I am reading this article and in the second page I am not being able to understand the argument.
In essence we have two random process $(X_t)_{t > 0}$, $(Y_t)_{t > 0}$, such that both have the same finite dimensional distributions. We are interested in the probability of the event $A:=\{$ path of $(Y_{t_i})_{i \in\mathbb{N}}$ is uniformly continuous $\}$. It is mentioned that this event depends only on countably many random variables $(Y_{t_i})_{i \in\mathbb{N}}$, and belongs to the cylindrical $\sigma$ field of $(X_t)_{t > 0}$.
If we let $B:=\{$ path of $(X_{t_i})_{i \in\mathbb{N}}$ is uniformly continuous $\}$, then it apparently follows that $P(A)=P(B)$. Could someone help me understand why is this claim is true and why does it belong to the $\sigma$ field? Is this the consequence of some well known theorem? I have read the pages on Wikipedia about cylinder sets and cylinder $\sigma$ algebras but still remain quite confused.
Thanks in advance.