By Horbachuk O.L., Maturin Yu.P.

Horbachuk O.L., Maturin Yu.P. jewelry and houses of lattices of I-radicals Buletinul Academiei de Stiinte a Republicii Moldova

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**Example text**

It is for every f ∈ Cb (X). We call µ easy to see that µ ˜ is a bounded linear functional on Cb (X). 2. Invariant Probabilities 19 that if µ ˜(Sf ) = µ ˜(f ) for every f ∈ C0 (X), then T µ = µ. ) The Lasota–Yorke lemma is an extension of the above observation to arbitrary positive linear functionals on Cb (X). More precisely, let φ : Cb (X) → R be a positive linear functional. Then φ is bounded (continuous), so by the Riesz representation theorem we may think of the restriction µφ of φ to C0 (X) as an element of M(X).

Clearly, a Markov operator is positive. Moreover, if T : E → E is a Markov operator, then T = 1 (so, T is a contraction) because the positivity of T implies that T = sup{ T u | u ∈ E, u ≥ 0, u ≤ 1}. 2 are particular cases of Markov operators as deﬁned here. 3. In order to oﬀer the proof, we need some preparation. Let E be a Riesz space. A vector subspace V of E is called an (order) ideal of E if the following two conditions are satisﬁed: (i) If u ∈ V , then |u| ∈ V . (ii) If u ∈ V , v ∈ E, and 0 ≤ v ≤ u, then v ∈ V .

The idea of such a decomposition has appeared in Krylov and Bogolioubov [33]; the case of a Markov–Feller pair (S, T ) deﬁned on a compact metric space (X, d) was dealt with by Beboutoﬀ [5] and Yosida [68]. 3) and maps Cc (X) into Cc (X). ) The features of the decomposition will “come to life” in Chapter 2 where we will show that these features are preserved in our setting. Let (S, T ) be a Markov–Feller pair deﬁned on a locally compact separable metric space (X, d). Set D(S, T ) = x∈X the sequence (An (S)f (x))n∈N converges to zero whenever f ∈ C0 (X) , Γ0 (S, T ) = X \ D(S, T ), and Γc (S, T ) = x ∈ Γ0 (S, T ) the sequence (An (S)f (x))n∈N converges for every f ∈ C0 (X) .