We know that each ultrafilter $p$ on $\mathbb{N}$ can be identified with a finitely additive $\{0,1\}$-valued probablity measure $\mu_{p}$ on the power set of $\mathbb{N}$.

Now my question is which conditions should be added to a topological space $(X,T)$ to provide a situation for it's Stone-Chech compactification to identify each ultrafilter $p$ on $X$ with a measure $\mu_{p}$ on the power set of $X$ (or the sigma-algebra of Borel subsets of $X$)?!

Also it's possible for this measure to has some properties.

Rings of Continuous Functions$\endgroup$ – Gerald Edgar Jul 26 '18 at 15:40