1. S. Saks, Théorie de l'intégrale, p. 169–175.
2. Long ago I thought that this theorem was true and I suggested to Mr Gillis that he should investigate from this side sets of upper density 1/2. By an ingenious method Gillis arrived at some partial results concerning this class of sets. He has also studied sets of directions from which an irregular set can be projected into a set of positive measure and he has constructed a set for which the set of such directions has the power of continuum in any angle, however small. I have quoted his papers in my paper II.