Boolean duals of virtual groups

The purpose of this paper is to formulate the notion of virtual group and that of a homomorphism between two in such a way that no null sets are involved. That is, we replace the point set by the measure algebra of Borel sets modulo null sets, and the various functions involved must be described by objects associated to these measure algebras, such as σ-homomorphisms. In the process it is necessary to redefine functions satisfying algebraic identities almost everywhere to get new ones satisfying the identities in a stricter sense. Similar refinements on quasiinvariance of measures are also required.