In this talk basic general properties of f -divergences, including their axiomatic, and some important classes of f -divergences are presented. Without essential loss of insight we restrict ourselves to discrete probability distributions and note that the extension to the general case relies strongly on the Lebesgue-Radon-Nikodym Theorem.

In this paper, we examine the Radon-Nikodym property and its relation to the Bishop-Phelps theorem for complex Banach spaces. We also show that the Radon-Nikodym property implies the Bishop-Phelps property in the complex case. 1. Introduction. Let X be a complex Banach space and let C be a closed convex subset of X. The set of support points of C, written as supp C, is the collection of all poi...

To demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the Banach-Zareckitheorem is presented on the basis of the Radon-Nikodym theoremwhich emphasizes on measure-type properties of the Lebesgueintegral. The Banach-Zarecki theorem says that a real-valuedfunction $F$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

Let X and Y be Polish spaces with non-atomic Borel measures μ and ν of full support. Suppose that T and S are ergodic non-singular homeomorphisms of (X, μ) and (Y, ν) with continuous Radon-Nikodym derivatives. Suppose that either they are both of type III1 or that they are both of type IIIλ, 0 < λ < 1 and, in the IIIλ case, suppose in addition that both ‘topological asymptotic ranges’ (defined ...

Introduction. Let C(S) be the ring of continuous real valued functions on a compact Hausdorff space S. Stone [5] shows that each bounded subset of C(S) has a least upper bound (in C(S)) if and only if the closure of each open subset of S is open ; in this event we call C(S) a Stone algebra. Throughout this paper C(S) is a Stone algebra. It is convenient to adjoin an object +00, not in C(S), and...

Abstract Let Ω be a compact Hausdorff space, let E be a Banach space, and let C(Ω, E) stand for the Banach space of all E-valued continuous functions on Ω under supnorm. In this paper we study when nuclear operators on C(Ω, E) spaces can be completely characterized in terms of properties of their representing vector measures. We also show that if F is a Banach space and if T : C(Ω, E) → F is a ...

We present a micro-macro strategy able to describe the dynamics of crowds in heterogeneous media. Herein we focus on the example of pedestrian counterflow. The main working tools include the use of mass and porosity measures together with their transport as well as suitable application of a version of Radon-Nikodym Theorem formulated for finite measures. Finally, we illustrate numerically our m...

Two generalizations of Komll os' theorem for functions and multifunctions with values in a Banach space are presented. The rst generalization is partly new and the second one is a Komll os-type result of a completely new nature. It requires the Radon-Nikodym property for the Banach space and its dual. In both cases our approach relies on using Komll os' theorem by means of a diagonal extraction...

Araci et al. introduced a $$p$$ -adic $$(\rho,q)$$ -analogue of the Haar distribution. By means distribution, they constructed -Volkenborn integral. In this paper, by virtue Mahler expansion continuous functions, author gives Radon-Nikodym theorem with respect to -distribution on $$\Bbb Z_p$$ .