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Let (Ω,Σ, μ) be a complete finite measure space and X a Banach space. We show that the space of all weakly μ-measurable (classes of scalarly equivalent)X-valued Pettis integrable functions with integrals of finite variation, equipped with the variation norm, contains a copy of `∞ if and only if X does. 1. Preliminaries. Along this paper X will be a Banach space over the field K of real or complex numbers. If (Ω,Σ) is a measurable space, we denote by ca(Σ,X) the Banach space over K of all X-valued countably additive measures F on Σ equipped with the semivariation norm ‖F‖, and by bvca(Σ,X) the Banach space of allX-valued countably additive measures F of bounded variation on Σ with the variation norm |F |. Let ca+(Σ) denote the set of all positive and finite countably additive measures defined on Σ. Let us recall some useful facts. If (Ω,Σ, μ) is a finite measure space, a weakly μ-measurable function f : Ω → X is said to be Dunford integrable if x∗f ∈ L1(μ) for every x∗ ∈ X∗, and if f is Dunford integrable and E ∈ Σ the map x∗ 7→ E x ∗f dμ, denoted by (D) E f dμ, is a continuous linear form on X∗. If (D) E f dμ ∈ X for each E ∈ Σ then f is said to be Pettis integrable and one writes (P ) E f dμ instead of (D) E f dμ. The Pettis space of all weakly measurable (classes of scalarly equivalent) Pettis integrable functions f : Ω → X will be denoted by P1(μ,X) and the subspace of all those strongly measurable (classes of) functions by P1(μ,X). These spaces are provided with the semivariation norm ‖f‖P1(μ,X) = sup { Ω |x∗f(ω)| dμ(ω) : x∗ ∈ X∗, ‖x∗‖ ≤ 1 } . Neither P1(μ,X) nor P1(μ,X) is in general a Banach space, although they are barrelled normed spaces [4]. According to a result of Pettis, if f : Ω → X 2010 Mathematics Subject Classification: Primary 28B05; Secondary 46B03.

Copies of l∞in the space of Pettis integrable functions with integrals of finite variation

Copies of l_∞in the space of Pettis integrable functions with integrals of finite variation