Category-measure duality: convexity, midpoint convexity and Berz sublinearity
File(s)
Author(s)
Bingham, NH
Ostaszewski, AJ
Type
Journal Article
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.
Date Issued
2017-10-01
Date Acceptance
2017-07-01
Citation
Aequationes Mathematicae, 2017, 91 (5), pp.801-836
ISSN
0001-9054
Publisher
Springer Verlag
Start Page
801
End Page
836
Journal / Book Title
Aequationes Mathematicae
Volume
91
Issue
5
Copyright Statement
© 2017 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000410734400001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
Dependent Choice
Subadditive
Sublinear
Midpoint convex
Density topology
Steinhaus-Weil property
Baire topology
Quasi cover
Quasi sigma-continuity
Left Haar null
HAAR NULL SETS
ABELIAN POLISH GROUPS
SUBADDITIVE FUNCTIONS
REGULAR VARIATION
AUTOMATIC-CONTINUITY
LINEAR-SPACES
MEAGER SETS
THEOREM
BAIRE
EXTENSIONS
Publication Status
Published
Date Publish Online
2017-07-01