Examples of doubly stochastic measures supported on the graphs of two functions

摘要

A doubly stochastic measure (DSM) is a measure mu on the unit square so that mu([0, 1] x A) = mu(A x [0, 1]) = m(A) where m is Lebesgue measure. The set of DSMs forms a convex set in the space of measures. It is known that DSMs supported on the union of two graphs of invertible functions are extreme points of that convex set (Seethoff and Shiflett, 1977/78). In general, there are few examples of extreme points in the literature. There are examples of so-called hairpins where the functions involved are inverses of each other, but there are also examples of the union of the graphs of a function and its inverse does not support a DSM (Sherwood and Taylor, 1988). In this paper, for a function f in a certain class, we find companion functions g so that the union of the graphs of f and g support a DSM even though the union of the graphs of f and f-inverse do not.