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On the Deterministic and Asymptotic σ-Algebras of a Markov Operator
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- Journal:
- Canadian Mathematical Bulletin / Volume 32 / Issue 1 / 01 March 1989
- Published online by Cambridge University Press:
- 20 November 2018, pp. 64-73
- Print publication:
- 01 March 1989
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- Article
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Let P be a Markov operator on L ∞(X, Σ, m) which doesnot disappear (i.e., P1A ≡ 0 => 1A ≡ 0 ) . We study the relationshipbetween the σ-algebras
(the deterministicσ-algebra), and the asymptoticσ-algebra
When m is a σ-finite invariant measure,
measurable iff p*npnf = f, and also iff Pnf has the same distribution as f . The case of a convolution operator on a locally compact group is considered.
measurable iff 