The speed measure and absolute continuity for curves in metric spaces

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–Nikodým 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.