Projective limit of spaces of probability measures

Question

Consider a projective system $\dots X_{n+1} \to X_n \to \dots \to X_1$ of completely regular Hausdorff spaces with projective limit $X$. Then the linking mappings $f_n$ induce a projective system (in the category of sets) of spaces of probability measures $\dots P(X_{n+1}) \to P(X_n) \to \dots \to P(X_1)$ with the canonical pushforward linking mappings $(f_n)_*$. What is the corresponding projective limit? For simplicity, let us first restrict to products $X_n = Y^n$. In general, a compatible system of probability measures on $Y^n$ need not have an extension to a probability measure on $X$, unless $Y$ is say Polish (by the Kolmogorov extension theorem), in which case then the projective limit is precisely $P(X)$. Is a characterization of the projective limit of the $P(X_n)$ known for the more general setup?

Answer 1

Just some night thoughts on your question, but too long for a comment.

1. If all of your $X$’s are compact, then everythig is fine and the desired projective limit is just the family of probability measures on the (compact) projective limit of the $X$’s (I am assuming, by the way that the image of $X_n$ is equal to $X_{n-1}$ for each $n$).

2. Back to the general case—then each probability measure on a component can be regarded as a finitely additive one on the corresponding Stone-Čech compactification. Now these compactifications also form a projective system and so have a compact space $\hat X$ as limit. An element in your projective limit determines a thread in the system of compactifications and so a probability measure on $\hat X$.

3. Hence your space can be identified as a space of probabilities on $\hat X$.

4. The question is how to identify just which space this is. At this point, the answer depends on the point that I raised in my comment. You would require conditions on a probability on $\hat X$ which ensure that its images in the component spaces satisfy the regularity conditions you are interested in.

5. I will conclude with the remark that there are known conditions for probabilities on a Stone-Čech compactification to correspond to $\sigma$-additive, $\tau$-additive or tight measures on the underlying completely regular space.

Not an answer but I hope it helps.