Cardinality and structure of extreme infinite doubly stochastic matrices

Abstract.

The main results of this article are contained in Sections 2 and 3. In Section 2 we show that for non-trivial infinite systems of marginals $\cal I$ there are always continuum many extreme matrices that are doubly stochastic with respect to $\cal I$ (cf. Theorem 2.1). Corollary 2.2 contains the corresponding result for the substochastic case. In Section 3 we show that the entries of an extreme doubly stochastic matrix are always in the closure of the additive subgroup of $\Bbb R$ that is generated by the respective marginals (cf. Theorem 3.2). Theorem 3.4 deals with the corresponding substochastic case.