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Notation used:

$S$- Seperable metric space

$\mathcal{C}_u(S)$- Class of uniformly continuous functions from $S$ into $\mathbb{R}$

$\mathcal{P}(S)$- Space of probability measures on $(S,\mathcal{B}(S))$, where $\mathcal{B}(S)$ is the Borel $\sigma$-algebra on $S$

How do I prove the following statement (any references ?):-

There exists a countable subset $\mathcal{C}_0$ of $\mathcal{C}_u(S)$ such that for every sequence $(\nu_n)$ in $\mathcal{P}(S)$ \begin{equation} \lim\limits_n\int\limits_Sc \ d\nu_n=\int\limits_S c\ d\nu_0, \text{ for every } c\in\mathcal{C}_0 \end{equation} implies that $\nu_n$ converges narrowly to $\nu_0$ in $S$. In particular, $\mathcal{C}_0$ seperates the points of $\mathcal{P}(S)$.

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Maybe Dudley's book Real Analysis and Probability or Billingsley's Convergence of probability measures can be useful. –  Davide Giraudo Feb 11 '13 at 20:25
1  
What do you exactly mean by narrow convergence? $\int_S fd\nu_n\to \int_S fd\nu_0$ for all $f$ continuous with compact support, or what? –  Davide Giraudo Feb 15 '13 at 21:30

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