L _p -spaces, 1 ≤ p ≤ ∞

Abstract

We develop the theory of L _p -spaces (1 ≤ p < ∞) for the Bartle-Dunford-Schwartz integral with respect to a Banach space-valued σ-additive vector measure m de- fined on a δ-ring of sets and obtain results analogous to those known for such spaces in the theory of the abstract Lebesgue and Bochner integrals. For this we adapt some of the techniques employed by Dobrakov in the study of integration with respect to operator-valued measures. Though a few of these results are already there in the literature for p = 1 (sometimes with incorrect proofs as observed in Chapter 2)), they are treated here differently with simpler proofs.