I have a question about Volume 5 of "A comprehensive introduction to differential geometry" by Spivak.
Starting on p. 308, he wants to show that: given a compact, connected Lie group $G$ and a closed Lie subgroup $H\subset G$, the de Rham cohomology of $G/H$ is isomorphic with the cohomology of the $G$-invariant forms on $G/H$.
In the part below, he claims that one can integrate a family of differential forms over an open subset of $G$. Why does the integral over $U$ exist? For instance, look at the simple case where each $\eta_a$ is a constant function (i.e. $0$-form), smooth in $a$. For sure one can find smooth functions on $U\subset G$ whose integral is infinite, no?
