×

A generalization of Steinhaus’ theorem to coordinatewise measure preserving binary transformations. (English) Zbl 0358.28002

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

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
Full Text: DOI