The problem of continuation periodic in the mean (Q1593910)

The functional \(\langle f,d{\mathcal L}\rangle= \int^a_{-a} f(t) d{\mathcal L}(t)\) is called a Cantor-Lebesgue functional if \(d{\mathcal L}\) is the Lebesgue function constructed for a perfect nowhere dense set \(E\subset [-a,a]\) with a constant partition ratio \(\xi\), \(0<\xi<{1\over 2}\). If \(\xi\) is a rational number, it is stated without proof that there is for \(d{\mathcal L}\) no effect of continuation periodic in the mean.