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It is known that the class of nonatomic measures on compact metric space without isolated point $X$ is dense in $\mathcal{M}(X)$, where $\mathcal{M}(X)$ is the set of Borel probability measures endowed with the weak$^*$ topology.

In my research, $X$ is a compact uniform space. Is it true that the class of non-atomic measures on compact uniform $X$ is dense in $\mathcal{M}(X)$?

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$X=\omega_1+1$ is an uncountable compact uniform space. I don't know of any non-atomic Borel measure defined on $X$. The standard Dieudonné measure is not. Ulam proved there cannot be a total non-atomic measure. So I think this $X$ will give you a counterexample.

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