A set \(A\subset\mathbb{R}^2\) is purely unrectifiable if \(\mathcal H^1(\text{graph}(\gamma)\cap A)=0\) for every Lipschitz curve \(\gamma\) and \(A\) is uniformly purely unrectifiable if for every \(K\geq 0\) and every \(\varepsilon>0\) there is an open set \(U\) with \(A\subset U\) such that \(\mathcal H^1(\text{graph}(g)\cap U)\leq\varepsilon\) for every \(K\)-Lipschitz function \(g\) in any rotated Cartesian coordinates. It is not known whether these notions coincide for \(G_\delta\) set (they coincide for \(F_\sigma\) sets).
The author considers \(G_\delta\) sets which are purely unrectifiable graphs of lower semicontinuous functions. The main result of the paper is the following. Let \(f:[0,1]\to[0,1]\) be a lower semicontinuous function with purely unrectifiable graph. Then for any \(\varepsilon>0\) there is an open set \(U\supset\text{graph}(f)\) with every vertical section of one-dimensional Lebesgue measure at most \(\varepsilon\).