An application of partitions of unity: integrating over open sets. In Spivak's "Calculus on Manifolds", Spivak first defines integration over rectangles, then bounded Jordan-measurable sets (for functions whose discontinuities form a Lebesgue null set).
He then uses partitions of unity to define integration over arbitrary open sets (top of p.65). Used in the proof of this assertion is the claim that if:
i) $\Phi$ is subordinate to an admissible cover $J$ of our open set $A$.
ii) $f:A\rightarrow \mathbb{R}$ is locally bounded in $A$.
iii) The set of discontinuities of $f$ is Lebesgue-null. 
then each $\int_A \phi\cdot|f|$ exists.
I cannot see how this statement makes sense, seeing as the integral is thus far only defined for bounded Jordan-measurable sets. Perhaps I am missing something simple here?
 A: 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".
A: 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). 
