Measures on hyperspaces

Abstract:

From a measure space, $(X, \mu ^{\mathbb {X}})$ we define a measure $\mu ^{\mathbb {P}(\mathbb {X})}$ on the power set of $X$. If $(X, \tau )$ is a compactum, whose topology $\tau$ is compatible with the measure $\mu ^{\mathbb {X}}$ on $X$, then the measure $\mu ^{\mathbb {P}(\mathbb {X})}$ restricts to a natural measure on the hyperspace of closed sets of that given compactum. Surprisingly, under very mild conditions, $\mu ^{\mathbb {P}(\mathbb {X})}$ is always supported on the hyperspace of finite subsets.