Approximate modularity: Kalton's constant is not smaller than 3

Michal Gnacik, Marcin Guzik, Tomasz Kania

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Abstract

Kalton and Roberts [Trans. Amer. Math. Soc., 278 (1983), 803-816] proved that there exists a universal constant K ⩽ 44.5 such that for every set algebra F and every 1-additive function ƒ : F → R there exists a finitely-additive signed measure μ defined on F such that |ƒ(A) - μ(A)| ⩽ K for any A ε F. The only known lower bound for the optimal value of K was found by Pawlik [Colloq. Math., 54 (1987), 163-164], who proved that this constant is not smaller than 1.5; we improve this bound to 3 already on a non-negative 1-additive function.
Original languageEnglish
Article number0
Pages (from-to)661-669
Number of pages9
JournalProceedings of the American Mathematical Society
Volume149
Issue number2
DOIs
Publication statusPublished - 22 Oct 2020

Keywords

  • 1-additive set function
  • Kalton's constant
  • Ulam-Hyers stability
  • approximate modularity

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