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?
Best Answer
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".