Vol. 16, No. 1, 1966

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Approximation theorems for Markov operators

James Russell Brown

Vol. 16 (1966), No. 1, 13–23

Let (X,,m) be a totally σ-finite measure space. A Markov operator (with invariant measure m) is a positive operator T on L(X,,m) such that T1 = 1 and Tf dm = f dm for all f L1(X,,m) L(X,,m). If φ is an invertible measure-preserving transformation of (X,,m), then φ determines a Markov operator Tφ by the formula Tφf(x) = f(φx). The set M of all Markov operators is convex and each Tφ is an extreme point.

In case (X,,m) is a finite, homogeneous, nonatomic measure space, M may be identified with the set of all doubly stochastic measures on the product space (X ×X,ℱ×ℱ,m×m). The main result of the present paper is that M is compact in the weak operator topology of operators on L2(X,,m) and that the set Φ of operators Tφ is dense in M. It follows that M is the closed convex hull of Φ in the strong operator topology. We shall further show that Φ is closed in the uniform operator topology and that the closure of Φ in the strong operator topology is the set Φ1 of all (not necessarily invertible) measure-preserving transformations of (X,,m).

Mathematical Subject Classification
Primary: 60.60
Received: 21 September 1964
Published: 1 January 1966
James Russell Brown