Recently reading the Marcelo Vianna's Book, Foundations of Ergodic Theory, I came across to the following:
Let be $M$ a metric space, we have the following characterization of convergence of sequence of borelian measures in $M$ with respect to the weak topology: $$ \mu_n\to \mu \iff \int f d\mu_n\to \int fd\mu, \forall~ f\in C_b(M) $$ where $C_b(M)$ means the set of all bounded continuous functions.
It is that correct? For a compact $M$ I know this is correct but for non compact $M$ I think this is false... Can anyone provide me a clarification
Edit: In order to clarify my question, $\mathcal{M}$ means the set of all Borelians finite measures in $M$, and the weak topology that I am talking about is the topology generated by the functionals $\varphi_{\mu}$ of the form $$ \varphi_{\mu}:\mathcal{M} \ni \mu \mapsto \int f d\mu \in \mathbb{R} $$ where $f\in C_{b}(M)$.
If $\mu_n(M)=\mu(M)<\infty$ for all $n$, then the assertion is true (in both directions), since these are essentially probability measures. $M$ does not have to be compact. See the Portmanteau Theorem.