Let $I$ be a (possibly infinite) set. To each finite subset $J$ of $I$, we associate a joint probability distribution $X_J$ whose sample space is $\mathbb R^J$ (i.e. the outcomes are $|J|$-tuples). Moreover, if $J' \subseteq J \subseteq I$, then $X_{J'}$ and $X_J$ are compatible in the sense that if $F \sim X_{J}$, then $F {\restriction_{J'}} \sim X_{J'}$. I have two question:
- Is there a probability distribution $X_I$ over $\mathbb R^I$ such that $F \sim X_I$ implies $F {\restriction_{J}} \sim X_{J}$?
- Is this $X_I$ unique?
Essentially, we are trying to describe the probability distribution of $|I|$ random variables in terms of the distributions on finite sets of those variables.
This appears to be some sort of inverse limit, or maybe has something to do with Sheafs, but I do not really know the details in this context.
Kolmogorov's Extension Theorem deals with the existence of the product measure on $\mathbb{R}^\mathbb{N}$, which is built from a sequence of consistent, finite-dimensional measures. See Theorem 2.1.21 in the former reference or Theorem 9.1 in the latter.
https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf http://bass.math.uconn.edu/math5160f14/post9.pdf
Does this answer your question?