[Math] Measure of the support of a Borel probability on a metric space

Question

Does the support of a Borel probability measure always have full measure in a metric space?

I know this is true for separable metric spaces, and locally compact metric spaces. Is it true in general?

Answer 1

Following Pietro's lead, let me observe that if there is a measurable cardinal, then there is a counterexample.

Suppose that $\kappa$ is a measurable cardinal. Then there is a $\kappa$-additive 2-valued measure $\mu$, measuring all subsets of $\kappa$, giving them either measure $0$ or $1$, giving measure $1$ to the whole space and giving measure $0$ to any set of size less than $\kappa$ (among others). If we give $\kappa$ the discrete topology, then every set is closed (and hence Borel), and the support is empty.