Let $(X, d)$ be a metric space and $\mathcal P(X)$ the space of all Borel probability measures on $X$. We endow $\mathcal P(X)$ with the Lévy–Prokhorov metric $d_P$. Let $\mathcal C_b(X)$ be the space of all continuous bounded real-valued functions on $X$.
Then I have a theorem.
Let $\mu, \mu_1, \mu_2,\ldots \in \mathcal P(X)$ such that $\mu_n \to \mu$ in $d_P$. Then $\mu_n \to \mu$ weakly, i.e.,
$$
\int_X f \mathrm d\mu_n \to \int_X f \mathrm d\mu \quad \forall f \in\mathcal C_b(X).
$$
This suggests that there exist an example of $\mu, \mu_1, \mu_2,\ldots \in \mathcal P(X)$ such that $\mu_n \to \mu$ weakly but $\mu_n \not\to \mu$ in $d_P$.
Could you give me such an example?
Assume that $\mu, \mu_1, \mu_2,\ldots \in \mathcal P(X)$ such that $\mu_n \to \mu$ weakly. We only have that $\mu_n(A) \to \mu (A)$ for all Borel sets $A$ such that $\mu(\partial A)=0$.
It seems quite "unsatisfactory" to me that the limit $\mu$ is unique, but we can only recover information of $\mu$ for such Borel sets $A$ whose boundary has $0$ probability.
Can we recover $\mu(A)$ from a sequence $(\mu_n)$ for every Borel set $A$?
- In the same spirit as 2., now we assume $\mu, \mu_1, \mu_2,\ldots \in \mathcal P(X)$ such that $\mu_n \to \mu$ in $d_P$.
Can we recover $\mu(A)$ from a sequence $(\mu_n)$ for every Borel set $A$?
Best Answer
For 1) If the space is separable, Levy Prokhorov convergence is equivalent to weak convergence. In the no separable case (which is not very interesting for probability applications), I'm not even sure the direct implication is true either.
2,3)Let $\mu_n = 1/n(\delta_{1/n} +...+\delta_{n/n})$. Then $\mu_n\to \mu$ the Lebesgue measure on $[0,1]$ but $\mu_n(\mathbb Q)\not\to \mu(\mathbb Q)$