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,