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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.

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    $\begingroup$ 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. $\endgroup$
    – Gerald Edgar
    Jul 26, 2018 at 15:26
  • $\begingroup$ @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?! $\endgroup$
    – mahdi meisami
    Jul 26, 2018 at 15:34
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    $\begingroup$ This great book that every student of analysis should read ... L. Gillman & M. Jerison, Rings of Continuous Functions $\endgroup$
    – Gerald Edgar
    Jul 26, 2018 at 15:40
  • $\begingroup$ @GeraldEdgar Thanks for your suggestion $\endgroup$
    – mahdi meisami
    Jul 26, 2018 at 15:41
  • $\begingroup$ @GeraldEdgar what will be happen if we change to this: which conditions on Ston-Chech compatification of $X$ achieved from ultrafilter.? $\endgroup$
    – mahdi meisami
    Jul 26, 2018 at 16:46

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