Is the Birkhoff–von Neumann theorem true for infinite matrices?

Question

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?

Answer 1

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.