I am quite familiar with Ergodic Theory from the Math point of view, that is, when you have a dynamical system $f:M \to M$ with a measure which is preserved by $f$ and where $M$ is some measure space with "nice" properties depending on the context. So, now I want to understand everything more from the Probabilistic point of view.
To achieve this, I have been studying the basic concepts from that theory (sample space, events, random variables, stochastic processes, and so on), but there are a couple of things that I can't connect to the dynamical systems context. For instance,
(1) what would an example be of a dynamical system from the Prob. point of view? Is it a stochastic process , with $t \in K$, where the dynamics is given by the action of $K$? ($K$ might be $Z$ or $R$ for example). If so, what would the equivalent of M be in this context and the corresponding $f$? (I'm trying to get the equivalent concepts as in the dynamical systems setting with $f: M \to M$), what are the initial conditions here? (points that belong to $M$)
(2) If I finally have that function describing the dynamics (from the Prob point of view), what does a measure that is preserved mean? From Physics point of view, a measure that is preserved makes sense if we want to think as the mass is preserved under the dynamics, but what would an explanation of that kind of measure be be in the Prob point of view?
Hope I have been clear in my questions and if not please let me know so I can reformulate in some other way.
Thank you!