I have been thinking about the following problem. Let me know if the notation below makes sense.
Let $\mathcal{P}$ denote the set of Borel probability measures on a metric space $(\mathbb{R}^{k}, ||\cdot ||)$. Consider two collections of measures $\mathbf{P}_1 \equiv \{P_1^{\theta}\}_{\theta \in \Theta}$ and $\mathbf{P}_2 \equiv \{P_2^{\theta}\}_{\theta \in \Theta}$, where $P^\theta_i \in \mathcal{P}$ for all $\theta \in \Theta$ and $i=1,2$ ($\Theta$ is some parameter space).
I am interested in defining a notion of "distance" for the collections $\mathbf{P}_1$ and $\mathbf{P}_2$. For a fixed $\theta \in \Theta$, I can start by considering the bounded lipschitz metric on $\mathcal{P}$:
\begin{equation} \beta (\mathbf{P}_1, \mathbf{P}_2; \theta) \equiv \sup \Big\{ \Big| \int f dP_1^{\theta}-\int f dP^\theta_2 \Big| \: : \: ||f||_{\text{BL}} \leq 1 \Big\} \end{equation}
where $||f||_{\text{BL}} \equiv \max \{ |f(x)| , \inf_{c}\{c : |f(x,y)| \leq c d(x,y) \}$ (see Van der Vaart and Wellner p. 73).
It seems that I can then define the following notion of distance between the collections $\mathbf{P}_1$ and $\mathbf{P}_2:$
$$ \sup_{\theta \in \Theta} \beta(\mathbf{P}_1, \mathbf{P}_2; \theta),$$
and use this notion to say that a sequence of collections $\mathbf{P}_n$ converges to a collection $\mathbf{P}$ (uniformly in $\Theta$) if:
$$\sup_{\theta \in \Theta} \beta (\mathbf{P}_n, \mathbf{P};\theta) \rightarrow 0,$$
as $n \rightarrow \infty$. The question that I have is the following. Suppose that $g$ is a uniformly continuous function. Is it true that
$$\mathbf{P}_n \rightarrow \mathbf{P} \implies \sup_{\theta} \sup \Big\{ \Big| \int f \circ g dP_n^{\theta}-\int f \circ g dP^\theta \Big| \: : \: ||f||_{\text{BL}} \leq 1 \Big\} \rightarrow 0.$$
I would appreciate any comments or ideas. If interested, I can share my attempt to proof this claim when $\Theta$ has finitely many elements.