Kuczma, Marcin E. A generalization of Steinhaus’ theorem to coordinatewise measure preserving binary transformations. (English) Zbl 0358.28002 Colloq. Math. 36, 241-248 (1976). This paper concerns the question under what conditions a transformation \(f \colon X\times X \to X\) allows interior points in \(f(A \times B)\) for any two measurable sets \(A,B\in 2^{x}\). A classical result of H. Steinhaus [Fundamenta Math. 1, 93–104 (1920; JFM 47.0179.02)] proofs this in case of \(X=\) real line, Lebesgue measurability and \(f(x,y)=x+y\). The author generalizes this problem to topological spaces \(X\), Borel measurability with \(\sigma\)-finite regular measures on the Borel sets and continuous transformations fulfilling certain invertibility and measurability conditions. Further bibliographical remarks on the whole subject are made. Reviewer: H. Michel Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 5 Documents MSC: 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) 28D05 Measure-preserving transformations Citations:JFM 47.0179.02 PDFBibTeX XMLCite \textit{M. E. Kuczma}, Colloq. Math. 36, 241--248 (1976; Zbl 0358.28002) Full Text: DOI