The Birkhoff–von Neumann theorem states that every $n \times n$ doubly stochastic matrix is a convex combination of permutation matrices. Is this true for $\mathbb{N} \times \mathbb{N}$ matrices as well? If so, can you provide a reference?
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Rodrigo de Azevedo
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Yoav Rodeh
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3Late to the party but a reference for this other than the one by bof is the 1960 paper by Kendall on "On Infinite Doubly‐Stochastic Matrices and Birkhoffs Problem 111" - essentially, the statement generalizes to $\mathbb N\times\mathbb N$ if the convex hull is replaced by the closed convex hull in the weak topology. – Frederik vom Ende Oct 02 '18 at 13:23
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I don't know the exact answer to your question but I found a reference that should be useful. In L. Mirsky's book Transversal Theory (Academic Press, 1971, ISBN 0-12-498550-5), on p.213,
I read:
Birkhoff's theorem has been extended to infinite d.s. matrices, with the notion of convex closure in a suitable topological vector space replacing that of convex hull. For references to work in this field, see Mirsky (1).
The reference Mirsky (1) is to the following paper:
L. Mirsky, Results and problems in the theory of doubly-stochastic matrices,
Z. Wahrscheinlichkeitstheorie 1 (1962–3), 319–334.
bof
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