An integral representation theorem is proved. Each continuous function from a totally disconnected compact space to the probability measures on a complete metric space is shown to be the resolvent of a probability measure on the space of continuous functions from to .
We define a notion of volume for sets definable in an o-minimal structure on an archimedean real closed field. We show that given a parametric family of continuous functions on the positive cone of an archimedean real closed field definable in an o-minimal structure, the set of parameters where the integral of the function converges is definable in the same structure.
Let X, Y be uncountable Polish spaces and let μ be a complete σ-finite Borel measure on X. Denote by K and L the families of all meager subsets of X and of all subsets of Y with μ measure zero, respectively. It is shown that the product of the ideals K and L restricted to C-sets of Selivanovskiĭ is σ-saturated, which extends Gavalec's results.
The two main results of this paper are the following: (a) If X is a Banach space and f : [a,b] → X is a function such that x*f is Denjoy integrable for all x* ∈ X*, then f is Denjoy-Dunford integrable, and (b) There exists a Dunford integrable function which is not Pettis integrable on any subinterval in [a,b], while belongs to for every subinterval J in [a,b]. These results provide answers to two open problems left by R. A. Gordon in [4]. Some other questions in connection with Denjoy-Dundord...
Differentiation of integrals of functions from the class with respect to the basis of convex sets is established. An estimate of the rate of differentiation is given. It is also shown that there exist functions in , N ≥ 3, and with ω(δ)/δ → ∞ as δ → +0 whose integrals are not differentiated with respect to the bases of convex sets in the corresponding dimension.
Special flows over some locally rigid automorphisms and under L² ceiling functions satisfying a local L² Denjoy-Koksma type inequality are considered. Such flows are proved to be disjoint (in the sense of Furstenberg) from mixing flows and (under some stronger assumption) from weakly mixing flows for which the weak closure of the set of all instances consists of indecomposable Markov operators. As applications we prove that ∙ special flows built over ergodic interval exchange...
We study the Hausdorff dimension of measures whose weight distribution satisfies a Markov non-homogeneous property. We prove, in particular, that the Hausdorff dimensions of this kind of measures coincide with their lower Rényi dimensions (entropy). Moreover, we show that the packing dimensions equal the upper Rényi dimensions. As an application we get a continuity property of the Hausdorff dimension of the measures, when viewed as a function of the distributed weights under the norm.