# Proving weak convergence of random probability measures

I don't understand the following as I read along a proof in a paper:

We denote by $$\mathcal{P}({M})$$ the space of probability measures on a metric space $$M$$, equipped with the weak topology.

Let $$E$$ be a metric space. Let $$\{ \mu_n \}$$ be a sequence of random measures on $$E$$, i.e. for each $$n$$, $$\mu_n$$ is a $$\mathcal{P} (E)$$-valued random variable. Also, let $$\mathbb{Q}$$ be a deterministic probability measure on the same probability space. We can then treat $$\delta_{\mathbb{Q}}$$ be a constant $$\mathcal{P}(E)$$-valued random variable.

Since both $$\text{Law} ( \mu_n)$$ and $$\text{Law} ( \delta_{\mathbb{Q}})$$ are measures on $$\mathcal{P}(E)$$, the paper defines that $$\mu_n$$ converges in law to $$\mathbb{Q}$$ if $$\text{Law} ( \mu_n) \implies \text{Law} ( \delta_{\mathbb{Q}}). \quad \quad \quad \quad \, \, \, \, (*)$$ However, in the paper, the fact that $$\mu_n$$ converges in law to $$\mathbb{Q}$$ is concluded by establishing that $$\mathbb{E} \bigg[ \bigg| \int_E f \,d \mu_n - \int_E f \,d \mathbb{Q} \bigg| \bigg] \rightarrow 0, \quad \quad \quad (**)$$ for all continuous bounded functions $$f$$ on $$E$$.

By definition of $$(*)$$, this is equivalent to saying that $$\int_{\mathcal{P}(E)} f \, d\text{Law} ( \mu_n) \rightarrow \int_{\mathcal{P}(E)} f \, d \text{Law} ( \delta_{\mathbb{Q}}),$$ for all continuous bounded functions $$f$$ on $$\mathcal{P}(E)$$. How does this follow from $$(**)$$? Any ideas?

• You forgot to cite the paper itself for us to look at. – MathematicalPhysicist Jan 24 '16 at 20:16

Maybe my answer is not the simplest, but what we have is: for all $f\in \mathcal C_b$, let $\Phi_f : \nu \in \mathcal P(E) \mapsto \int_E f d\nu$ (is continuous and bounded),
$$E[ \Phi_f(\mu_n)] \to E[ \Phi_f(\mu)]$$
In fact, there is some stuff I never really understand that are Convergent determining function... You can found that in the book of Dawson if I remember. The set $\{ \Phi_f \, : \, f\in \mathcal C_b \} \subset \mathcal C_b(\mathcal P(E),\mathbb R)$ if it separates the points, etc (I imagine your $E$ is a Polish). It is a direction I think...