How to implement conditional probability distribution on set-valued Random variables? A Random Set is a set-valued RV, i.e. a map $X:\Omega\to\mathcal{C}$ from a probability space $(\Omega,\Lambda,P)$ to the family of measurable closed sets $\mathcal{C}$ on a $\sigma-$algebra $\Lambda$ built from the elementals $\omega\in\Omega$ (with $\Omega$ countable), and such RV takes set values:
$$X^{-1}(K)=\{ \omega|X(\omega)\cap K\neq\emptyset \},$$
where $K$ is a closet set called the "trap" or structuring set. For simplicity, I denoted $X^{-1}(K)\in\mathcal{X}$ by $x_i$, and $Y^{-1}(K)\in\mathcal{Y}$ by $y_j$, and $X(\omega)$ by $X$.
I'm trying to implement conditional probability distribution $Y|X$. So, for $Y=y_j$, given $X=x_i$, we can use the Bayes theorem:
$$P(Y=y_j|X=x_i)=\frac{P(X=x_i|Y=y_j)P(Y=y_j)}{\sum_{y_j'\in\mathcal{Y}}P(X=x_i|Y=y_j')P(Y=y_j')}=\frac{f(x_i,y_j)}{\sum_{y_j'\in\mathcal{Y}}f(x_i,y_j')},$$
where the question is, how to define $f(x_i,y_j)$ as $X,Y$ take values on the sets of sets $\mathcal{X}=\{x_1,\dots,x_\ell\}$ and $\mathcal{Y}=\{y_1,\dots,y_{\ell'} \}$, respectively?
I've tried with this:
$$(1)\hspace{10mm}f(x_i,y_j)=\sum_{(x_i',y_j')} |x_i\cap x_i' |\cdot|y_j\cap y_j'|. $$
However, I'm not sure how to verify here the usual notion of quatifying the observation of the outcome $(x_i,y_j)$ over the set of outcomes $\{ (x_i', y_j')\} $. Please, someone can help me to get the right expression for Eq. (1)?
 A: Since the expressions for conditional probability involves a sum to marginalise the joint probabilities, you will need to specify the support of the random sets you are dealing with, which will require you to refer to classes of sets (i.e., sets of sets).  Thus, the first thing that will be useful here is to use standard fonts for elements, sets, and classes.  For example, you might denote a specific element value by $x$, a set of these values by $\mathcal{X}$, and a class of these sets by $\mathscr{X}$.  Once you are using notation that clearly distinguishes between these things, it is fairly simple to formulate the conditional probability formula for random sets.
For simplicity, I will consider the case where your random sets are discrete random variables (i.e., both have countable support).  Consider the case where we have two random sets $\mathcal{X} \in \mathscr{X}$ and $\mathcal{Y} \in \mathscr{Y}$ where the latter classes are their respective supports.  Then the rule of conditional probability says that:
$$f(\mathcal{Y}| \mathcal{X}) = \frac{f(\mathcal{X}, \mathcal{Y})}{\sum_{\mathcal{Y} \in \mathscr{Y}} f(\mathcal{X}, \mathcal{Y})}.$$
Without specifying how these random sets are constructed from underlying random elements, that is really as far as you can go.  If you are willing to specify how random sets are constructed from some underlying random elements then you might be able to split the term $f(\mathcal{X}, \mathcal{Y})$ into a product of probabilities for those random elements.  At the moment you appear to be trying to split this term out into a sum, but you are trying to do this without giving any information on how the random sets are consructed; consequently, your attempts are all abortive.
