Equivalence of D and D' properties in Banach spaces

Abstract

This work explores the equivalence of two sequential properties, $D$ and $D'$, for dual Banach spaces under the weak* topology. Property $D$ ensures that any totally scalarly measurable function is also scalarly measurable, while property $D'$ states that every weakly* sequentially closed subspace of $X^*$ is weakly* closed. These properties, which are central to the study of the interplay between topology and measurability in Banach spaces, was left as an open question. By examining the topological and measurable structures induced by the Baire $\sigma$-algebra, we prove that properties $D$ and $D'$ are indeed equivalent. The proof utilizes the relationship between total sets, weak* closures, and scalar measurability, extending previous results on sequential properties of dual Banach spaces. Additionally, we revisit the failure of property $D$ in nonseparable Banach spaces with $M$-basic $\ell_1^+$-systems, providing a topological reinterpretation of this phenomenon. These findings contribute to a deeper understanding of the weak* topology and measurable mappings in Banach space theory.