I've been thinking about integration on manifolds and am pondering the notion of partitions of unity.
Given a partition of unity $ \Phi$ of an open set $A \subset \mathbb{R}^n$ subordinate to an open cover $O$, and $f: A \to \mathbb{R}$ is locally bounded in $A$, and $\sum\_{\phi \in \Phi} \int\_A \phi \cdot \left| f \right|$ converges, we define

$$ \int\_A f := \sum\_{\phi \in \Phi} \int\_A \phi \cdot f. $$
My question is very simple: How can we guarantee that in the overlap of any two open sets we are not counting them twice? I'm guessing this has something to do with the condition that $\sum\_{\phi \in \Phi} \phi(x) = 1$ but would prefer to have it spelt out by someone who understands this topic better.