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:
1. Is there a standard name for this property?


2. Is it true that continuum+choice implies property (\*) -- is there a reference?
3. Is it true that if (either? both?) continuum+choice don't hold, (\*) can fail? Again, reference please!



Non-separable metric probability space
Apparently, a relevant source here is H. J. Keisler and A. Tarski, From accessible to inaccessible cardinals. Fundamenta mathematicae, vol. 53 (1964), pp. 225--308, a review of which is given at <https://philpapers.org/rec/KEIFAT> . 


I don't have the paper by Keisler and Tarski at this point with me. I found a reference to Keisler and Tarski in Добавление III (Appendix III) to the Russian translation of Convergence of Probability Measures by Billingsley; apparently, this Добавление does not appear in the English original of the Billingsley book. 


So, this is what I have gathered: 


1. The condition that the support of $\mu$ is contained in a separable subspace of $X$ is equivalent to the support itself being separable; see e.g. <https://math.stackexchange.com/questions/516886/prove-that-a-subset-of-a-separable-set-is-itself-separable> . Measures with a separable support were simply called *separable* in that Добавление.
2. Theorem 2 stated and proved in that Добавление says that all probability measures on a metric space $X$ are separable iff every discrete subset of $X$ is of non-measurable cardinality.


A subset $S$ of a metric space is called *discrete* if every point of $S$ is in some open ball not containing other points of $S$. A cardinality $k$ is called *measurable* if for some set $U$ of cardinality $k$ (or, equivalently, for all sets $U$ of cardinality $k$) there is a non-atomic probability measure on the power set $2^U$.


Obviously, $\aleph\_0$ is non-measurable. If the continuum hypothesis holds, then the continuum cardinality is also non-measurable. It is also noted in that Добавление that the existence of measurable cardinalities is a famous unsolved problem, called the measure problem. 


