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

Abstract

Let $X$ be a Banach space, $\Sigma$ be a $\sigma$-algebra, and $m:\Sigma\to X$ be a (countably additive) vector measure. It is a well known consequence of the Davis-Figiel-Johnson-Pelcz\'{y}nski factorization procedure that there exist a reflexive Banach space $Y$, a vector measure $\tilde{m}:\Sigma \to Y$ and an injective operator $J:Y \to X$ such that $m$ factors as $m=J\circ \tilde{m}$. We elaborate some theory of factoring vector measures and their integration operators with the help of the isometric version of the Davis-Figiel-Johnson-Pelcz\'{y}nski factorization procedure. Along this way, we sharpen a result of Okada and Ricker that if the integration operator on $L_1(m)$ is weakly compact, then $L_1(m)$ is equal, up to equivalence of norms, to some $L_1(\tilde m)$ where $Y$ is reflexive; here we prove that the above equality can be taken to be isometric.