On Section Sets of Neighborhoods of Graphs of Semicontinuous Functions

Abstract

We prove that for any lower semicontinuous function \(f:[0,1]\to[0,1]\) with purely unrectifiable graph and for any \(\varepsilon>0\) there is an open set \(U\supset \text{graph} f\)\, with every vertical section set of one-dimensional Lebesgue measure at most \(\varepsilon.\)