I have attached to this post a short treatment of the Kolmogorov extension theorem for measures.
In the following, I did not understand what is meant by the $A$ that I circled in red. I suppose that $\mathbb{R}^{\infty}:=\prod^{\infty}_{i=1}\mathbb{R}_i$, but I cannot believe that $A$ is just some subset of this set, i.e. $A$ should be measurable or open or something like that. But on the other hand, I don't if we are talking here about the canonical product topology inducing our notion of measurability on $\mathbb{R}^{\infty}.$ Thus, I am wondering whether anybody could clarify a bit the nature of $A$?
