Example of Non-separable stochastic process. This question is related to the link:
https://www.encyclopediaofmath.org/index.php/Separable_process
The link provided a basic definition of separable Stochastic process. I felt all the process under study seems to be separable. Is it true? Is there any example of  Non-separable stochastic process which is of some importance?
 A: Let's consider the following setting: $([0,1],\mathscr L, \mu_L)$ with Lebesgue Set $\mathscr L$ and $t\in[0,1]$. Define the stochastic process as$Y_t=\mathbb 1_{\{\omega\}}(t)$, where $\{\omega\}$ is a singleton. This process just take value 1 at $t=\omega$.(for example one possible sample path is: For $\omega=1$. $Y_t(\omega)=1$ if $t=1$ and $Y_t(\omega)=0$ if $t \neq 1$).
Now we want to show it is not separable:
Consider the close set $A=\{1\}$, the set $\{\omega\in\Omega,Y_t(\omega)\in \{1\}\}$ becomes $\{\omega \in \Omega ,\omega=t\}$ and the union $ \cup_{t\in T\cap O}\{\omega\in\Omega,Y_t(\omega)\in \{1\}\}$ for some open interval $O$ is just the set ${T\cap O}$. Similar for a countable set $S_1,S_2,...\subset T$. The union $ \cup_{t\in (S_j)_{j\in\mathbb N} \cap O}\{\omega\in\Omega,Y_t(\omega)\in \{1\}\}$ is just $(S_j)_{j\in\mathbb N}\cap O$. The difference between the two set is not a subset of null set.
The advantage that we are dealing with separable process is that we can operate with countable operations (Union and common sigma-Algebra operations). This simplify our problem for stochastic process in continous time a lot. There is a theorem by Loeve, that under certain condition one can always find a equivalent seperable process.
A: Could you show me the reference of any process that is right continuous and has left limits is separable? I now have a continuous stochastic process, and want to show it is separable. It seems to be true but I couldn't find a reference or proof of it. Thanks.
Tao
A: I know the question is very old, but I wanted to share one nice example of why the separability assumption may be too strong in some cases.
The example is a very natural "zeros of Brownian motion" process. Here is an excerpt from Claude Dellacherie, Paul-André Meyer, Probabilities and Potential (1978), page 96 (see also page 85):

As the authors mention, the example is

...by no means "artificial": it has been studied by Paul Lévy in a series of works which are considered masterpieces of probability theory.

A: Any process that is right continuous and has left limits is separable,
the poisson jump process is also separable. Just notice that you can know the value of $X(t,\omega)$ by looking at a sequence of rational numbers $q_n \downarrow t$.
To get a non separable process you need to consider a process that has more irregularity, say that is not right continuous. This might occur if you have a subset that is considered Bad and you tele-transport it to infinity as it enters the Bad region. This often happens when we are studying meta-stability, but still you can modify your process (ignoring the times you spent in the bad region - this is called the trace of the process).
You could also find a non separable process if it took value on a non separable space (it can't be a real valued process )
One often deals with separable process since it is a good property when we are dealing with denumerable set operations (as it is the case for a $\sigma$-algebra).
