2002
DOI: 10.4064/fm173-2-3
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Sets with doubleton sections, good sets and ergodic theory

Abstract: Abstract.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.Introduction. Given non-empty sets X and Y , a subset of X × Y is called a matching if it is the graph of a one-to-one map o… Show more

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Cited by 6 publications
(5 citation statements)
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“…In the last section §6 we discuss a matching problem in the Borel and measurable contexts related to earlier work of Laczkovich [30] and Kłopotowski-Nadkarni-Sarbadhikari-Srivastava [29].…”
Section: Also the Following Are Equivalentmentioning
confidence: 99%
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“…In the last section §6 we discuss a matching problem in the Borel and measurable contexts related to earlier work of Laczkovich [30] and Kłopotowski-Nadkarni-Sarbadhikari-Srivastava [29].…”
Section: Also the Following Are Equivalentmentioning
confidence: 99%
“…In the paper Kłopotowski-Nadkarni-Sarbadhikari-Srivastava [29], the authors argue that the graph shown below which consists of 4 "copies" of the preceding graph (actually, the authors discard finitely many connected components rather than adding dots at the corners, but it is clear that one graph admits a Borel matching if and only if the other does), and in which every vertex has degree 4 provides a counterexample for k D 4 (and similarly for all even k). They also raised the question of whether there is a counterexample for k D 3.…”
Section: ˛1mentioning
confidence: 99%
“…If S is a good Borel set and if f a complex valued Borel function on S, the question whether one can choose the functions U i , 1 ≤ i ≤ n, in (14) in a Borel fashion has, in general, a negative answer [6]. We discuss conditions under which an affirmative answer is possible.…”
Section: Descriptive Set Theoretic Considerationsmentioning
confidence: 99%
“…More generally it can be shown, as in the case n = 2 (see [5,6]), that if S is a good Borel set in which any two points are related and there is a uniform upper bound for U 1 ,U 2 , . .…”
Section: Simplicial Measures and Sums Of Algebrasmentioning
confidence: 99%
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