On the boundary of the group of transformations leaving a measure quasi-invariant

Let $A$ be a Lebesgue measure space. We interpret measures on $A\times A\times R_+$a{s}^{\mathrm{\prime }}map{s}^{\mathrm{\prime }}from$A$to$A$,whichspread$A$ along itself; their Radon-Nikodym derivatives also are spread. We discuss basic properties of the semigroup of such maps and the action of this semigroup in the spaces $L^p(A)$.