Spivak's book "A Comprehensive Introduction to Differential Geometry, Volume 5" says the following on page 309 (I'll simplify here a bit): Let $G$ be a compact, connected Lie Group and $a \mapsto \eta(a)$ a smooth family of $k$-forms on $G$, where $\eta(a)$ is defined for all $a$ in an open subset $U \subset G$. I think with smooth family he means that $$ a \mapsto \eta(a)_p(X_1(p),\dots,X_k(p))$$ is a smooth function for any $p$ and $X_i$'s on $U$. Then he says we can form the following integral: $$ \int_U \eta(a) \,da = \int_U (a \mapsto \eta(a)_p(X_1(p),\dots,X_k(p)))\, da ,$$ which is again a $k$-form on $G$.
If $U = G$ one can prove that this is a $k$-form by applying the dominated convergence theorem and checking that this is a smooth function on a neighborhood of $p$. But how do I do the case where $U$ is a proper open subset? I don't see why the integral exists - why does this function not blow up if you approach the boundary of $U$?