Measures on hyperspaces

From a measure space, ( X , μ X ) (X, \mu ^{\mathbb {X}}) we define a measure μ P ( X ) \mu ^{\mathbb {P}(\mathbb {X})} on the power set of X X . If ( X , τ ) (X, \tau ) is a compactum, whose topology τ \tau is compatible with the measure μ X \mu ^{\mathbb {X}} on X X , then the measure μ P ( X ) \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, μ P ( X ) \mu ^{\mathbb {P}(\mathbb {X})} is always supported on the hyperspace of finite subsets.