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: Does there exists 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.