Minimal left covering net of a compact group ($G=FV,|F|=\min \{|E|:G=EV\}$)

For an open identity neighbourhood $V$ in a compact topological group, a finite left covering net is a finite set $F$ such that the left translates $fV$, $f\in F$, cover the group. Compactness gives such finite sets; minimum cardinality gives a minimal one. Left translation preserves minimality because $G=FV$ implies $G=gFV$.

Two minimal left covering nets for the same identity neighbourhood $V$ can be matched bijectively so that paired translates intersect. Join $f\in F$ to $f^{\prime }\in F^{\prime }$ when $fV\cap f^{\prime }V\ne \varnothing$. For $S\subseteq F$, its neighbours $N(S)$ cover every point covered by $S$, so $(F\setminus S)\cup N(S)$ still covers the group. Minimality forces $|N(S)|\ge |S|$. Hall's marriage theorem supplies the matching, and an intersection $fv=f^{\prime }{v}^{\prime }$ gives $f^{-1}{f}^{\prime }=v{v}^{\prime -1}\in V{V}^{-1}$. Averaging a continuous function over matched nets therefore gives values differing by at most its uniform oscillation on right displacements in $V{V}^{-1}$. This is the finite combinatorial step in constructing Haar measure on a compact group.