Recently, M. Badger and R. Schul proved that for a $1$-rectifiable Radon measure $\mu$, the density weighted Jones' square function $$ J_{ 1}(x) = \mathop{ \sum_{ Q \in \mathcal{ D}}}_{ \ell(Q) \leq 1} \beta_{ 2,\mu}^{ 2}(3Q)\frac{ \ell(Q)}{ \mu(Q)} 1_{ Q}(x) $$ is finite for $\mu$-a.e. $x$. Answering a question of Badger-Schul, we show that the converse is not true. Given $\epsilon > 0$, we construct a Radon probability measure on $[0,1]^{ 2} \subset \mathbb{ R}^{ 2}$ with the properties that $J_{ 1}(x) \leq \epsilon$ for all $x \in \operatorname{ spt} \mu$, but nevertheless the $1$-dimensional lower density of $\mu$ vanishes almost everywhere. In particular, $\mu$ is purely $1$-unrectifiable.
Categories: math.CA math.MG