We study several properties of the Banach lattices L p (m) and L p w (m) of p-integrable scalar functions and weakly p-integrable scalar functions with respect to a countably additive vector measure m. The relation between these two spaces plays a fundamental role in our analysis. Mathematics Subject Classification 2000: 46G10, 46E30

For a given measurable space (Ω, Σ), and a vector measure m : Σ −→ X with values in a Banach space X we consider the spaces of p-power integrable and weakly integrable, respectively, functions with respect to the measure m, L p (m) and L p w (m), for 1 p < ∞.In this note we describe the real interpolated spaces that we obtain when the K -method is applied to any couple of these spaces.

Let (Ω, Σ) be a measurable space and m : Σ → X be a vector measure with values in the complex Banach space X. We apply the Calderón interpolation methods to the family of spaces of scalar p−integrable functions with respect to m with 1 ≤ p ≤ ∞. Moreover we obtain a result about the relation between the complex interpolation spaces[θ] for a Banach couple of interpolation (X 0 , X 1 ) such that X 1 ⊂ X 0 with continuous inclusion.

We study the structure of bounded sets in the space L ' ( E } of absolutely integrable Lusin-measurable functions with values in a locally convex space E. The main idea is to extend the notion of property (B) of Pietsch, defined within the context of vector-valued sequences, to spaces of vector-valued functions. We prove that this extension, that at first sight looks more restrictive, coincides with the original property ( B ) for quasicomplete spaces. Then we show that when dealing with a locally convex space, property (B) provides the link to prove the equivalence between Radon-Nikodym property (the existence of a density function for certain vector measures) and the integral representation of continuous linear operators T:L' -. E, a fact well-known for Banach spaces. We also study the relationship between Radon-Nikodym property and the characterization of the dual of L'{E} as the space LODIE;}.

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