Supports of Extremal Doubly and Triply Stochastic Measures - Master’s Project

Doubly stochastic measures are Borel probability measures on the unit square which push forward via the canonical projections to Lebesgue measure on each axis. The set of doubly stochastic measures is convex, so its extreme points are of particular interest. I review necessary and sufficient conditions for a set to support an extremal doubly stochastic measure, and include a proof that such a set can be decomposed into a countable collection of graphs and antigraphs of functions, called a ‘limb-numbering system.’ I also investigate how this structure partially generalizes to triply stochastic measures on the unit cube. A doubly stochastic matrix is a real matrix whose entries are positive and whose rows and columns individually sum to one. A classical theorem first due to Birkhoff [1], but also attributed to von Neumann [2], states that the set of doubly stochastic matrices is the convex hull of the set of n× n permutation matrices. This, along with the Krein-Milman theorem [3], tells us that a matrix is doubly stochastic if and only if it is a convex combination of permutation matrices. In his 1949 book Lattice Theory, Birkhoff proposed the problem of extending this to an infinite dimensional analog [4]. Known as Birkhoff’s Problem 111, this project has been taken up at various points since it’s formulation; one approach is to consider doubly stochastic measures, probability measures on the unit square which project to the Lebesgue measure on each axis [5][6][7][8]. This paper is structured as follows. In the first section, we introduce basic definitions and survey relavent results from the literature relating to doubly stochastic measures. In the second section we ask and examine analagous questions for triply stochastic measures. Proofs of two of the major theorems from section 1 can found in an appendix. 1 Extremal doubly stochastic measures