Let $X_n$ be a random variable, says $X_n:\Omega\to \Bbb Z$ for concreteness. Let $f:\Bbb Z\to B$ be a Borel measurable function into some Banach space $B$. What would be a good mental picture when I think about $f(X_n)$?
More specifically, I am interested in the case when $\{X_n\}_{n=0}^{\infty}$ is a Markov chain or $\{f(X_n)\}_{n=0}^{\infty}$ a martingale.
Technically, I mean of course we can simply say that $f(X_n)$ is just a sequence of measurable functions from $\Omega$ to $B$, but it is just painful to imagine it as a function on $\Omega$ since $\Omega$ is usually implicit anyway. It seems somehow unnatural to me.