The speed measure and absolute continuity for curves in metric spaces

TL;DR

This paper introduces a speed measure for curves in metric spaces, characterizes their continuity and absolute continuity, and extends classical theorems relating metric speed and measure derivatives.

Contribution

It defines the speed measure for bounded variation curves in metric spaces and extends the Banach-Zaretsky theorem to this setting.

Findings

Characterization of continuity and absolute continuity via the speed measure

Identification of the Radon-Nikodym derivative as the metric speed

Extension of the Banach-Zaretsky theorem

Abstract

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