first time asked a question here. I am reading Gray [1], Kallenberg [2] and the excellent lecture notes on stochastic processes [3]. All three define stochastic processes in general as a family $\{ X_t \}_{t\in\mathcal{I}}$ of random variables $X_t$ defined on a probability space $(\Omega, \mathcal{F}, P)$. I am interested in the case where the process is (strictly) stationary. Specifically when all finite distributions are invariant to shifts for $t_i,\tau \in \mathcal{I}$. $$P(X_{t_1} \in A_1,...,X_{t_n}) = P(X_{t_1+\tau} \in A_1,...,X_{t_n+\tau}) $$
Kallenberg provides a proof [2, Lemma 25.1] of the existence of a measure preserving transformation $T$. However this proof and the proof [3, Theorem 52]. These proofs are demonstrated when the index set is the set of integers -- or more specifically positive integers.
Is there are reference that has a similar proof for continuous-time signals? Specifically I would expect that for any stationary continuous-time process there should exist a measure-preserving flow $\{T_s\}_{s \in \mathbb{R}}$ such that the random process at time $t$ can be written as $X_t = f \circ T_t$ where $f$ is a measurable function and $T_t$ is a measure-preserving map on $\Omega$.
[1] Gray, Robert M. Probability, Random Processes, and Ergodic Properties. Boston, MA: Springer US, 2009. https://doi.org/10.1007/978-1-4419-1090-5.
[2] Kallenberg, Olav. Foundations of Modern Probability. Vol. 99. Probability Theory and Stochastic Modelling. Cham: Springer International Publishing, 2021. https://doi.org/10.1007/978-3-030-61871-1.
[3] Rohilla Shalizi, Cosma, and Aryeh Kontorovich. Almost None of the Theory of Stochastic Processes, 2007.