Invariant random subgroup version of fundamental theorem of group actions

Definition
Suppose $$G$$ is a locally compact group and $$H$$ is a random subgroup of $$G$$. The following are equivalent:


 * 1) $$H$$ is an invariant random subgroup of $$G$$.
 * 2) $$H$$ is the stabilizer of a random point for some measure-preserving action of $$G$$ on some measure space.

Related facts

 * Fundamental theorem of group actions