Amenable Invariant Random Subgroups

We show that an amenable Invariant Random Subgroup of a locally compact second countable group lives in the amenable radical. This answers a question raised in the introduction of the paper "Kesten's Theorem for Invariant Random Subgroup" by Abert, Glasner and Virag. We also consider, in the opposite direction, property (T), and prove a similar statement for this property. The Appendix by Phillip Wesolek proves that the set of amenable subgroups is a Borel subset in the Chabauty topology.