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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 $ν$ 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.

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Sebastian Boldt, Peter Stollmann, Felix Wirth. 2026-03-07. The speed measure and absolute continuity for curves in metric spaces. https://arxiv.org/abs/2507.23584

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