Proof: The proof is identical to the finite case of Theorem 1.22.

If we wish to constructively prove that the extreme points of D are exactly the permutation operators, we must begin with a doubly stochastic matrix P with an entry pmn (0, ], From this we need to construct doubly stochas tic matrices A = (aij) and B = (bij) such that 1/2 (aij + bij) = pij Ai, j . The following proof of the following Lemma was suggested by a technique used by Mauldon and it is essential in this construction.