Let $X$ be a separable topological space and let $\left\{X_i\right\}_{i \in I}$ be a non-empty collection of distinct dense $G_{\delta}$ subsets satisfying $ \cap_{i \in I} X_i $ is dense in $X$.
Does there exist a finite and strictly positive Borel measure $\mu$ on $X$ satisfying $$ \mu\left( X- \cup_{i \in I} X_i \right)=0 ? $$