Saeki’s improvement of the Vitali-Hahn-Saks-Nikodým theorem holds precisely for Banach spaces having cotype

Abstract:

We prove that a Banach space $X$ has nontrivial cotype if and only if given any $\sigma$-field $\Sigma$ and any sequence ${\mu _n}:\Sigma \to X$ of strongly additive vector measures such that for some $\gamma \geq 1,\lim {\sup _{n \to \infty }}\left \| {{\mu _n}\left ( E \right )} \right \| \leq \gamma \lim {\inf _{n \to \infty }}\left \| {{\mu _n}\left ( E \right )} \right \| < \infty$ for each $E \in \Sigma$ then $\left \{ {{\mu _n}:n \in \mathbb {N}} \right \}$ is uniformly strongly additive.