Radon measures and Lipschitz graphs

Abstract

For all 1 ⩽ m ⩽ n − 1 , we investigate the interaction of locally finite measures in R n with the family of m-dimensional Lipschitz graphs. For instance, we characterize Radon measures μ, which are carried by Lipschitz graphs in the sense that there exist graphs Γ 1 , Γ 2 , ⋯ such that μ ( R n ∖ ⋃ 1 ∞ Γ i ) = 0 , using only countably many evaluations of the measure. This problem in geometric measure theory was classically studied within smaller classes of measures, for example, for the restrictions of m-dimensional Hausdorff measure H m to E ⊆ R n with 0 < H m ( E ) < ∞ . However, an example of Csörnyei, Käenmäki, 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.