ON TYPICAL MARKOV OPERATORS ACTING ON BOREL MEASURES

Generic properties of different objects (functions, sets, measures, and many others) have been studied for a long time (see [1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 15, 16]). We say that some property is generic (or typical) if the subset of all elements satisfying this property is residual. Recall that a subset of a complete metric space is residual if its complement can be represented as a countable union of nowhere dense sets. Generic properties of Markov operators have been recently examined by Lasota and Myjak [11, 12]. Indeed, they have shown that the typical Markov operator corresponding to an iterated function system is asymptotically stable and its invariant measure is singular with respect to the Lebesgue measure (see [12]). This result has been recently extended to learning systems and stochastic perturbed dynamical systems (see [17, 18]). In [14], a more general result has been proved. Namely, most of the Markov operators in the class of all Markov operators acting on Borel measures inRd are asymptotically stable and have a singular stationary measure. Let (X ,ρ) be a complete and separable metric space. By B(x,r) we denote the open ball with center x and radius r > 0. Given a set A⊂ X and a number r > 0, we denote by diamA the diameter of the set A and by B(A,r) the r-neighbourhood of the set A, that is,


Introduction
Generic properties of different objects (functions, sets, measures, and many others) have been studied for a long time (see [1,2,3,4,5,7,8,9,10,13,15,16]).We say that some property is generic (or typical) if the subset of all elements satisfying this property is residual.Recall that a subset of a complete metric space is residual if its complement can be represented as a countable union of nowhere dense sets.
Generic properties of Markov operators have been recently examined by Lasota and Myjak [11,12].Indeed, they have shown that the typical Markov operator corresponding to an iterated function system is asymptotically stable and its invariant measure is singular with respect to the Lebesgue measure (see [12]).This result has been recently extended to learning systems and stochastic perturbed dynamical systems (see [17,18]).In [14], a more general result has been proved.Namely, most of the Markov operators in the class of all Markov operators acting on Borel measures in R d are asymptotically stable and have a singular stationary measure.
Let (X,ρ) be a complete and separable metric space.By B(x,r) we denote the open ball with center x and radius r > 0. Given a set A ⊂ X and a number r > 0, we denote by diamA the diameter of the set A and by B(A,r) the r-neighbourhood of the set A, that is, where ρ(x,A) = inf{ρ(x, y) : y ∈ A}.By Ꮾ(X) we denote the σ-algebra of all Borel subsets of X.By ᏹ we denote the family of all finite Borel measures on X, by ᏹ 1 the space of all µ ∈ ᏹ such that µ(X) = 1, and by ᏹ s = {µ 1 − µ 2 : µ 1 , µ 2 ∈ ᏹ} the space of all finite signed Borel measures on X.
Given µ ∈ ᏹ, we define the support of µ by the formula suppµ = x ∈ X : µ B(x,r) > 0 for every r > 0 . (1.2) As usual, by C(X) we denote the subspace of all bounded continuous functions.We consider this space with the supremum norm.
For f ∈ C(X) and µ ∈ ᏹ s , we will write We admit that ᏹ s is endowed with the Fortet-Mourier norm (see [6]) given by where An operator P : ᏹ → ᏹ is called a Markov operator if it satisfies the following conditions: (i) positive linearity: for λ 1 ,λ 2 ≥ 0 and µ 1 ,µ 2 ∈ ᏹ, (ii) preservation of measures: (1.6) A measure µ * is called invariant (or stationary) with respect to P if Pµ * = µ * .A Markov operator P is called asymptotically stable if there exists a stationary measure for every µ ∈ ᏹ 1 .Let ᏼ denote the set of all continuous Markov operators P : ᏹ → ᏹ, where ᏹ is endowed with the Fortet-Mourier metric.In this space, we introduce Clearly ρ is a distance and ᏼ with this distance is a complete metric space.
For A ⊂ X and s,δ > 0, define (1.9) The restriction of Ᏼ s to the σ-algebra of Ᏼ s -measurable sets is called the Hausdorff sdimensional measure.Note that all Borel sets are Ᏼ s -measurable.The value is called the Hausdorff dimension of the set A. As usual, we admit inf ∅ = +∞.
The Hausdorff dimension of a measure µ ∈ ᏹ 1 is defined by the formula We are in a position to formulate the main result of our note.
Theorem 1.1.Let ᏼ 0 denote the set of all P ∈ ᏼ such that P is asymptotically stable and its invariant measure µ P ∈ ᏹ 1 satisfies dim H (µ P ) = 0 and suppµ P = X.Then ᏼ 0 is residual in ᏼ.

