Let $M^{\pm}$ be Polish spaces and $\mu^{\pm}$ be Borel probability measures on $M^+$ and $M^-$ respectively. If $G: M^+ \rightarrow M^-$ with the constraint $G_\#\mu^+ = \mu^-$, why does the following follow:
"Let’s consider the constraint $G_\#\mu^+ = \mu^-$, assuming moreover that $\mu^{\pm} = f^{\pm}dVol^{\pm}$ on $\mathbb{R}^n$ or on Riemannian manifolds $M^{\pm}$. Then if $\phi \in C(M^-)$ is a test function, it follows that
$$ \int_{M^+} \phi(G(x))f^+(x)dVol^+(x) = \int_{M^-} \phi(y)f^-(y)dVol^-(y)."$$
I've not come across test functions before so this isn't following for me. Many thanks.
(Link: page 2:4)
I believe a test function is just a function that satisfies the necessary properties that make sense in the discussed context. For instance when considering the weak formulation of a PDE one may consider multiplying the PDE by a "test function" $v\in C^\infty_c(\Omega)$, and then integrating by parts. If we consider the case of the Poisson equation (with homogeneous Dirichlet boundary conditions), we see that we can actually multiply by a "test function" $v\in H^1_0(\Omega)$ in order to find the weak formulation.
In this case it seems to be saying that $\phi\in C(M^-)$ is a sufficient assumption to make sense of the integral equality (though I believe $\phi\in L^1(M^-)$ would also do the job, and this is what we see in Cedric Villani's book on optimal mass transport).