Measures on effect algebras

We study measures defined on effect algebras. We characterize real-valued measures on effect algebras and find a class of effect algebras, that include the natural effect algebras of sets, on which σ-additive measures with values in a finite dimensional Banach space are always bounded. We also prove that in effect algebras the Nikodym and the Grothendieck properties together imply the Vitali-Hahn-Saks property, and find an example of an effect algebra verifying the Vitali-Hahn-Saks property but failing to have the Nikodym property. Finally, we define the concept of variation for vector measures on effect algebras proving that in effect algebras verifying the Riesz Decomposition Property, the variation of a finitely additive vector measure is a finitely additive positive measure.

Lattice Copies ofℓ2inL1of a Vector Measure and Strongly Orthogonal Sequences

Letmbe anℓ2-valued (countably additive) vector measure and consider the spaceL2(m) of square integrable functions with respect tom. The integral with respect tomallows to define several notions of orthogonal sequence in these spaces. In this paper, we center our attention in the existence of stronglym-orthonormal sequences. Combining the use of the Kadec-Pelczyński dichotomy in the domain space and the Bessaga-Pelczyński principle in the range space, we construct a two-sided disjointification method that allows to prove several structure theorems for the spacesL1(m) andL2(m). Under certain requirements, our main result establishes that a normalized sequence inL2(m) with a weakly null sequence of integrals has a subsequence that is stronglym-orthonormal inL2(m∗), wherem∗is anotherℓ2-valued vector measure that satisfiesL2(m) = L2(m∗). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to anℓ2-valued positive vector measure contains a lattice copy ofℓ2.

ON CONNECTIONS BETWEEN DELTA-CONVEX MAPPINGS AND CONVEX OPERATORS

We study conditions under which every delta-convex (d.c.) mapping is the difference of two continuous convex operators, and vice versa. In particular, we prove that each d.c. mapping $F:(a,b)\to Y$ is the difference of two continuous convex operators whenever $Y$ belongs to a large class of Banach lattices which includes all $L^{p}(\mu)$ spaces ($1\leq p\leq\infty$). The proof is based on a result about Jordan decomposition of vector-valued functions. New observations on Jordan decomposition of finitely additive vector-valued measures are also presented.

The bounded vector measure associated to a conical measure and pettis differentiability

Let X be a locally convex space. Kluvánek associated to each X-valued countably additive vector measure a conical measure on X; this can also be done for finitely additive bounded vector measures. We prove that every conical measure u on X, whose associated zonoform Ku is contained in X, is associated to a bounded additive vector measure σ(u) defined on X, and satisfying σ(u)(H) ∈ H, for every finite intersection H of closed half-spaces. When X is a complete weak space, we prove that σ(u) is countably additive. This allows us to recover two results of Kluvánek: for any X, every conical measure u on it with Ku ⊆ X is associated to a countably additive X-valued vector measure; and every conical measure on a complete weak space is localizable. When X is a Banach space, we prove that σ(u) is countably additive if and only if u is the conical measure associated to a Pettis differentiable vector measure.

On copies of the null sequence Banach space in some vector measure spaces

In this note we extend a result of Drewnowski concerning copies of C0 in the Banach space of all countably additive vector measures and study some properties of complemented copies of C0 in several Banach spaces of vector measures.