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Vector Measures, Integration and Related Topics
Let X and Y be Banach spaces and (Ω, Σ, µ) a finite measure space. In this note we introduce the space L p [µ; L (X, Y )] consisting of all (equivalence classes of) functions Φ : Ω → L (X, Y ) such that ω → Φ(ω)x is strongly µ-measurable for all x ∈ X and ω → Φ(ω)f (ω) belongs to L 1 (µ; Y ) for all f ∈ L p ′ (µ; X), 1/p + 1/p ′ = 1. We show that functions in L p [µ; L (X, Y )] define operator-valued measures with bounded p-variation and use these spaces to obtain an isometric characterizationdoi:10.1007/978-3-0346-0211-2_6 fatcat:2emzjejpuve5fhwcnofpnhlmai