ABSTRACT

To continue the historical development of Birkhoff's Problem 111, we move from the countably infinite to the uncountable case. Until now, the study has been able to work with operators induced by matrices, and stochastic properties were defined in terms of finite or infinite row and column sums. In the uncountable setting, we have no matrix and no summation. However, in both the finite and countably infinite settings, we were careful to point out that with every stochastic and doubly stochastic matrix, there were associated bounded linear operators defined on l1 and l . It is exactly these operators that survive the move to the uncountable case, with sums becoming integrals and lp spaces becoming Lp spaces.