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The speed measure and absolute continuity for curves in metric spaces

Sebastian Boldt, Peter Stollmann, Felix Wirth

Abstract

We define the speed measure $ν$ for mappings $γ:I\to X$ from an interval to a metric space that are locally of bounded variation. We characterize continuity and absolute continuity of $γ$ in terms of $ν$ and identify the Radon-Nikodým derivative of $ν$ with respect to Lebesgue measure as the metric speed of $γ$. In doing so we prove an extension of the Banach-Zaretsky theorem.

The speed measure and absolute continuity for curves in metric spaces

Abstract

We define the speed measure for mappings from an interval to a metric space that are locally of bounded variation. We characterize continuity and absolute continuity of in terms of and identify the Radon-Nikodým derivative of with respect to Lebesgue measure as the metric speed of . In doing so we prove an extension of the Banach-Zaretsky theorem.

Paper Structure

This paper contains 4 sections, 11 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. The set-up
  3. Absolute continuity and the Banach-Zaretsky Theorem
  4. The Lebesgue Decomposition Theorem and the metric derivative

Key Result

Proposition 2.4

Let $\gamma\in \mathsf{BV}_{loc}(I;X)$ and $t\in I$. Let $(s_n)_{n\in\mathbb N}$ be a sequence in $(-\infty,t)\cap I$ with $s_n\to t$ as $n\to\infty$. Then, If $X$ is complete, this limit agrees with $\mathbf{d}(\gamma(t-),\gamma(t))$. An analogous statement holds if $(s_n)_{n\in\mathbb N}$ is a sequence in $(t,+\infty)\cap I$. Hence, $\gamma$ is left-, respectively right-continuous at $t$$\iff$$

Theorems & Definitions (26)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • Definition 2.6
  • Theorem 2.7
  • proof
  • ...and 16 more