Let $(\mathbf{Y}, \mathcal{Y})$ be a Polish space. Then, there exists a sequence of measurable functions $(\phi_{n})_{n\in\mathbb{N}}$ from $(\mathbf{Y}, \mathcal{Y})$ to $\mathbb{R}$ such that, for two probability measures $\nu_{1}$ and $\nu_{2}$ on $(\mathbf{Y}, \mathcal{Y})$, one has: $$ \left(\int_{\mathbf{Y}}\phi_{n}d\nu_{1}\right)_{n\in\mathbb{N}} = \left(\int_{\mathbf{Y}}\phi_{n}d\nu_{2}\right)_{n\in\mathbb{N}} \implies \nu_{1} = \nu_{2} $$ I think I must exploit the separability property of space $\mathbf{Y}$,but I can't see how to do so. Any hint/help would be appreciated.