Let $X,Y: \Omega \rightarrow S$ be two random variables. Now, I was wondering what exactly does it mean that $X,Y$ are identically distributed? Does this mean that $P(X=Y)=1$? Identically distributed random variables are apparently pretty common in probability theory, but I could not find a definition for general $S.$
My intuition tells me that $P(X=Y)=1$ is an even stronger condition, but I am not sure about this.