Beyond Lebesgue and Baire: generic regular variation

N. H. Bingham¹ ; A. J. Ostaszewski²

¹ Mathematics Department Imperial College London London SW7 2AZ, UK
² Mathematics Department London School of Economics Houghton Street London WC2A 2AE, UK

Colloquium Mathematicum, Tome 116 (2009) no. 1, pp. 119-138

Résumé

We show that the No Trumps combinatorial property (NT), introduced for the study of the foundations of regular variation by the authors, permits a natural extension of the definition of the class of functions of regular variation, including the measurable/Baire functions to which the classical theory restricts itself. The “generic functions of regular variation” defined here characterize the maximal class of functions to which the three fundamental theorems of regular variation (Uniform Convergence, Representation and Characterization Theorems) apply. The proof uses combinatorial variants of the Steinhaus and Ostrowski Theorems deduced from NT in an earlier paper of the authors.

Détail

DOI : 10.4064/cm116-1-6

Keywords: trumps combinatorial property introduced study foundations regular variation authors permits natural extension definition class functions regular variation including measurable baire functions which classical theory restricts itself generic functions regular variation defined here characterize maximal class functions which three fundamental theorems regular variation uniform convergence representation characterization theorems apply proof uses combinatorial variants steinhaus ostrowski theorems deduced earlier paper authors