Can every stochastic $(\psi_n)$ process be mimicked by a measure preserving dyamics $f$ and a observable $\psi$, giving $\psi_n = \psi \circ f^n$?

Question

Let $(X, \mathcal{A}, \mu)$ be a measure space and $(\psi_n)_{n \ge 0}$ a sequence of $\bar{\mathbb{R}}$ valued measurable functions on $X$. Is it always possible to equip the measure space with a measurable and measure-preserving dynamical system $f: X \rightarrow X$ and measurable observable $\psi: X \rightarrow \bar{\mathbb{R}}$ such that $\psi_n = \psi \circ f^n, $\mu$-a.e., \forall n \ge 0$? Thanks :)

Answer 1

How about if we allow a different measure space?
Let the measure space be $\Omega = \overline{\mathbb R}^\infty$, the map $f =$ left shift on $\Omega$, map $\psi : \Omega \to \overline{\mathbb R}$ selects the first coordinate. The measure on $\Omega$ is made from the joint distributions of the random variables $\psi_n$. In fact there is a canonical measure-preserving map $X \to \Omega$ onto a subset of (outer) measure $1$ by $x \mapsto (\psi_n)_n$. The difference from what is asked is that the left shift acts on $\Omega$ not on $X$.