Let $\mu$ be a probability measure on a **metric** space $M$ (with the Borel $\sigma$-algebra). If $\mu(A)\in \{0,1\}$ for all measurable set $A\subset M$, then:

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> Is it true that $\mu$ is a Dirac measure?
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I think the answer should be negative, but I do not know of a counterexample. There is an answer [here](https://math.stackexchange.com/questions/291501/two-valued-measure-is-a-dirac-measure) for the general case where we don't know anything about the topology (metrizable in our case). There you can see also that the result holds for all Polish spaces.