While reading the usual construction of the integral on a manifold (as in Isaac Chavel, "Riemannian Geometry", page 147) I encountered the following point that I do not understand: Chavel considers an arbitrary covering with charts $U_a$ to which he subordinates a partition of unity $\phi _a$ and then goes on to define $\int \limits _U \omega = \sum \limits _a \int \limits _{U_a \cap U} \phi _a \omega$ for every open $U$.
My question is: why does the above sum exist (in principle, it may have arbitrarily many terms)? I can understand why it exists for a compactly-supported $\omega$ (as Guillemin and Pollack choose to do in "Differential Topology"), but Chavel makes no such assumption.
Please note that I am not talking about the convergence of that sum, because it is not even a series; the problem is that it may have uncountably many non-zero terms, which makes it an undefined mathematical object (think of what happens when $U$ is the whole manifold).