Every uncountable compact uniform space has a nonatomic measure?

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 $$\mathcal{M}(X)$$?

$$X=\omega_1+1$$ 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.
• In [Rao, Rao, "Borel $\sigma$-algebra on $[0, \omega_1]$" (1971)] it is proven that on $X = \omega_1 + 1 = [0, \omega_1]$ there is no non-atomic Borel measure. Moreover, every finite positive Borel measure is of the form $c \mu + \tau$ where $\mu$ is the Dieudonné measure and $\tau$ is concentrated on a countable subset of $[0, \omega_1]$. – yada Aug 1 '19 at 7:51