I wonder if the following situations have already been considered by mathematicians :
Random variables with values in a set of binary operations endowed with a certain topology (or just with a $\sigma$-algebra structure). For example, let $X$ be a set of cardinality at least 3 endowed with a $\sigma$-algebra structure, and $\{ \otimes ^i \}_{i \in I}$ be commutative (closed and total) binary operations on $X$. We give $I^2$ a topology whose subbase is $\{ U_{a,b,c} \}_{a,b,c \in X}$, where $U_{a,b,c} = \{ (i,j) \in I^2 / (a \otimes ^i b) \otimes ^j c = (a \otimes ^j b) \otimes ^i c \}$. Then we give $I$ one of the 2 topologies :
1) the final topology with respect to the projection onto the first factor $\pi_1 : I^2 \to I$ ;
2) the initial topology with respect to the family of injections $\alpha_j : I \to I^2$ for $j \in I$, where $\alpha_j(i) = (i,j)$.
Suppose that the map $(a,b,i) \to a \otimes ^i b$ from $X^2 \times I$ to $X$ is measurable (with respect to the product $\sigma$-algebra structure).
Then if $a_1, \cdots, a_n \in X$, and $X_1, \cdots, X_{n-1}$ are random variables with values in $I$, we can try (under certain conditions) to compute the distribution law of the random variable $Y_n = (\cdots ((a_1 \otimes ^{X_1} a_2) \otimes^{X_2} a_3) \cdots ) \otimes ^{X_{n-1}} a_n$, and study some type of convergence of the sequence $Y_n$ when $n \to \infty$.
- Random variables with values in a space of topologies of a certain set $X$ (i.e given an adequate topology, or just a $\sigma$-algebra structure). For example, we can endow a (sub)set of wild topologies $E$ of $\mathbb{C}$ such for for all $T \in E$, $\overline{\mathbb{D}}$ is compact in $E$ (by $\overline{\mathbb{D}}$, I mean the closed unit (euclidian) disk), with a topology whose subbase is $\{ U_{a,b} \}_{a \neq b \in \mathbb{C}}$, where $U_{a,b} = \{$ the subset of topologies that separate $a$ and $b$}. Then, given a random variable $Y$ with values in $E$, what is the distribution law of $Y' = $ minimal number of strict open sets for the topoloy $Y$ that cover $\overline{\mathbb{D}}$ (0 if there is no covering of $\overline{\mathbb{D}}$ by strict open sets).
These are random examples that I came up with to illustrate my question.
Thanks.
Edit :
@NateEldredge : I only have an intuition from physics. For the first point, the elements of $X$ can be certain sets of particles. $\{ \otimes ^i \}_{i \in I}$ are the possible interactions (possibly interactions at a distance) between sets of particles, i.e. $a \otimes ^i b$ is the set of particles that arise from the interaction of type $i$ between $a$ and $b$. I want that two couple of interactions $(i,j)$ and $(i',j')$ be close to each other if $(a \otimes ^i b) \otimes ^j c = (a \otimes ^j b) \otimes ^i c$ and $(a \otimes ^{i'} b) \otimes ^{j'} c = (a \otimes ^{j'} b) \otimes ^{i'} c$, so this is why I define $U_{a,b,c}$ in this way. As I am defining it, the topology on $I$ is then a natural one. Finally $(\cdots ((a_1 \otimes ^{X_1} a_2) \otimes^{X_2} a_3) \cdots ) \otimes ^{X_{n-1}} a_n$ is what we get from a finite set of sets of particles when the interactions are random, and the limit $n \to \infty$ is the case when we have a countable set of sets of particles and a countable number of random interactions.
