Debruijn 1966 almost additive functions

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corollary_section_6: Finite outer product measure of the exceptional pairs still permits almost-everywhere correction on a group of infinite measure.

hartman_theorem: Additivity outside a fixed null set in each input forces additivity for every pair of real numbers.

main_theorem: A real function additive for almost every pair agrees almost everywhere with an everywhere additive function.

theorem_1: Extends almost-everywhere additivity from Lebesgue null sets to thin and light subsets of arbitrary abelian groups.

theorem_2: A Cauchy equation holding away from a thin set in each input holds on the whole abelian group.

theorem_3: Gives quantitative bounds for correcting an additive equation whose exceptional set has bounded outer product measure.

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de Bruijn, N. G., "On almost additive functions." Colloquium Mathematicum 15 (1966), 59--63. DOI: 10.4064/cm-15-1-59-63.

De Bruijn answers Erdős's Problem P 310 affirmatively. Here "almost all pairs" means outside a null set for two-dimensional Lebesgue measure, while equality of two one-variable functions almost everywhere means outside a null set for one-dimensional Lebesgue measure. Section 2 proves that if $f(x+y)=f(x)+f(y)$ for almost every $(x,y)\in\mathbb R^2$, then $f$ agrees almost everywhere with an additive function $h$.

The proof first uses Fubini's theorem to obtain a null set $M$ such that every vertical section above $x\notin M$ is null. For each fixed $x$, an auxiliary $x_1$ is then chosen outside $M\cup(x-M)$; this choice depends on $x$. It shows that $f(x+y)-f(y)$ is almost everywhere constant as a function of $y$, and that constant defines $h(x)$. To prove additivity, the paper chooses one pair $(w,z)$ outside five null sets: two coordinate cylinders, the inverse image of a one-dimensional null set under $(w,z)\mapsto w+z$, the original exceptional set, and a translate of that exceptional set. This supplies five compatible identities whose cancellation gives $h(a+b)=h(a)+h(b)$.

Section 3 derives Hartman's earlier theorem: if a null set $S\subset\mathbb R$ is excluded from each input separately and the equation holds whenever $x,y\notin S$, then it holds for every pair. Sections 4 and 5 abstract the argument to an ideal of "thin" subsets of an abelian group and the associated "light" subsets of its square. Section 6 gives a quantitative form and, for groups of infinite measure, a corollary allowing the exceptional subset of the square to have finite outer product measure.

For Jurkat's work, the introduction cites only his 1964 notice, which contained no proof. Jurkat's published 1965 paper does contain an independent proof; its conull sumset construction is treated separately in Jurkat 1965.

Source: https://www.impan.pl/en/publishing-house/journals-and-series/colloquium-mathematicum/all/15/1/96207/on-almost-additive-functions. No notice is printed on the scan's first and last pages; the publisher's article page offers the PDF as a "Free download under CC-BY license", a Creative Commons Attribution license with no version or URL named (read 2026-10-02).

Bears on. #1126

Results.

• [[analysis/debruijn_1966_almost_additive_functions/main_theorem|Main theorem (Section 2)]]: almost-everywhere additivity can be corrected on a null set.
• [[analysis/debruijn_1966_almost_additive_functions/hartman_theorem|Hartman's theorem (Section 3)]]: excluding a null set from each input does not create new solutions.
• Theorem 1: group-theoretic extension through thin and light sets.
• Theorem 2: the corresponding group-theoretic form of Hartman's theorem.
• Theorem 3: quantitative correction under an exceptional set of bounded outer measure.
• [[analysis/debruijn_1966_almost_additive_functions/corollary_section_6|Section 6 corollary]]: finite outer product measure suffices when the group has infinite measure.