Real interpolation method on spaces of scalar integrable functions with respect to vector measures

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 w p ( 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.