Existence of an invariant measure on a topological quasigroup
Let \(G\) be a unital quasigroup, i.e. a set with a multiplication \(G^2\ni (a,b)\mapsto ab\in G\) and a neutral element such that for any \(a,b\in G\) both the equations \(ax=b\) and \(ya=b\) have a unique solution. Assume that \(G\) is endowed with a Tychonoff topology such that the multiplication, the right division and the left division are jointly continuous. The main theorem says: If \(G\) admits a nontrivial left invariant \([0,+\infty]\)-valued \(\sigma\)-additive Borel measure \(\mu\) such that the family of all bounded uniformly continuous functions of \(L^2(G,\mu,\mathbb{C})\) is dense in \(L^2(G,\mu,\mathbb{C})\), then \(G\) is a dense unital subquasigroup of a locally compact unital quasigroup.
scientific article; zbMATH DE number 7187687
scientific article