For a Polish space $X$, let $K(X)$ be the set of compact subsets of $X$. Given the topology with basis $\{K\in K(X):K\subset U_0, K\cap U_1\neq\emptyset,\ldots,K\cap U_n\neq\emptyset\}$ for open sets $U_i$, $K(X)$ is also Polish. A compatible metric is the Hausdorff metric
I was wondering if in cases where there is a natural measure on $X$, say Lebesgue measure on $[0,1]$ or the uniform measure on $2^{\omega}$, if there is a natural measure on $K(X)$? By natural I suppose I mean that it can somehow be described in terms of the measure on $X$.
