Radon-Nikodym derivatives in vector integration
For a Banach space and a probability space , a new proof is given that a measure , with , has RN derivative with respect to iff there is a compact or a weakly compact such that is a finite valued countably additive measure. Here we define where is a finite disjoint collection of elements from , each contained in , and satisfies . Then the result is extended to the case when is a Frechet space.
It is proved that if a Frechet space has property, then also has property, for .
Let (T,F,μ) be a separable probability measure space with a nonatomic measure μ. A subset K ⊂ L(T,Rⁿ) is said to be decomposable if for every A ∈ F and f ∈ K, g ∈ K one has . Using the property of decomposability as a substitute for convexity a relaxation theorem for fixed point sets of set-valued function is given.
In this paper we consider some spaces of differentiable multifunctions, in particular the generalized Orlicz-Sobolev spaces of multifunctions, we study completeness of them, and give some theorems.
We give a quantitative characterization of the pairs of weights for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak type inequality for . More precisely, given any measurable set , the estimate holds if and only if the pair belongs to , that is, for every dyadic cube and every measurable set . The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the non-dyadic...
We prove an existence theorem for the equation x' = f(t,xₜ), x(Θ) = φ(Θ), where xₜ(Θ) = x(t+Θ), for -r ≤ Θ < 0, t ∈ Iₐ, Iₐ = [0,a], a ∈ R₊ in a Banach space, using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function f satisfies some conditions expressed in terms of the measure of weak noncompactness.
The McShane and Kurzweil-Henstock integrals for functions taking values in a locally convex space are defined and the relations with other integrals are studied. A characterization of locally convex spaces in which Henstock Lemma holds is given.