Functions with measurable differences

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evidence/: Retains the bounded independent review of the Theorem 3 statement.

theorem_3: Laczkovich's main theorem: a function on the reals whose every shift difference is Lebesgue measurable is the sum of a measurable function, an additive function and a function whose every shift difference vanishes almost everywhere, as Erdős conjectured.

theorem_4: Laczkovich's answer to Carroll's question: for every p > 0, a function on the reals whose every shift difference is periodic mod 1 and p-th power integrable over [0,1] splits as such a function plus an additive function plus a function with almost everywhere vanishing shift differences.

theorem_5: Laczkovich's double difference theorem: if f(x+y)-f(x)-f(y) is Lebesgue measurable as a function of two variables, then f is a Lebesgue measurable function plus an additive function, with no third summand.

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M. Laczkovich, Functions with measurable differences, Acta Mathematica Academiae Scientiarum Hungaricae 35 (1980), 217–235. Published volume record. The copy read for this card is a 19-page article extract, printed pp. 217–235, containing physical pp. 223–241 of the complete 484-page volume scan. The article introduction is printed p. 217 / extract p. 1, and Theorem 3 is printed p. 224 / extract p. 8. The extract prints no copyright line; the publisher's article page for DOI 10.1007/BF01896840 names the rights holder "© Akadémiai Kiadó" under Rights and permissions and offers the article as subscription content with no Creative Commons or open-access statement (read 2026-10-02), every other right reserved.

Theorem 3 says the class $L$ of Lebesgue measurable functions has the weak difference property. With the introduction's definition, if every real-shift difference of $f:\mathbb R\to\mathbb R$ is measurable, there is a pointwise decomposition $f=g+H+S$, with $g$ measurable, $H$ additive, and $\Delta_tS=0$ almost everywhere for each fixed real $t$. Exceptional null sets may depend on $t$. The theorem does not require or promise continuous $g$. Positive-shift measurability suffices by the elementary negative-shift identity recorded on Problem 908.

De Bruijn 1951 printed p. 195 and Mátrai 2003 printed pp. 1–2 corroborate this measurable formulation. Erdős 1982 printed p. 76 instead writes continuous $g$ and attributes a proof to Laczkovich. Problem 908 takes the measurable formulation as its corrected Statement, and Theorem 3 is not read as proving the continuous-summand wording.

Bears on. #908: Theorem 3 answers the corrected Statement, with a measurable summand, in the affirmative; it gives no continuous summand, the site's wording.

Results. Page numbers are the printed ones (pp. 217--235).

• Theorem 3 (p. 224): the class of Lebesgue measurable functions has the weak difference property; the paper's main result.
• Theorem 4 (p. 228): the classes $L_p(0,1)$ have the weak difference property for every $p>0$, answering Carroll's question for $0<p<1$.
• Theorem 5 (p. 229): if $f(x+y)-f(x)-f(y)$ is Lebesgue measurable on the plane, then $f$ is a measurable function plus an additive function; the paper's Section 3 applications (Theorems 7--9, pp. 232--233) rest on it.

Proof obligation. Compile and independently review the complete proof of Theorem 3 and its named same-paper dependencies. The exact statement and opening definitions were independently reviewed on 6 September 2026; the filed formulation review retains that report. That bounded review does not reconstruct the complete source proof or establish formal verification or additional publication acceptance. Primary reinspection: 6 September 2026 UTC.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.