Beyond Lebesgue and Baire II: Bitopology and measure-category duality

Abstract

We re-examine measure-category duality by a bitopological approach, using both the Euclidean and the density topologies of the line. We give a topological result (on convergence of homeomorphisms to the identity) obtaining as a corollary results on infinitary combinatorics due to Kestelman and to Borwein and Ditor. We hence give a unified proof of the measure and category cases of Uniform Convergence Theorem for slowly varying functions. We also extend results on very slowly varying functions of Ash, Erdős and Rubel.

Beurling regular variation, Bloom dichotomy, and the Gołąb–Schinzel functional equation

“…We summarize from [BinO7] the combinatorial framework needed here: Baire and measurable cases are handled together by working bi-topologically, using the Euclidean topology in the Baire case (the primary case) and the density topology in the measure case; see [BinO2], [BinO5], [BinO4]. We work in the a¢ ne group Af f acting on (R; +) using the notation n (t) = c n t + z n ; where c n !…”

Kingman, category and combinatorics

“…We give a new uni…ed proof of the measure and Baire cases in Section 5, based on almost complete metrizability (see below); earlier uni…cation was achieved through a bitopological approach (as here to the Kingman Theorem) in [BOst11]. This result is a theorem about additive in…nite combinatorics.…”

“…2. The Lemma addresses d-open sets but also holds in the metric topology (the proof is similar but simpler), and so may be restated bitopologically (from the viewpoint of [BOst11]) as follows.…”

“…In Section 3 we generalize to a Baire/measurable setting the Kingman Theorem (originally stated for open sets). Our (bi-)topological approach (borrowed from [BOst11]) allows the two cases to be treated as one that, by specialization, yields either case. (This is facilitated by the density topology).…”

Automatic continuity via analytic thinning

“…3]) and rediscovered by Trautner [Trau]. Much more is true; see [BOst6], [BOst9], [BOst11]. Following J.-P. Kahane [Kah], the term 'quasi all' below refers to 'all off some meagre set'.…”