Let
$$\mathcal{C}(\Bbb R^n) = \{A \subseteq \Bbb R^n \mid A \neq \emptyset, A \textrm{ compact}\}.$$
I am curious about whether there are some notable probability spaces defined on $\mathcal{C}(\Bbb R^n).$ Formally, does there exists a well known (relative to the people in the filed) $\sigma$-algebra $\mathcal{A}$ of $\mathcal{C}(\Bbb R^n)$ and a probability measure $\Bbb P: \mathcal{A} \to [0,1]$ such that $(\mathcal{C}(\Bbb R^n), \mathcal{A}, \Bbb P)$ is a probability space? This sounds like something that is well studied, but I was unable to find on my search. In case it exists, a description of the probability space would be highly appreciated. Otherwise, it will be enough to have any reference in the literature.