Unimodularity of Invariant Random Subgroups

An invariant random subgroup H≤G is a random closed subgroup whose law is invariant to conjugation by all elements of G. When G is locally compact and second countable, we show that for every invariant random subgroup H≤G there almost surely exists an invariant measure on G/H. Equivalently, the modular function of H is almost surely equal to the modular function of G, restricted to H. We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact invariant random subgroups