Proposition 7. Assuming that:

$t$ , $α$ and $G$ are such that for any Lipschitz function $f : G \to ℝ$ with median $M$ we have
$\frac{| {x \in G : | f (x) - M | > t} |}{| G |} \leq α,$

Then for all

A \subseteq G

with

\frac{| A |}{| G |} \geq \frac{1}{2}

, we have

\frac{| A_{(t)} |}{| G |} \geq 1 - α