Equivalence of D and D' properties in Banach spaces

TL;DR

This paper proves the equivalence of two key properties, D and D', in dual Banach spaces under the weak* topology, clarifying their relationship and implications for measurability and topology.

Contribution

It establishes the equivalence of properties D and D' in dual Banach spaces, resolving an open question and extending understanding of weak* topology and measurability.

Findings

Properties D and D' are equivalent in dual Banach spaces.

Failure of property D in nonseparable spaces is topologically reinterpreted.

The results deepen understanding of weak* topology and scalar measurability.

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.