On the interior of certain measurable sets (Q2723170)

This paper moves around the result of \textit{H. Steinhaus} [``Sur les distances des points dans les ensembles de mesure positive'', Fundam. Math. 1, 93-104 (1920; JFM 47.0179.02)] which says that if a subset \(E \subset \mathbb{R}\) has positive Lebesgue measure then \(E-E\) has nonempty interior, some of its generalizations and some of its applications. Thus, Steinhaus' theorem is proved in section 3 of the paper, \textit{J. H. B. Kemperman}'s generalization [``A general functional equation'', Trans. Am. Math. Soc. 86, 28-56 (1957; Zbl 0079.33402)] in section 4 and \textit{Chae-Peck}'s generalization [``A generalization of Steinhaus-Kemperman theorem'', Notices Am. Math. Soc. 709, B24 (1973)] in section 5. Section 6 contains a proof of the extension to \(\mathbb R^n\) of \textit{M. Fréchet}'s result [``Pri la funkcia ekvacio \(f(x+y)=f(x) + f(y)\)'', Enseignement Math. 15, 390-393 (1913; JFM 44.0399.01)] that a measurable additive function \(\mathbb R \to \mathbb R\) must be linear. The paper is interesting because the author has worked a lot to show us what comes from what, who did what and how he did it. NEWLINENEWLINENEWLINEIn this regard, I've just missed a mention to \textit{W. Sierpinski}'s paper [``Sur l'équation fonctionnelle \(f(x+y)=f(x)+f(y)\)'', Fundam. Math. 1, 116-121 (1920; JFM 47.0235.01)], which provides a more general proof for Fréchet's result and without using the axiom of choice; and to \textit{S. Banach}'s paper [``Sur l'équation fonctionnelle \(f(x+y)=f(x)+f(y)\)'', Fundam. Math. 1, 123-124 (1920; JFM 47.0235.02)].NEWLINENEWLINEFor the entire collection see [Zbl 0948.00025].