Does the support of a Borel probability measure always have [full measure](https://en.wikipedia.org/wiki/Almost_everywhere) in a metric space?


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



Measure of the support of a Borel probability on a metric space
Following Pietro's lead, let me observe that if there is a
[measurable cardinal](http://en.wikipedia.org/wiki/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.


