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

N H Bingham,A J Ostaszewski

doi:10.1007/s00010-017-0486-7

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 {mathbb {Q}} to prove that the graph of a measurable sublinear function that is {mathbb {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.

Highlights

The Berz theorem of our title is his characterization of a function S : R → R which is sublinear, that is—it is subadditive ([50, Ch. 3], [87]): S(u + v) S(u) + S(v), and homogeneous with respect to non-negative integer scaling
The results above for the Baire/measurable functions on R are to be expected: they follow from the classical Bernstein–Doetsch continuity theorem for locally bounded, midpoint convex functions on normed vector spaces, to which we turn in Sect. 5
Fundamental for our purposes is the Steinhaus–Weil property1 [21, 22]—that the difference set A − A has a non-empty interior for any non-negligible set A with the Baire property, briefly: Baire set—as opposed to Baire topology

Summary

Introduction

The Berz theorem of our title is his characterization of a function S : R → R which is sublinear, that is—it is subadditive ([50, Ch. 3], [87]): S(u + v) S(u) + S(v), and homogeneous with respect to non-negative integer scaling. HB is derivable from the Prime Ideal Theorem, PI, an axiom weaker than AC: for literature see again [81,82]; HB for separable normed spaces is not provable from DC [32, Cor. 4] For more on this (with references), see Appendix 1 of the fuller arXiv version of this paper. The sector between the lines c±x in the upper half-plane is a two-dimensional cone This suggests the generalization to Banach spaces that we prove in Sect. The results above for the Baire/measurable functions on R are to be expected: they follow from the classical Bernstein–Doetsch continuity theorem for locally bounded, midpoint convex functions on normed vector spaces, to which we turn in Sect. It is here that further set-theoretic assumptions become crucial; in brief, measure theory needs stronger assumptions

Topological preliminaries

Sublinearity and Berz’s Theorem

Banach-space versions

Convexity

Complements