= Hall matching of maximal separated sets
{c}
Two maximum-<cardinality> $\varepsilon$-separated subsets $A,B$ of a <compact metric space> have a bijection that moves each point by less than $\varepsilon$. Join $a$ to $b$ when their distance is below $\varepsilon$. If $S\subseteq A$ had fewer than $|S|$ neighbours, replacing these neighbours in $B$ by $S$ would create a larger separated set. The <Hall marriage theorem> therefore supplies a <perfect matching>. This allows empirical averages on a compact <isometry group> to be made approximately invariant without assuming a prior <Haar measure> construction.
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