In a nutshell
Are there CDFs for infinite sequences of random variables with similar properties to the CDFs for finite-dimensional random vectors?
In more detail
Every random variable has a corresponding CDF. The same applies to (finite dimensional) random vectors. The CDF of a RV uniquely determines the RV's distribution. Conversely, it is possible to establish a set of criteria irrespective of an underlying RV in such a way that every CDF associated with a given RV satisfies the criteria and conversely to every function F satisfying the criteria there corresponds a RV whose CDF is F.
Does the same apply in the case of infinite dimensions? Given an infinite sequence of random variables (defined on the same probability space), is there a CDF associated with this sequence that uniquely determines the joint distribution? If so, is it possible to list a set of criteria which every CDF arising from an infinite sequence of RVs satisfies and such that every function F that satisfies these criteria can be associated with an infinite dimensional sequence of RVs whose CDF is F?