2020/02/18 by Naples, Lisa
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2002.07570
In geometric measure theory, there is interest in studying the interaction of measures with rectifiable sets. Here, we extend a theorem of Badger and Schul in Euclidean space to characterize rectifiable pointwise doubling measures in Hilbert space. Given a measure μ, we construct a multiresolution family \mathscrCμ of windows, and then we use a weighted Jones' function J2(μ, x) to record how well lines approximate the distribution of mass in each window. We show that when μ is rectifiable, the mass is sufficiently concentrated around a lines at each scale and that the converse also holds. Additionally, we present an algorithm for the construction of a rectifiable curve using appropriately chosen δ-nets. Throughout, we discuss how to overcome the fact that in infinite dimensional Hilbert space there may be infinitely many δ-separated points, even in a bounded set. Finally, we prove a characterization for pointwise doubling measures carried by Lipschitz graphs.