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

Let $A$ be a Lebesgue measure space. We interpret measures on $A\times A\times \mathbb R^\times$ as ‘maps’ from $A$ to $A$, which ‘spread’ $A$ along itself; their Radon-Nikodym derivatives are also spread. We discuss the basic properties of the semigroup of such maps and the action of this semigroup on the spaces $L^p(A)$.
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