Doubly stochastic measures with mass on the graphs of two functions (Q1099865)

A doubly stochastic measure is a Borel probability measure P on the unit square \(I^ 2\) such that \(P(I\times A)=P(A\times I)=m(A)\), where m is the Lebesgue measure and A is a Borel subset of [0,1]. Let L and H be two nondecreasing maps of [0,1] into itself such that \(L\circ H(t)<t\), \(H\circ L(t)<t\) on (0,1). The authors have shown that the union of the graphs of L and H can support at most one doubly stochastic measure.