Looking at this question/answers:
Volume of a paracompact manifold
it is used a partition of unity $\{\eta_j : j \in J\}$ corresponding to the coutable, locally finite open cover $\{ U_j : j \in J\}$, where $J$ is a countable set, to find a finite volume for a paracompact connected manifold $M$. At some point it is said that the following integral $$ \int_M \sum_{j\in J} \eta_j \omega_j \tag{1} $$ can be infinite even though the integral of each term in the sum is finite, $$ \int_{B_j} \eta_j \omega_j \leq Vol(B^n) , \tag{2} $$ with $B^n$ the closed unit ball in $\mathbb R^n$. Then, some $j$-dependent coefficients are introduced in $(1)$ to make the series convergent. What I don't understand is why do we need these coefficients? I mean, by the properties of the partition of unity, all but a finite number of the functions $\eta_j$ are zero in every open set of the cover, which is also locally finite, so the sum in $(1)$ contains actually a finite number of terms and it should be thus finite provided that $(2)$ is satisfied, isn't it?