Rectifiability of pointwise doubling measures in Hilbert Space

Abstract

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 $\mu$, we construct a multiresolution family $\mathscr{C}^\mu$ of windows, and then we use a weighted Jones' function $\hat{J}_2(\mu, x)$ to record how well lines approximate the distribution of mass in each window. We show that when $\mu$ 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 $\delta$-nets. Throughout, we discuss how to overcome the fact that in infinite dimensional Hilbert space there may be infinitely many $\delta$-separated points, even in a bounded set. Finally, we prove a characterization for pointwise doubling measures carried by Lipschitz graphs.