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.

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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.

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