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

TL;DR

This paper studies a semigroup of measure-based maps on Lebesgue spaces, exploring their properties and actions on $L^p$ spaces, with implications for understanding transformations that preserve measure classes.

Contribution

It introduces a novel interpretation of measures as maps on Lebesgue spaces and analyzes their algebraic and functional properties.

Findings

Characterization of the semigroup of measure-based maps.

Analysis of the Radon-Nikodym derivatives of these maps.

Description of the action of this semigroup on $L^p$ spaces.

Abstract

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 $L^p(A)$.