Questions about the intuition on or physical meaning of probability space, random number, and stochastic process I have a question about the physical meaning of, or the mathematical intuition on, the mathematical concept of probability space, random number, and stochastic process. Mainly, I would like to ask about the exact counterpart of the events, random numbers, and stochastic processes in particular examples.
The book I first encountered those concepts (the theory of open quantum systems by H.-P. Breuer et al.) states that a random number is a map $X: \Omega\times T\rightarrow R$ with respect to the probability space $(\Omega, A, \mu)$, and the probability distribution is defined by $P(x)=\mu(X(t)^{-1}(x))$.
I am having trouble matching this definition with the physical stochastic process. The problem is because, according to the above definition, the probability of events($\mu(B), B\in\Omega$) does not vary in time, but instead, the map from the probability space to $R$ varies in time. I first thought that microstates with equal a priori probability should be events, but I am having difficulty figuring out what " events " should be in some examples. For instance, if I think of a taxi driver randomly picking up a passenger, then obviously, the number of passengers should be a random number, but I don't know what should be corresponding events. Could someone please help me understand the concept?
Edit))
Symbol Definitions
$\Omega$ : sample space
$A$ : $\sigma$-algebra on $\Omega$
$\mu$ : Probability measure
 A: Perfect knowledge of the state of a system at one time determines what it will do later. In your thermodynamic example, you can take $\Omega$ to be the set of possible states of the system at a chosen fixed time, and $\mu$ determines the probability distribution of those states at that time. These data do not change with time, but they still determine the values and probability distributions of observations, which do vary with time.
In the case of the taxi driver, you have do something similar: an outcome in $\Omega$ represents the state of the entire world at a chosen time $t_0$, including the driver, passengers, and any factors that might later affect when passengers get on or off the taxi. $X(x,t)$ then predicts the number of passengers in the taxi at time $t$ by running the state at $t_0$ forward (or backwards) in time. The intuition in both the thermodynamic and macroscopic cases is that $\mu$ represents ignorance of a "real, complete" condition of the world, which deterministically produces all later (and previous!) observations. This is in contrast to a picture where the initial condition might be known but the evolution can be nondeterministic.
