ISSN:
1420-8903
Keywords:
Primary 28A35, 28A33, 60A10
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary A measure μ on the unit squareI } I is doubly stochastic ifμ(A } I) = μ(I } A) = the Lebesgue measure ofA for every Lebesgue measurable subsetA ofI = [0, 1]. By the hairpinL ∪L −1, we mean the union of the graphs of an increasing homeomorphismL onI and its inverseL −1. By the latticework hairpin generated by a sequence {x n :n ∈ Z} such thatx n-1 〈 xn (n ∈ Z), $$\mathop {\lim }\limits_{n \to - \infty } $$ x n = 0 and $$\mathop {\lim }\limits_{n \to \infty } $$ x n = 1, we mean the hairpinL ∪L −1 , whereL is linear on [x n-1 ,x n ] andL(n) =x n-1 forn ∈ Z. In this note, a characterization of latticework hairpins which support doubly stochastic measures is given. This allows one to construct a variety of concrete examples of such measures. In particular, examples are given, disproving J. H. B. Kemperman's conjecture concerning a certain condition for the existence of doubly stochastic measures supported in hairpins.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01836092
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