Auxiliary results
In this section, we recall auxiliary results which are useful in the proof of the main theorem.Lemma 2.1 has been already proved in [19].On the other hand, Lemma 2.2 has been used in [14].Since the proofs of both lemmas may be easily presented here, they are included in this section.
for every Borel set A ⊂ X.
Proof.Consider the function f : X → [0,ε] given by the formula whence the statement of Lemma 2.1 follows.

4)
Then P is asymptotically stable.
Proof.Fix x ∈ X, P ∈ ᏼ, and ε > 0. Let {x m } m≥1 be a dense subset of X.Then Define the sets (2.12) The sets D i , i ≥ 1, are disjoint and cover X. Obviously diam D i ≤ ε, i = 1,2,.... Consider the operator Q : ᏹ → ᏹ given by where δ x means the δ-Dirac measure at x. Clearly Q is a Markov operator.

Tomasz Szarek 493
Now fix an x ∈ X and consider the operator Q : ᏹ → ᏹ given by where The proof of the above lemma can be found in the literature under slightly weaker assumptions.Namely, it has been shown that if lim r→0 (logµ(B(x,r)))/logr = 0 for all x of some compact set Y , then dim H Y = 0 (see [20]).For the convenience of the readers, we will give the proof of the lemma.

Proof of the main theorem
Proof.Fix x ∈ X and n ∈ N. Fix Q ∈ ᏼ x .By µ Q denote the unique invariant measure with respect to Q.By Lemma 2.
where α Q corresponds to Q according to Lemma 2.3.Set Let ε Q,n > 0 be such that ρ P kQ,n ,Q kQ,n < d 0 (3.4) for every Since the set Q∈ᏼx m B(Q,ε Q,n ) for every m,n ∈ N is open and dense, the set ᏼ is residual in ᏼ.Fix P ∈ ᏼ and m ∈ N. Let {Q n } n≥1 be a sequence of elements of ᏼ xm such that For abbreviation, we set Let µ 1 ,µ 2 ∈ ᏹ 1 .By Lemma 2.3 and conditions (3.2), (3.4), we have (3.7) Since µ 1 ,µ 2 ∈ ᏹ 1 were arbitrary and ᏹ 1 equipped with the Fortet-Mourier distance is complete, P admits an invariant measure.Moreover, P is asymptotically stable.Let µ * ∈ ᏹ 1 be its invariant measure.First we check that x m ∈ suppµ * .By Lemma 2.3(iii) and the choice of Q n , n ∈ N, we have Fix ε > 0. Let n 0 ∈ N be such that 2/n 0 < ε.By Lemma 2.1 and the definition of d 0 , we obtain We have shown that x m ∈ suppµ * for every m ∈ N. Since {x m } m≥1 is dense, suppµ * is closed, and x m ∈ suppµ * for every m ∈ N, we obtain suppµ * = X.This completes the proof.
Observe that Q is a Markov operator as well.Moreover, from Lemma 2.1, it follows that Q is asymptotically stable.Further, observe that x ∈ suppµ Q .Since µ Q is a purely atomic measure, condition (ii) holds.Finally, from the proof of Lemma 2.2, it follows that condition (iii) is satisfied and ρ(P,Q) < 3ε/2.Using a standard Vitali argument, one can prove the following lemma.Lemma 2.4.Let µ ∈ ᏹ 1 and Y ⊂ X be compact.If