On thick subgroups of uncountable \(\sigma \)-compact locally compact commutative groups (Q837565)

The author shows that an uncountable group, \(G\), can be written as the union of a countable family \(\{G_j:j\in J\}\) of subgroups such that \(|G/G_j|=|G|\) for all~\(j\). From this he deduces that every uncountable \(\sigma\)-compact locally compact commutative group, \(G\), has a countable family \(\{H_j:j\in J\}\) of subgroups each of full outer Haar measure and such that \(|G/H_j|=|G|\). Furthermore, the Haar measure of~\(G\setminus\bigcup_jH_j\) is zero; whenever \(\mu\)~is a \(G\)-invariant extension of the completion of Haar measure then one of the \(H_j\) is \(\mu\)-nonmeasurable; and if \(H_j\)~is nonmeasurable with respect to a \(G\)-invariant extension of Haar measure then there is a further extension \(\mu'\) of~\(\mu\) for which \(\mu'(H_j)=0\).