Tube-measurability (Q930123)

A tube in \(\mathbb{R}^n\) is given by a line and an \(\epsilon>0\) and then defined as \(T=\{x:d(x,L)<\varepsilon\}\). Its \textit{content} \(c(T)\) is defined to be the \(n-1\)-dimensional measure of its cross section. The tube-measure (related to Tarski's plank problem, see [\textit{T. Bang}, Proc. Am. Math. Soc. 2, 990--993 (1951; Zbl 0044.37802)]) of a subset~\(E\) of~\(\mathbb{R}^n\) is then \(\mu(E)=\inf\{\sum_nc(T_n): E\subseteq\bigcup_nT_n\}\). The authors prove that the only \(\mu\)-measurable sets are the \(\mu\)-null sets and their complements.