Let $X$ be a $T_0$ topological space (not $T_1$), and let $\Sigma_X$ be the Borel $\sigma$-algebra. Assume that $(X,\Sigma_X)$ is a standard measurable space, i.e., measurably isomorphic to the Borel $\sigma$-algebra $(Y,\Sigma_Y)$ of a complete separable metric space $Y$.

**Question (amended to honour Dieter Kadelka):** Is it true that there does NOT exist a positive measure $\mu$ on $(X,\Sigma_X)$ with empty support, $\mathrm{supp}\mu=\emptyset$?

Note that the measurable isomorphism between $X$ and $Y$ is a priori merely Borel measurable, and can potentially map open sets to sets without interior and vice versa.

Thank you.

**Discussion:** The question arises from the discrepancy in definitions of the support of a Borel measure on a topological space. I think that the standard definition is this, in which case, unless $X$ is a very good space, you don't have to expect $\mu(X\setminus\mathrm{supp}\mu)=0$. However, in Propositiom 8.6.8 in Dixmier's "$C^*$-algebras", the author defines the support of a measure as the smallest closed subset with negligible complement (the standard analysis definition). Now I wonder if the two definitions coincide in this context. Note that the space in question is only $T_0$ in general, possesses a dense locally compact open subset (Proposition 4.4.5 in the book), and the Borel structure is measurable isomorphic to that of a complete separable metric space (Proposition 4.6.1 in the same book).

**Discussion 2:** Well, there is more to the spectrum of a separable postliminal $C^*$-algebra - it is locally qusicompact, second countable etc. But the question remains as it is: does the mere Borel isomorphism to a Polish space rule out empty support?