I have been considering the following question:


Let $X$ be a compact, metrizable space with the following property: every (regular) Borel probability measure on $X$ is atomic, i.e. for each $\mu\in\text{Prob}(X)$ there exists $x\_\mu\in X$ such that $\mu(\{x\_\mu\})>0$. Does it follow that $X$ is countable?


It is trivial to see that countable spaces satisfy this property, due to the fact that measures are countably additive. From the discussion in [this MSE post](https://math.stackexchange.com/questions/3182794/every-uncountable-compact-uniform-space-has-a-nonatomic-measure?fbclid=IwAR3IEwV0SvqUY4uNDBlZPCDBRFp7iJqiG400YOf31irdLqWNkAjHCkp6tnc), it seems that $\omega\_1+1$ does indeed have the above property. However, $\omega\_1+1$ is not metrizable. I would like to know if there are counter-examples coming from metrizable spaces and, in the case that such examples exist, I would also be interested in the same question if we restrict our attention to zero dimensional compact metrizable spaces (i.e. having a basis consisting of clopen sets).


Any comments or references are appreciated. Also, since I'm new here it could be that this is too elementary for an MO post. If that's the case I will migrate this question to MSE.



The class of spaces where every Borel measure is atomic
Yes. Every uncountable Polish space is isomorphic as a measurable space to the unit interval by Kuratowski's isomorphism theorem and admits, therefore, a nonatomic probability measure. On a Polish space, every finite Borel measure is regular.


