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Claim. Let $f:\mathbb{R}\to\mathbb{R}$ be such that, for every real $t$, the difference $\Delta_tf(x)=f(x+t)-f(x)$ is Lebesgue measurable in $x$. Then there is a pointwise decomposition $f=g+H+S$ with $g$ measurable, $H$ additive and, for every fixed real $t$, $\Delta_tS=0$ outside a null set that may depend on $t$. This is Theorem 3 of Laczkovich's paper, which says that the class of Lebesgue measurable functions has the weak difference property; the paper's digest is the [[../library/analysis/laczkovich_1980_functions_measurable_differences/_index|source card]]. It answers the corrected Statement of Problem 908, whose summand $g$ is measurable, in the affirmative. The problem assumes measurability of the differences for $t>0$ only; the identity $\Delta_{-t}f(x)=-\Delta_tf(x-t)$ supplies the negative shifts. Its proof is not reconstructed in this corpus.

Acceptance. The result is refereed: M. Laczkovich, Functions with measurable differences, Acta Mathematica Academiae Scientiarum Hungaricae 35 (1980), 217–235, received by the journal on August 10, 1978. It is reviewed in the sense of a documented independent acceptance: Thomas Bloom, the curator of erdosproblems.com, marks Problem 908 proved and credits this paper for the affirmative answer. No Lean proof is recorded.

The page is dated by the first day of the issue month: the publisher's record for the article gives volume 35, issue 1-2, March 1980.