I'm trying to define a set-valued Random Variable (RV) whose set values are not real points. Instead, I'm trying to define something more general:
Definition (Random Set): A Random Set is a set-valued RV, i.e. a map $W:\Omega\to\mathcal{C}$ from a probability space $(\Omega,\Lambda,P)$ to the family of closed sets $\mathcal{C}\ni W_1,\dots,W_n$ generating a $\sigma-$algebra $\Lambda$ from the elementals $\omega\in\Omega$, and such RV takes set values:
$$W^{-1}(K)=\{ \omega|W(\omega)\cap K\neq\emptyset \},$$
where $K$ is a closet set called "trap" set.
My doubt with this definition is that all defintions I've found are similar except by the fact that $\mathcal{C}$ is a family of measurable locally compact second countable Hausdorff (LCSCH) sets. I'm not introucing such kind of mathematical structure in the above definition, being thus much more general. So I'm, not sure wheter we actually can define an RV in that manner, or what is the minimun needed structure to define RVs for $\mathcal{C}\ni W_i$, where $W_i$s are simple closed sets (e.g. $\omega$s can be letters of the DNA code, so $W_i$s can be codons, genes, etc). Molchanov mentions it, but he does not provide a definition.
Thank you
What you describe is what is called a closed-valued measurable correspondence in [Aliprantis & Border 2006, Infinite Dimensional Analysis]. There are a lot of results available for such correspondences that do not require any local compactness assumption, though one usually takes values in Polish spaces. One of the most important results in this setting is the Kuratowski - Ryll-Nardzewski measurable selection theorem. Note that the measurability assumption in that theorem is weaker than the one you use; see Lemma 18.2 in the book of Aliprantis and Border.