Radon measures and Lipschitz graphs

Abstract

For all $1\leq m\leq n-1$, we investigate the interaction of locally finite measures in $\mathbb{R}^n$ with the family of $m$-dimensional Lipschitz graphs. For instance, we characterize Radon measures $\mu$, which are carried by Lipschitz graphs in the sense that there exist graphs $\Gamma_1,\Gamma_2,\dots$ such that $\mu(\mathbb{R}^n\setminus\bigcup_1^\infty\Gamma_i)=0$, using only countably many evaluations of the measure. This problem in geometric measure theory was classically studied within smaller classes of measures, e.g.~for the restrictions of $m$-dimensional Hausdorff measure $\mathcal{H}^m$ to $E\subseteq \mathbb{R}^n$ with $0<\mathcal{H}^m(E)<\infty$. However, an example of Cs\"{o}rnyei, K\"{a}enm\"{a}ki, Rajala, and Suomala shows that classical methods are insufficient to detect when a general measure charges a Lipschitz graph. To develop a characterization of Lipschitz graph rectifiability for arbitrary Radon measures, we look at the behavior of coarse doubling ratios of the measure on dyadic cubes that intersect conical annuli. This extends a characterization of graph rectifiability for pointwise doubling measures by Naples by mimicking the approach used in the characterization of Radon measures carried by rectifiable curves by Badger and Schul.