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

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