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The observable diameter of metric measure spaces and the existence of points of positive measures

Shun Oshima

Abstract

A metric measure space is a metric space with a Borel measure. In Gromov's theory of metric measure spaces, there are important invariants called the partial diameter and the observable diameter. We obtain the result that the partial diameter or the observable diameter equals zero if and only if there exists a point that has positive measure.

The observable diameter of metric measure spaces and the existence of points of positive measures

Abstract

A metric measure space is a metric space with a Borel measure. In Gromov's theory of metric measure spaces, there are important invariants called the partial diameter and the observable diameter. We obtain the result that the partial diameter or the observable diameter equals zero if and only if there exists a point that has positive measure.

Paper Structure

This paper contains 6 sections, 31 theorems, 115 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. multivariable observable diameter of mm-spaces
  4. Proof of Main Theorem
  5. Properties of the multivariable diameters
  6. Another definition of the multivariable partial and observable diameter

Key Result

Proposition 2.5

Any mm-space has a parameter.

Theorems & Definitions (81)

  • Definition 2.1: mm-space
  • Definition 2.2: mm-isomorphism
  • Definition 2.3: Lipschitz order
  • Definition 2.4: Parameter
  • Proposition 2.5: ShioyaMMG
  • Definition 2.6: Box distance between mm-spaces
  • Proposition 2.7: ShioyaMMG
  • Definition 2.8: Observable diameter
  • Remark 2.9
  • Proposition 2.10: ShioyaMMG
  • ...and 71 more