I would like to investigate if an analog of the classical Portmanteau theorem holds for vague convergence of Radon measures.
Here are the definitions I'm using.
Let $X$ be a Hausdorff locally compact topological space, and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra.
A positive measure $\mu$ on $(X, \mathcal{B}(X))$ is said to be a Radon measure if :
(i) $\mu(K) < \infty$ for all compact subsets $K \subset X$, (ii) $\mu(O) = \sup \, \{\mu(K) \, / \, K \subset O, K \rm{\: is \: compact \:} \}$ for all open subsets $O \subset X$, and
(iii) $\mu(A) = \inf \, \{ \mu(O) \, / \, A \subset O, O \rm{\: is \: open \:} \}$ for every Borel subset $A \subset X$.
I will say that a sequence $(\mu_n)_n$ of Radon measures on $X$ converges vaguely to a Radon measure $\mu$ if $\lim_{n \to \infty} \int_X f \, d \mu_n = \int_X f \, d \mu$ for all $f \in C_c(X)$, where $C_c(X)$ denotes the set of all continuous real-valued functions defined on $X$, with a compact support.
Now, consider the following propositions :
$(P_1)$ $(\mu_n)_n$ converges vaguely to $\mu$;
$(P_2)$ $\mu(O) \le \underline{ \lim } \, \mu_n(O)$ for all open subsets $O \subset X$;
$(P_3)$ $\mu(K) \ge \overline{ \lim } \, \mu_n(K)$ for all compact subsets $K \subset X$;
$(P_4)$ $\mu(A) = \lim \, \mu_n(A)$ for all Borel subsets $A \subset X$ with a compact closure, and satisfying $\mu(\partial A) = 0$.
I'm able to prove that $(P_1) \Leftrightarrow ((P_2) + (P_3)) \Rightarrow (P_4)$.
My questions are :
1) Are $(P_2)$ and $(P_3)$ equivalent in full generality ? If $X$ is compact, this is obvious by taking complementary sets.
2) Can one prove that $(P_4) \Rightarrow (P_1)$ ? I succeeded to prove that $(P_3) + (P_4) \Rightarrow (P_1)$. (in fact, instead of $(P_3)$, I only need that $\sup_n \mu_n(K) < \infty$ for all compact subsets $K \subset X$)
Thanks.
Best Answer
ad 1) No, neither implication holds; let $X = \mathbb{R}$. First take $\mu_n = \delta_1$ and $\mu = 0$. Then $(P_2)$ is satisfied, whereas $(P_3)$ is not. Second, take $\mu_n=0$ and $\mu = \delta_2$. Then $(P_3)$ holds, but $(P_2)$ does not.
ad 2) A statement of the equivalences between $(P_1)$, $((P_2)+(P_3))$ and $(P_4)$ can be found in Theorem 3.2 in Resnick, Heavy-Tail Phenomena, Springer 2007. $(P_2)$ should additionally have the criterion that $O$ is relatively compact. Resnick's book is not explicit about the proof; the space $X$ is required to be locally compact and separable, see page 48.
The book can be found here, https://books.google.de/books?id=p8uq2QFw9PUC&lpg=PP1&dq=resnick%202007%20probability&pg=PA52#v=onepage&q=theorem%203.2&f=false
EDIT: another source which I recently encountered: Lindskog, Resnick and Roy, Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps, available at http://projecteuclid.org/euclid.ps/1413896892. In Section 2 a certain type of convergence is the topic and in Theorem 2.1 you might find a portmanteau theorem.