Factorization of vector measures and their integration operators
Let $X$ be a Banach space and $\nu $ a countably additive $X$-valued measure defined on a $\sigma $-algebra. We discuss some generation properties of the Banach space $L^1(\nu )$ and its connection with uniform Eberlein compacta. In this way, we provide a new proof that $L^1(\nu )$ is weakly compactly generated and embeds isomorphically into a Hilbert generated Banach space. The Davis–Figiel–Johnson–Pełczyński factorization of the integration operator $I_\nu : L^1(\nu )\to X$ is also analyzed. As a result, we prove that if $I_\nu $ is both completely continuous and Asplund, then $\nu $ has finite variation and $L^1(\nu )=L^1(|\nu |)$ with equivalent norms.