We have a function $D(Z;\omega):\mathcal{Z}\times\Omega\rightarrow\mathbb{R}$. $\mathcal{Z}$ is Riesz space, $\Omega$ is sample space. For each given $\omega\in\Omega$, $z$ satisfies the partial order property (this is why they lie in Riesz space). I was wondering how to understand the sampling procedure of $D(Z;\omega)$ from measure-theoretic perspective?
Starting from a simple version, if we don't have $Z$, the function $D(\omega):\Omega\rightarrow\mathbb{R}$ is a random variable. We can characterize its sampling process using its probability distribution. But how to generalize this notion to $D(Z;\omega)$?
I am not quite familiar with functional analysis (Some terminology may be used incorrectly). That would be great if someone could recommend some books or materials at any level. Thanks in advance!
Updated: I realize that it is a stochastic process in which we replace $t$ with $Z$.