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A Characterization of Maps of Bounded Compression

Lorenzo Dello Schiavo

Abstract

A measurable map between measure spaces is shown to have bounded compression if and only if its image via the measure-algebra functor is Lipschitz-continuous w.r.t. the measure-algebra distances. This provides a natural interpretation of maps of bounded compression/deformation by means of the measure-algebra functor and corroborates the assertion that maps of bounded deformation are the natural class of morphisms for the category of complete and separable metric measure spaces.

A Characterization of Maps of Bounded Compression

Abstract

A measurable map between measure spaces is shown to have bounded compression if and only if its image via the measure-algebra functor is Lipschitz-continuous w.r.t. the measure-algebra distances. This provides a natural interpretation of maps of bounded compression/deformation by means of the measure-algebra functor and corroborates the assertion that maps of bounded deformation are the natural class of morphisms for the category of complete and separable metric measure spaces.
Paper Structure (6 sections, 5 theorems, 9 equations)

This paper contains 6 sections, 5 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Some categories
  3. Measure algebras
  4. Main result
  5. A natural choice of morphisms for metric measure spaces
  6. Proofs

Key Result

Theorem 1

A map has bounded compression if and only if its image via the measure-algebra functor is Lipschitz-continuous with respect to the measure-algebra distances.

Theorems & Definitions (8)

  • Theorem
  • Proposition 1
  • Remark 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • proof : Proof of Proposition \ref{['p:USp']}
  • proof : Proof of Theorem \ref{['t:Main']}