# Identification of ultrafilters with measures

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.

• Needs more explanation. Does the definition of "ultrafilter" on a topological space have something to do with the topology? There is a measure characterization of realcompact spaces that may (or may not) be what you want here. – Gerald Edgar Jul 26 '18 at 15:26
• @GeraldEdgar I have encountered with this question in reading of Vitaly Bergelson survey on Ergodic Ramsey theory during the process of defining Stone-Cech compactification. Where can i find what you said at the end of your comment?! – mahdi meisami Jul 26 '18 at 15:34
• This great book that every student of analysis should read ... L. Gillman & M. Jerison, Rings of Continuous Functions – Gerald Edgar Jul 26 '18 at 15:40
• @GeraldEdgar Thanks for your suggestion – mahdi meisami Jul 26 '18 at 15:41
• @GeraldEdgar what will be happen if we change to this: which conditions on Ston-Chech compatification of $X$ achieved from ultrafilter.? – mahdi meisami Jul 26 '18 at 16:46