Difference functions of periodic measurable functions

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Tamás Keleti, Difference functions of periodic measurable functions, Fundamenta Mathematicae 157 (1998), 15–32. Published PDF. Retained PDF. The retained PDF has 18 physical pages; its first and last printed pages are 15 and 32. The file's text layer carries no copyright or license line; the publisher's issue listing marks the article "Free download under CC-BY license", as it marks every article in the issue, and names no Creative Commons version or URL (https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/157/1, read 2026-10-02; the article's own page was not opened), so the term is the Creative Commons Attribution license with its version unstated; the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

Notation 2.1, printed p. 18 / PDF p. 4, defines $C^*$ as functions equal almost everywhere to a continuous function. Corollary 2.18, printed p. 23 / PDF p. 9, classifies the exact essentially-continuous-difference shift subgroups for essentially bounded measurable periodic functions ($L_\infty$) not in $C^*$ as the finite subgroups of $\mathbb T$. Its reverse-inclusion proof explicitly uses $q_n(x)=\operatorname{sgn}(\sin(2\pi n x))$, with shift set $\{0,1/n,\ldots,(n-1)/n\}$ modulo one.

Theorem 2.9, printed p. 21 / PDF p. 7, assumes $f:\mathbb R\to\mathbb R$ measurable and every $\Delta_tf$ essentially continuous, and concludes $f$ essentially continuous. Theorem 2.13, printed p. 22 / PDF p. 8, gives the weak difference property for $C^*$, using the definition on p. 21. No mere-measurability-to-$C^*$ inference is made.

Bears on. #908, a stronger-hypothesis variant.

Proof scope. The full proof of Corollary 2.18's other inclusion and the general Theorems 2.9 and 2.13 remain source-proof obligations. No complete general source proof or formal verification is recorded. Relevant pages were locally re-read.