Two sufficient conditions for rectifiable measures (Q2796713)

The authors prove two conditions for rectifiability of a locally finite Borel measure \(\mu\) on \({\mathbb R}^n\). The first condition describes the rectifiability in terms of upper and lower limits of ratio \(\frac{\mu(B(x,r)}{\omega_{m}r^m}\) for \(r\to 0\) (the Hausdorff densities) and \(L^{p}\)-beta numbers. The second one is a condition of finiteness of the sum \(\sum\frac{\mathrm{diam }Q}{\mu(Q)}\chi_{Q}(x)\) for \(\mu\)-almost all \(x\); here \(\{Q\}\) is a system of half-open dyadic cubes in \({\mathbb R}^n\) of side length at most 1.