This paper by J. Diestel and B. Faires explores four related but distinct topics in the theory of vector measures. The first section discusses the conditions under which a Banach space \(X\) ensures that bounded additive \(X\)-valued maps on \(\sigma\)-algebras are strongly bounded, specifically that \(X\) cannot contain a copy of \(L_\infty\). The second section focuses on the Jordan decomposition for measures with values in \(L_1\)-spaces and \(c_0(\Gamma)\) spaces. The third section addresses the integrability of scalar functions with respect to vector measures, generalizing a result by D. R. Lewis on the equivalence of scalar integrability and non containment of \(c_0\). The final section examines the Radon-Nikodym theorem for vector measures, providing new sufficient conditions for a Banach space to have the Radon-Nikodym property, such as having an equivalent very smooth norm. The paper concludes with several open questions and remarks, highlighting ongoing research directions in the field.This paper by J. Diestel and B. Faires explores four related but distinct topics in the theory of vector measures. The first section discusses the conditions under which a Banach space \(X\) ensures that bounded additive \(X\)-valued maps on \(\sigma\)-algebras are strongly bounded, specifically that \(X\) cannot contain a copy of \(L_\infty\). The second section focuses on the Jordan decomposition for measures with values in \(L_1\)-spaces and \(c_0(\Gamma)\) spaces. The third section addresses the integrability of scalar functions with respect to vector measures, generalizing a result by D. R. Lewis on the equivalence of scalar integrability and non containment of \(c_0\). The final section examines the Radon-Nikodym theorem for vector measures, providing new sufficient conditions for a Banach space to have the Radon-Nikodym property, such as having an equivalent very smooth norm. The paper concludes with several open questions and remarks, highlighting ongoing research directions in the field.