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 $\kappa$, and giving measure $0$ to any set
of size less than $\kappa$. If we give $\kappa$ the
discrete topology, then every set is closed (and hence
Borel), and the support is empty.