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.

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.

Remarks on the range of a vector measure

A long-standing problem is the characterization of subsets of the range of a vector measure. It is known that the range of a countably additive vector measure is relatively weakly compact and, in addition, possesses several interesting properties (see [2]). In [6] it is proved that if m: Σ → Χ is a countably additive vector measure, then the range of m has not only the Banach–Saks property, but even the alternate Banach-Saks property. A tantalizing conjecture, which we shall disprove in this article, is that the range of m has to have, for some p > 1, the p-Banach–Saks property. Another conjecture, which has been around for some time (see [2]) and is also disproved in this paper, is that weakly null sequences in the range of a vector measure admit weakly-2-summable sub-sequences. In fact, we shall show a weakly null sequence in the range of a countably additive vector measure having, for every p < ∞, no weakly-p-summable sub-sequences.