Uniformity and self-neglecting functions: II. Beurling Regular Variation and the class {\Gamma}

TL;DR

This paper extends the theory of Beurling regular variation, proving a uniform convergence theorem for measurable functions, and characterizes the gamma class within de Haan's framework.

Contribution

It introduces a uniform convergence theorem for Beurling regular variation functions and extends the gamma class of de Haan theory.

Findings

Proved a new uniform convergence theorem for measurable Beurling regular variation functions.

Characterized and represented functions within the extended gamma class.

Extended de Haan's gamma class to include these generalized functions.

Abstract

Beurling slow variation is generalized to Beurling regular variation. A Uniform Convergence Theorem, not previously known, is proved for those functions of this class that are measurable or have the Baire property. This permits their characterization and representation. This extends the gamma class of de Haan theory studied earlier.