Hausdorff Measures and Functions of Bounded Quadratic Variation

TL;DR

The paper introduces a Hausdorff measure associated with functions of bounded quadratic variation and uses these measures to analyze the structure of certain subspaces of the quadratic variation space.

Contribution

It establishes a Lipschitz and onto correspondence between functions of bounded quadratic variation and Hausdorff measures, and applies this to subspace structure analysis.

Findings

The map from functions to measures is locally Lipschitz.

The measure map is onto the positive cone of measures.

Characterizes subspaces containing $c_0$ or $S^2$.

Abstract

To each function $f$ of bounded quadratic variation ($f\in V_2$) we associate a Hausdorff measure $\mu_f$. We show that the map $f\to\mu_f$ is locally Lipschitz and onto the positive cone of $\mathcal{M}[0,1]$. We use the measures $\{\mu_f:f\in V_2\}$ to determine the structure of the subspaces of $V_2^0$ which either contain $c_0$ or the square stopping time space $S^2$.