I am confused by the difference between "$f$ integrable" and "$\int_Af$ exist", in Spivak's notion of extended integral. Here's his definition. Note that it has a flaw: each $\varphi $, in the partition of unity $\Phi$ subordinate to $O$, should be compactly supported in order to have $\int_A \varphi |f|$ exist (such $\varphi$'s existence is proven).
So, $f$ integrable, implies $\sum_{\varphi\in\Phi}|\int_A\varphi f|$ converge, then implies $\int_Af=\sum_{\varphi\in\Phi}\int_A\varphi f$ exist. It seems to me that each "implies" can't go backwards, so doesn't it means that $\int_Af$ may exist though $f$ is not integrable? (strange...) Or the condition on $\varphi$ actually make the "implies" go backward, so that $f$ integrable iff $\int_Af$ exist? Thanks.