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arXiv · 2507.23584

The speed measure and absolute continuity for curves in metric spaces

Sebastian Boldt·Peter Stollmann·Felix Wirth
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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.

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Sebastian Boldt, Peter Stollmann, Felix Wirth. 2025-07-31. The speed measure and absolute continuity for curves in metric spaces. https://arxiv.org/abs/2507.23584

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