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

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

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