Probability on Topological Space

Question

Suppose you have a topological space X, assuming it is Hausdorff, compact, connected space. Is it possible to equip it with probability measure? I am curious if one could create probabilistic topological space whose points and open sets encode some information about probability distribution and measure.

Answer 1

I think you can find an answer to your question here: Chapter 2 - Probability measures on topological spaces, by G. Kallianpur, ‎1995. Unfortunately, I do not know the title of the book.

For measures (not necessarily probability measures) we can always consider the Borel $\sigma$-algebra $\mathcal{B}(X)$ on the topological space $X$, and then consider on it some measure.

Answer 2

The Dirac measure $\mu_x$ always works for any $x \in X$: $\mu_x(A)=0$ if $x \notin A$, $\mu_x(A)=1$ if $x \in A$. It "measures" all sets and is a probability measure. It has no relation at all with the topology on $X$.

There is a whole subfield of topology/measure theory interactions (see Fremlin's books on measure theory, one book out of the five is about that). This is mostly about Borel or Baire measures with different varieties of "regularity" constraints (like being able to approximate measures of sets by measures of open sets or measures of compact sets), or finding topological constraints on spaces that follow from having a certain type of Borel measure defined on it. It's a large field (though somewhat obscure); like I said check out Fremlin volume 4, found here (You can also buy it as a book, but my shelf space is limited so I only know the digital versions.)

Answer 3

One branch of mathematics where probability measures on topological spaces receive a lot of attention is known as topological dynamics, and particularly the sub-branch of topological dynamics concerned with ergodic theory.

Such measures come up in the following context: for every compact metrizable space $X$ and every continuous self-map $f : X \to X$ there exists a Borel probability measure $\mu$ on $X$ which is $f$-invariant in the sense $\int_{f^{-1}(A)} \mu = \int_A \mu$ for every Borel subset $A$. The set of all invariant probability measures for $f$ then has the structure of a generalized kind of simplex known as a Choquet simplex. The "vertices" of this simplex are exactly the ergodic invariant probability measures for $f$ (almost by definition). Every other $f$-invariant probably measure is a kind of convex sum (better: convex integral) of ergodic measures, similar to how every point in a finite dimensional simplex is a convex sum of the vertices.

When you apply this to the identity map then you get an interesting situation, if somewhat tautological: the Choquet simplex structure on the set of all Borel probability measures on $X$ has "vertices" which are exactly the Dirac $\delta$-masses of the individual points of $X$. Every other Borel probability measure is a kind of convex integral of these Dirac masses.