I am following this source: http://www.hss.caltech.edu/~kcb/Notes/Kolmogorov.pdf and agree with everything done in sections 1-3.
In section 4, I cannot fill in the detail for for Lemma 4, because I cannot deal with the finite unions.
Also, I tried to apply the result of section 3 directly to $R^\mathbb{N}$ but I could not find a compact class with the desired property. I tried the set of finite-dimensional cylinders where the nontrivial section is compact, but I just can't see that this is a compact class. (I.e. sets of the form $K \times R \times R...$ where $K$ is a compact subset of some $R^n$.)
For some sort of self-containedness, a compact class is one where every sequence in the class has the property that all finite intersections being nonempty implies the whole intersection of the sequence is nonempty.