  Let $A$ be a Lebesgue measure space. We interpret measures on $A\times
A\times R_+$ as 'maps' from $A$ to $A$, which spread $A$ 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 $L^p(A)$.
