Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if: the support of $\mu$ is contained in a separable subspace of $X$.
Questions:
Is there a standard name for this property?
Is it true that continuum+choice implies property (*) -- is there a reference?
Is it true that if (either? both?) continuum+choice don't hold, (*) can fail? Again, reference please!