Consider an uncountable discrete metric space $X $ (i.e., metrized by the Kronecker delta). Define a measure on $X$ putting for any $A\subset X,\\ $ $\mu(A)=1$ or $\mu(A)=0$ according whether $A$ belongs to a given non-principal filter, or not (sigma-additivity holds, for there are no disjoint subsets of positive measure). Then $\mu$ is a Borel probability measure with empty support.

Consider an uncountable discrete metric space $X $ (i.e., metrized by the Kronecker delta). Define a measure on $X$ putting for any $A\subset X,\\ $ $\mu(A)=1$ or $\mu(A)=0$ according whether $A$ belongs to a given non-principal ultrafilter $\mathcal{F}$, or not (sigma-additivity holds, for there are no disjoint subsets of positive measure). Then $\mu$ is a Borel probability measure with empty support.

[edit] Actually, this is additive, but to ensure sigma-additivity it would be needed that $\mathcal{F}$ be closed under countable intersections.

Consider an uncountable discrete metric space $X $ (i.e., metrized by the Kronecker delta). Define a measure on $X$ putting for any $A\subset X,\\ $ $\mu(A)=0$ or $\mu(A)=1$ according whether $\mathbb{card}(A)$ is countable or not. Then $\mu$ is a Borel probability measure with empty support.

Consider an uncountable discrete metric space $X $ (i.e., metrized by the Kronecker delta). Define a measure on $X$ putting for any $A\subset X,\\ $ $\mu(A)=1$ or $\mu(A)=0$ according whether $A$ belongs to a given non-principal filter, or not (sigma-additivity holds, for there are no disjoint subsets of positive measure). Then $\mu$ is a Borel probability measure with empty support.