# Union of distinct dense $G_\delta$ sets

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 $$\mu\left( X- \cup_{i \in I} X_i \right)=0 ?$$

• Satisfying what exactly? – Shervin Sorouri Sep 11 at 6:54
• The intersection is dense in $X$. – N00ber Sep 11 at 6:57
• If the intersection has a dense sequence $x_n$, then $\mu(A)=\sum_n{2^{-n}1(x_n \in A)}$ works. – Mindlack Sep 11 at 8:17
• But what is $x_n$ in this case? Since things may not be countable. – N00ber Sep 11 at 9:03