A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm173-2-3,
author = {A. K\l opotowski and M. G. Nadkarni and H. Sarbadhikari and S. M. Srivastava},
title = {Sets with doubleton sections, good sets and ergodic theory},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {133-158},
zbl = {1004.60002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm173-2-3}
}
A. Kłopotowski; M. G. Nadkarni; H. Sarbadhikari; S. M. Srivastava. Sets with doubleton sections, good sets and ergodic theory. Fundamenta Mathematicae, Tome 173 (2002) pp. 133-158. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm173-2-3/