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Let (X,ℱ,m) be a totally
σ-finite measure space. A Markov operator (with invariant measure m) is a positive
operator T on L∞(X,ℱ,m) such that T1 = 1 and ∫
Tf dm = ∫
f dm for all
f ∈ L1(X,ℱ,m) ∩ L∞(X,ℱ,m). If φ is an invertible measure-preserving
transformation of (X,ℱ,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,ℱ,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,ℱ,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,ℱ,m).
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