On the Banach lattice structure of $$L^1_w$$ of a vector measure on a $$\delta $$ -ring

Abstract

Abstract. We study some Banach lattice properties of the space L 1 w (ν) of weakly integrable functions with respect to a vector measure ν defined on a δ-ring. Namely, we analyze order continuity, order density and Fatou type properties. We will see that the behavior of L 1 w (ν) differs from the case in which ν is defined on a σ-algebra whenever ν does not satisfy certain local σ-finiteness property.

A Komlós Theorem for abstract Banach lattices of measurable functions

“…In fact this equality can be obtained in a more general setting (see [3,Theorem 3.2]). However, this equality is not enough for assuring the Komlós property for L 1 w (ν) as we will show after the next result.…”

“…In [3,11] the lattice properties of the spaces L 1 w (ν) are analyzed. In these papers, some weaker versions of σ -finiteness of ν became relevant for characterizing which lattice properties are satisfied by the spaces L 1 w (ν).…”

“…Assume that E is an ideal of L 1 w (ν) and that ν is a locally σ -finite vector measure. Consider a bounded sequence ( f n ) n in E. Then by Theorem 4.8 in [3] for each n there is a sequence (h n k ) k of R-simple functions that converges ν-a.e. In fact, as we show in the following proposition, local σ -finiteness of the vector measure implies (L 1 w (ν)) a = L 1 (ν).…”

Tensor product representation of Köthe-Bochner spaces and their dual spaces

“…This will allow to identify dual spaces of Köthe-Bochner spaces with spaces of operators with a particular factorization property. Our technique uses the nowadays well-known representation procedure for Banach function spaces by means of spaces of integrable functions with respect to vector measure (see [18,Ch.3] and [9,2] for more recent results). We show also some concrete representations of dual spaces of Köthe-Bochner spaces X(µ, Y ) for order continuous Banach function spaces X(µ) on a finite measure µ in terms of the elements of the dual space X(µ) * and some applications related with the natural tensor norms ∆ p on tensor products L p (µ) ⊗ Y .…”

The Köthe Dual of an Abstract Banach Lattice

“…Certainly, the spaces 1 (]) of integrable functions and 1 (]) of weakly integrable functions represent a large family of Banach lattices. Nowadays it is well known that each order continuous Banach lattice can be written (isometrically and in order) as an 1 (])-space of a certain vector measure ] on a -ring, and an equivalent result holds for Banach lattices with the Fatou property and some additional requirements with the spaces 1 (]) (see [1,Theorems 4 and 8] and [2]; see also [3, pp. 22-23]).…”