Let $X$ be a separable metric space. Denote by $B_q(x)$ the open ball of radius $q>0$ and center $x \in X$. Also, let $\mathscr{B}$ be the (Borel)-$\sigma$-algebra generated by the open sets.
Is it true that there exists a probability measure $\mu: \mathscr{B} \to [0,1]$ such that $\mu(B_q(x))>0$ for all $x \in X$ and $q>0$?
Equivalently, I am asking whether a strictly positive Borel probability measure necessarily exists; probably there is an obvious answer that I am missing..
Take $\mu=\sum_{k\geq 1} \delta_{x_k}/2^k$, where $(x_k)$ is a countable dense set.