TITLE: Two stochastically continuous processes with the same finite dimensional distribution on a dense setvhave the same fdd everywhere?
QUESTION [1 upvotes]: Two stochastically continuous  processes on $[0,T]$ with the same finite dimensional distribution on a dense subset of $[0,T]$ have the same finite dimensional everywhere?
The processes live on different spaces.
I suspect that this is true since Skorokhod seems to be using this in his book "Studies in the Theory of Random Processes" which I am self studying . Can somebody point me to some references where I could find a proof or give me a hint. 
Thank you

REPLY [1 votes]: If $X$ and $Y$ are processes agreeing on a dense set $S$, then
$$ \mathbf{P}(X_t < A) = \mathbf{P}( \limsup_{s < t} X_s < A) = \limsup_{s < t} \mathbf{P}(X_s < A) = \limsup_{s< t} \mathbf{P}(Y_s < A) = \mathbf{P}(Y_t < A) $$
The biggest jump in the calculation is moving the limsip out of the probability, but this follows by continuity, because $X_t$ is less than $A$ only when the $X_s$ to the left are eventually less than $A$. This idea can easily be generalized to finite sequences of random variables, so all f.d.d. agree.
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