$ for each $i \in I$, then each $$ \int_{A_i} \varphi_i f $$ exists (the integrand is supported in $A_i$, a bounded Jordan measurable set, the set of discontinuities has measure $0$, so we know how to integrate it). Then you simply define $$ \int_A f := \sum_{i \in I} \int_{A_i} \varphi_i f $$ and hope that this is independent of the particular cover $J$ (and it usually is).