The singularity of extremal measures (Q1124195)

A Borel measure P on \(I\times I\) is doubly stochastic if for each Borel subset A of I, \(P(A\times I)=P(I\times A)=Q(A)\), where Q is the Lebesgue measure on R. Lindenstrauss proved that every extreme doubly stochastic measure is singular with respect to the planar Lebesgue measure \(Q\times Q\). The author extends this result in the following sense. Let \(L_ 1,L_ 2,...,L_ m\) be the lines through the origin in the plane and P be a probability measure on the plane. Consider the convex set of probabilities on the plane whose projections onto \(L_ 1,L_ 2,...,L_ m\) agree with those of P. Then the author shows (actually in a more general context) that the extreme points of this convex set are singular with respect to the Lebesgue product measure.