Let $X$ be a compact metric space and $\nu$ a fully supported Borel probability measure on $X$. Suppose that for every continuous $f \in C(X)$ there exists a constant $C$ and $A \subseteq X$ with $\nu(A) > 0$ such that $f(x) = C$ for all $x \in A$.
My suspicion is this implies the measure has atoms but I'm not sure how to prove it.
