Category-measure duality: convexity, midpoint convexity and Berz sublinearity

Abstract

Abstract. Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice, DC rather than the Axiom of Choice, AC. Berz used the Hahn-Banach theorem over Q to prove that the graph of a measurable sublinear function that is Q + -homogeneous consists of two half-lines through the origin. We give a category form of the Berz theorem. Our proof is simpler than that of the classical measure-theoretic Berz theorem, our result contains Berz's theorem rather than simply being an analogue of it, and we use only DC rather than AC. Furthermore, the category form easily generalizes: the graph of a Baire sublinear function defined on a Banach space is a cone. The results are seen to be of automatic-continuity type. We use Christensen Haar null sets to extend the category approach beyond the locally compact setting where Haar measure exists. We extend Berz's result from Euclidean to Banach spaces, and beyond. Passing from sublinearity to convexity, we extend the Bernstein-Doetsch theorem and related continuity results, allowing our conditions to be 'local'-holding off some exceptional set.Mathematics Subject Classification. Primary 26A03, 39B62.

Set theory and the analyst

“…In spirit he follows Hamel's construction of a discontinuous additive function, and so ultimately this rests on transfinite induction of continuum length requiring continuum many selections. Our own proof [BinO6] (cf. [BinO7,10]) of Berz's theorem, taken in a wider context including Banach spaces, depends in effect on the Baire Category Theorem BC, or the completeness of R (in either of the distinct roles of 'Cauchy-sequential' and 'Cauchy-filter' completeness, the latter stronger in the absence of AC, see [FosM,§3] and also [DodM,§7,§2]): we rely on generalizations of the Kestelman-Borwein-Ditor Theorem, KBD, asserting that for any (category/measure theoretic) non-negligible set T and any null sequence z n → 0, for quasi all t ∈ T the t-translate of some subsequence z n(m) (dependent on t) embeds in T, i.e.…”

“…In other variants the quantification over x may also be thinned -see [BinO6]. In electing to study sublinear functions as possible realizations of norms, Berz ([Berz], [BinO6]) showed, for measurable f, that the graph of f is conical -comprises two half lines through the origin; however, his argument relied on AC, in the usual form of Zorn's Lemma, which he used in the context of R over the field of scalars Q. In spirit he follows Hamel's construction of a discontinuous additive function, and so ultimately this rests on transfinite induction of continuum length requiring continuum many selections.…”

“…It turns out that such sets have the Baire/measurable property -see [FenN], where these are generalized to the universally (=absolutely) measurable sets (cf. [BinO6,§2]); the idea is ascribed to Solovay in [Kan,Ch. 3 Ex.…”

General regular variation, Popa groups and quantifier weakening

“…See [BinO3] for conditions under which this property is implied by the interior-point property of the sets Σ δ x − Σ δ x (cf. [BarFN]); see also the rich list of examples below, which are used in [BinO8,10,11,13,14], [MilMO]. 2.…”

Beyond Haar and Cameron-Martin: The Steinhaus support

“…In an abelian (N-) divisible group G, since its group-norm ||.|| is subadditive, we follow Berz [Ber] (cf. [BinO3,5,6]) in calling it sublinear if ||ng|| = n||g|| (g ∈ G, n ∈ N) (so that ||g/n|| = ||g||/n), or equivalently and more usefully:…”