On a theorem of Volkmann (Q2496997)

This short paper generalizes a theorem of Volkmann (1960) concerning the Hausdorff measures on subfields of \(R\), and is informed by more recent results by Foran (1974) and by Edgar and Miller (2002). The proof is based on a mensural trichotomy law for invariant subsets of a locally compact group. Thus, the generalized result is, roughly, in a locally compact Hausdorff topological group with topology induced by a left invariant metric, \(H\) is a dense subgroup, \(X\) is a left \(H\)-invariant subset of \(G, g\) in a certain class of continuous, increasing functions on \(G\) defined with respect to the metric, for some \(s\)-dimensional Hausdorff measure \(\mu^*_g\), we have (i) \(\mu^*_g(X \bigcap S) = 0\) for all \(S\) in the \(\sigma\)-ideal generated by all compact subsets of \(G\); or, (ii) \(\mu^*_g(X \cap O) = +\infty\) for every non-empty open subset \(O\) of \(G\).