On some properties of the space of tensor integrable functions

Abstract

A b s t r a c t . The tensor integral of a vector valued function f : Ω → X with respect to a countably additive vector valued measure ν : Σ → Y has been defined by Stefansson in [14] and he has investigated many of its properties. The integral is an element of the injective tensor product X⊗ Y . We study the Banach space L1(ν, X, Y ) of all⊗-integrable functions and discuss many properties of this space. We also study the space w-L1(ν, X, Y ) of all weakly⊗-integrable functions.

A note on $L^p_{w}(ν,X,Y)$ spaces of vector-valued functions with respect to vector measures

“…In [CB07] additional properties of L 1 (ν, X, Y ) are shown as the fact of being a Banach lattice, a separable space and the density of the set of simple functions in L 1 (ν, X, Y ). Then in 2008 in [CB08], define the spaces L p (ν, X, Y ) for 1 < p < ∞ in the following way: Definition 2.6.…”