Let $H$ be a separable Hilbert space and let $\mu$ be an arbitrary probability measure on $H$. I would like to approximate this distribution by by a finitely supported discrete distribution is the following sense:

For every $\varepsilon>0$ I would like to have a finitely supported probability measure $\nu_{\varepsilon}$ so that for every open set $A$ we have $$|\mu(A)-\nu(A)|<\varepsilon.$$

Is such a statement true?

Let $H$ be a separable Hilbert space and let $\mu$ be an arbitrary probability measure on $H$. I would like to approximate this distribution by by a finitely supported discrete distribution is the following sense:

For every $\varepsilon>0$ I would like to have a finitely supported probability measure $\nu_{\varepsilon}$ so that for every open ball $B$ of diameter $1$ we have $$|\mu(B)-\nu(B)|<\varepsilon.$$

Is such a statement true? Is it true at least in $\mathbb{R}^d$?

Comment: The question has been changed due to a comment by Brendan Mckay's example that refuted the stronger version of the question.