Idempotent integration is an analogue of the Lebesgue integration where σ-additive measures are replaced by σ-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon–Nikodym derivatives, which turns out to be crucial in all of these applications, was prov...

Conditional expectation is one of the most useful tools of probability. The Radon-Nikodym theorem enables us to construct conditional expectations. Definition 1. A measure µ on (Ω, F) is σ-finite iff there is A n ∈ F satisfying A n ↑ Ω where µ(A n) < ∞ for all n. Definition 2. Let µ, ν be measures on (Ω, F). Then ν is absolutely continuous with respect to µ (ν << µ) iff for all A ∈ F, µ(A) = 0 ...

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.

These notes give various references to the literature that I will not, for reasons of brevity, give during the lectures themselves. 1. Gaussian Measures A comprehensive reference for Gaussian measures is [Bog98]. This book works in the generality of locally convex topological spaces, and has material on Gaussian measure on Banach and Hilbert spaces as particular cases. A similar book, focussed ...

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...

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 ...