In Lenaic Chizat's "Sparse Optimization on Measures with Overparametrized Gradient Descent" one finds the following definition: for a measure $\mu \in \mathcal P_2(\Omega)$ (the Wasserstein-2-space on $\Omega := \Theta \times \mathbb R_{\ge 0}$, where $\Theta$ is a compact Riemannian manifold) we define homogeneous projection operator $\mathsf{h} \colon \mathcal P_2(\Omega) \to M_+(\Theta)$ (where $M_+(\Theta)$ are the nonnegative Radon measures on $\Theta$), where $\mathsf h \mu$ is characterized by $$ \int_{\Theta} \psi(\theta) \, \text{d}(\mathsf h \mu)(\theta) = \int_{\Omega} r^2 \psi(\theta) \, \text{d}\mu(r, \theta) \qquad \forall \psi \in \mathcal C(\Theta; \mathbb R). $$
My question. Why is $\mathsf h \mu$ defined umambigously via integration against all continuous functions and why don't we have test will are Borel measurable functions?
This follows for example from the Riesz representation theorem, that asserts that any linear positive form on the space of compactly supported continuous functions defined on a locally compact set is associated to a unique Radon measure on that set.