A doubly stochastic measure on the unit square is a Borel probability measure whose horizontal and vertical marginals both coincide with the Lebesgue measure. The set of doubly stochastic measuresis convex and compact so itsextremal points are of particular interest. The problem number 111of Birkhoff (Lattice Theory 1948) is to provide a necessary and sufficient condition on the support of a doubly stochastic measure to guarantee extremality. It was proved by Bene? and ?t?pán that an extremal doubly stochastic measure is concentratedon a set which admits an aperiodic decomposition. Hestir and Williams later found a necessary condition whichis nearly sufficient by further refining the aperiodic structure of the support of extremaldoubly stochastic measures.Our objective in this work is to provide a more practical necessary and nearly sufficientcondition for a set to support an extremal doubly stochasticmeasure.
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