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Let P be a Markov operator on L∞(X, Σ, m) which does not disappear (i.e., P1A ≡ 0 => 1A ≡ 0 ) . We study the relationship between 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.
