Are the following definitions of vague convergence equivalent?
The sequence of s.p.m $\mu_n$ vaguely converges to $\mu$ iff
(1) There exists a dense subset $D$ of the real line $\mathbb{R}$ so that $ \forall a \text{ and } b \in D \text{ with } a <b, \mu_n((a,b]) \rightarrow \mu((a,b])$.
(2) For all $f\in C_K^+ (\mathbb R)$ we have $\int f \mu_n \rightarrow \int f \mu$.
A source connecting these two definitions is also welcome.
Proof idea: Use the fact that $L^2$ functions like indicators are dense in $C$. Also $C$ functions can be approxdimated by simple functions.
Best Answer
It is right in fact. The core of proof is the D-valued approximation lemma. You can even let $f \in C_0$. And you can find the total proof in A Course in Probability Theory written by Kai Lai Chung.