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Given a simple Lie group $G$, we show that the lattices in $G$ are weaklyuniformly discrete. This is a strengthening of the Kazhdan-Margulis theorem.Our proof however is straightforward --- considering general IRS rather thanlattices allows us to apply a compactness argument. In terms of p.m.p. actions,we show that for every $\epsilon$ there is an identity neighbourhood $U$ whichintersects trivially the stabilizers of $1-\epsilon$ of the points in everynon-atomic $G$-space.

Kazhdan-Margulis theorem for Invariant Random Subgroups