Properties of a special class of doubly stochastic measures

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 < x_n (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.