I am reading Chapter 7 of the book
Zhikov, V. V.; Kozlov, S. M.; Olejnik, O. A., Homogenization of differential operators and integral functionals. Transl. from the Russian by G. A. Yosifian, Berlin: Springer-Verlag. xi, 570 p. (1994). ZBL0838.35001.
There in the introduction of said chapter, on a probability space $(\Omega,\mathcal F,\mathbb P)$, a stochastic process $(\xi_t)_{t \in \mathbb R}$ is said to be (strictly) stationary if for any finite collection of points $t_1,\dots, t_k \in \mathbb R$ we have the joint distribution of $(\xi_{t_1+h},\cdots,\xi_{t_d+h})$ is independent of $h \in \mathbb R$. In the book the authors actually consider processes indexed by vectors $x \in \mathbb R^m$, but I assume there is no significant difference for the purpose of this question.
The authors quickly go to instead consider processes taking the form $\xi_t(\omega) = X(T(t)\omega)$, where $X$ is a random variable and $T$ is a so-called dynamical system (in particular it is measure preserving and satisfies a group property). They remark that such an $a$ and $T$ exists subject to "natural conditions such as stochastic continuity and separability," but provide no further details.
Question: How can the above remark be made precise? Is there a reference where this is proven in detail?
Note: In this book of Berlyand & Rybalko, in Section 8.1 the authors make the same comment, and cite the classical text of Doob. There a notion of separability is defined in Chapter 2, and in Chapter 10 a construction is sketched for discrete processes Additionally in Chapter 11 it is claimed that an analogous construction works in the continuum case, but I don't see where continuity and separability assumptions come into play, and I could not find any further results in the text that seemed relevant to my question.
Motivation: In stochastic homogenisation one studied random coefficient fields which are assumed to be stationary in some sense, but many authors seem to be relatively imprecise with what they mean. I am looking to understand what the different notions are and how they differ.
Edited to add: Reading [ZKO] further, I am unable to follow their proof that a dynamical system always satisfies the $L^2$ stochastic continuity property $$ \lim_{x \to 0} \lVert f(T(x)\,\cdot) -f \rVert_{L^2(\Omega)} = 0$$ for all $f \in L^2(\Omega)$. The authors appeal to "continuity of translations in $L^2_{\mathrm{loc}}(\mathbb R^d)$ to claim that $$ \lim_{x \to 0} \int_{B_1} \lVert f(T(x+y)\omega) - f(T(y)\omega)\rVert^2 \,\mathrm{d}y $$ $\mathbb P$-almost surely in $\omega$. Unless I have missed something, this does not appear to follow from the definition of measure preserving functions given in Section 7.1 of the text.
The reason I add this is because I suspect it is actually sufficient to assume a strictly stationary process is separable to represent it using a transformation $T$. This seems to involve working in the space of functions $\mathbb R \to \mathbb R$ such as the coordinate evaluation functions $f \mapsto f(t)$ corresponds to $\xi_t$. Then it seems plausible the separability condition is simply to ensure suitable measurability conditions hold true, but I would be interested in a reference that details this construction (and that confirms my above suspicion if it's correct).