Integration on compact manifold

Integration on a nice enough manifold of a function $f:M \to \mathbb{R}$ is defined $$\int f = \sum_{ i \in I} \int_{U_i}\phi_i f$$ where $\phi_i$ is a partition of unity subordinate to the open cover $U_i$ of $M$.

The index set $I$ can be infinite in general.

If $M$ is compact (bounded), can I take $I$ to be a finite set so that the sum is a finite sum?

• Yes, if $M$ is compact, you can always have a finite partition of unity subordinate to any open cover. – Daniel Fischer Jan 12 '14 at 23:37