Approximation theorems for Markov operators

Abstract

Let (X, ^ ,m) be a totally (/-finite measure space. A Markov operator (with invariant measure m) is a positive operator T on L^X, v ^r, m) such that Tl = 1 and f Tfdm = 1 fdm for all feLi(X, ^, m) n Loo(X, ^~, ra). If is an invertible measure-preserving transformation of (X, ^", m), then determines a Markov operator T by the formula T f(x) = f(x). The set f of all Markov operators is convex and each T is an extreme point.