Saeki's Improvement of the Vitali-Hahn-Saks-Nikodym Theorem Holds Precisely for Banach Spaces having Cotype

Paul Abraham
1992 Proceedings of the American Mathematical Society  
We prove that a Banach space X has nontrivial cotype if and only if given any cr-field I and any sequence pn:~L -> X of strongly additive vector measures such that for some y > 1 , limsup,_00 ||/zB(£)|| < yliminf"_oo ||//n(.E)|| < °° for each E £ I. then {pn:n £ N} is uniformly strongly additive.
doi:10.2307/2159310 fatcat:yu2zvsmnlrfv3fyagrvyettqae