1. CMB 2011 (vol 54 pp. 302)
|Structure of the Set of Norm-attaining Functionals on Strictly Convex Spaces |
Let $X$ be a separable non-reflexive Banach space. We show that there is no Borel class which contains the set of norm-attaining functionals for every strictly convex renorming of $X$.
Keywords:separable non-reflexive space, set of norm-attaining functionals, strictly convex norm, Borel class
Categories:46B20, 54H05, 46B10
2. CMB 2010 (vol 54 pp. 180)
|Additive Families of Low Borel Classes and Borel Measurable Selectors|
An important conjecture in the theory of Borel sets in non-separable metric spaces is whether any point-countable Borel-additive family in a complete metric space has a $\sigma$-discrete refinement. We confirm the conjecture for point-countable $\mathbf\Pi_3^0$-additive families, thus generalizing results of R. W. Hansell and the first author. We apply this result to the existence of Borel measurable selectors for multivalued mappings of low Borel complexity, thus answering in the affirmative a particular version of a question of J. Kaniewski and R. Pol.
Keywords:$\sigma$-discrete refinement, Borel-additive family, measurable selection