Let $S$ be a metric space with be Borel $\sigma$-algebra $\Sigma$. Let $\boldsymbol{P}(S)$ be the set of probability measures on $(S,\Sigma)$. According to wikipedia, "the weak topology is generated by the following basis of open sets:
$\left\{ U_{\phi, x, \delta} \,\left|\, \begin{array}{c} \phi \colon S \to \mathbb{R} \text{ is bounded and continuous,} \\ x \in \mathbb{R} \text{ and } \delta > 0 \end{array} \right. \right\}$
where
U_{\phi, x, \delta} := \left\{ \mu \in \boldsymbol{P}(S) \,:\, \left| \int_{S} \phi \, \mathrm{d} \mu - x \right| < \delta \right\}."
I want to verify that this is in fact a basis. It is easy to see that any $\nu\in\boldsymbol{P}(S)$ is contained in some basis element (for any $\delta$ and $\phi$, take $x=\int \phi d\nu$). The next property is that
- if $\nu$ belongs to the intersection of two basis elements $B_{1}$ and $B_{2}$, then there is a basis element $B_{3}\ni\nu$ such that $B_{3}\subseteq B_{1}\cap B_{2}$.
In other words, given that
(1) $\displaystyle\left| \int_{S} \phi_1 \, \mathrm{d} \nu - x_1 \right| < \delta_1$; and
(2) $\displaystyle\left| \int_{S} \phi_2 \, \mathrm{d} \nu - x_2 \right| < \delta_2$,
I need to find $\delta^\ast, x^\ast\in\mathbb{R}$ and a continuous, bounded, real-valued function $\phi^\ast$ such that
$\displaystyle\left| \int_{S} \phi^\ast \, \mathrm{d} \nu - x^\ast \right| < \delta^\ast$ implies (1) and (2).
I would appreciate any suggestions on how to proceed. Thanks!