Kazhdan-Margulis theorem for invariant random subgroups

Given a simple Lie group G, we show that the lattices in G are weakly uniformly discrete. This is a strengthening of the Kazhdan-Margulis theorem. Our proof however is straightforward - considering general IRS rather than lattices allows us to apply a compactness argument. In terms of p.m.p. actions, we show that for every ϵ>0 there is an identity neighbourhood Uϵ⊂G which intersects trivially the stabilizers of 1−ϵ of the points in every non-atomic probability G-space.