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?