  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 $S^2$.
