On the boundary of the group of transformations leaving a~measure quasi-invariant (Q2863129)

Let \(\mathbb {R}^{\times}\) be the multiplicative group of the positive real numbers. Let \(A\) be a space with continuous probability measure \(\alpha\). By \(Gms(A)\) the group of transformations is denoted leaving the measure \(\alpha\) quasi-invariant.NEWLINENEWLINEConsider two Lebesgue spaces with measures \((A, \alpha)\) and \((B, \beta)\). A measure \(\mathfrak{P}\) on \(A\times B\) is a polymorphism if: 1) the pushward of \(t\times \mathfrak{P}\) under projection onto \(A\) coincides with \(\alpha\); 2) the pushward of \(t\times \mathfrak{P}\) under projection onto \(B\) coincides with \(\beta\).NEWLINENEWLINEThe structure of the paper: Sections 2 and 3 contain preliminaries on Lebesgue spaces and Markov operators. In \S 4 some properties are described of the semi-ring of positive measures on \(\mathbb {R}^{\times}\). Polymophisms are defined in \S 5. In \S 6 the action of polymorphisms is discussed on spaces \(L^p\).