This is a continuation of these two questions that are asking the same thing as each other:
In the answers to these two questions it was concluded that in item (4) of Spivak's definition of a partition of unity, he should have required that the supports of the functions $\varphi \in \Phi$ be compact, not merely closed. This is so the integral $\int \varphi\cdot |f|$ would be defined for locally bounded and almost everywhere continuous $f$. However, it seems that the support of $\varphi$ must also be Jordan-measurable for this integral to be defined. So am I right in thinking that Spivak should have required in item (4) that the supports of the functions $\varphi \in \Phi$ be both compact and Jordan-measurable, or is Jordan-measurability unnecessary or implied by other conditions?