The speed measure and absolute continuity for curves in metric spaces

We define the speed measure $ν$ for mappings γ:I→ 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.