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Weak Baire measurability of the balls in a Banach space. (English) Zbl 1147.46016
Author’s abstract: Let \(X\) be a Banach space. The property \((\star)\) “the unit ball of \(X\) belongs to Baire(\(X\), weak)” holds whenever the unit ball of \(X^*\) is weak\(^*\)-separable; on the other hand, it is also known that the validity of \((\star)\) ensures that \(X^*\) is weak\(^*\)-separable. In this paper, we use suitable renormings of \(\ell^{\infty}({\mathbb N})\) and the Johnson–Lindenstrauss spaces to show that \((\star)\) lies strictly between the weak\(^*\)-separability of \(X^*\) and that of its unit ball. As an application, we provide a negative answer to a question raised by K. Musiał [Rend. Ist. Mat. Univ. Trieste 23, 177–262 (1991; Zbl 0798.46042).
Reviewer: Hans Weber (Udine)

46B26 Nonseparable Banach spaces
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
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