Isometric factorization of vector measures and applications to spaces of integrable functions

Abstract

Let X be a Banach space, Σ be a σ-algebra, and m:Σ→X be a (countably additive) vector measure. It is a well known consequence of the Davis-Figiel-Johnson-Pełczyński factorization procedure that there exist a reflexive Banach space Y, a vector measure m˜:Σ→Y and an injective operator J:Y→X such that m factors as m=J∘m˜. We elaborate some theory of factoring vector measures and their integration operators with the help of the isometric version of the Davis-Figiel-Johnson-Pełczyński factorization procedure. Along this way, we sharpen a result of Okada and Ricker that if the integration operator on L1(m) is weakly compact, then L1(m) is equal, up to equivalence of norms, to some L1(m˜) where Y is reflexive; here we prove that the above equality can be taken to be isometric.