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Claim. The statement of Problem 1126 holds: if $f(x+y)=f(x)+f(y)$ for almost every pair $(x,y)\in\mathbb R^2$, there is a function $g$ with $g(x+y)=g(x)+g(y)$ for all real $x,y$ and $f=g$ almost everywhere on $\mathbb R$. This is Theorem I of W. B. Jurkat, On Cauchy's functional equation, Proceedings of the American Mathematical Society 16 (1965), no. 4, 683–686; see its library card and the theorem's page. The theorem allows $f$ to be undefined on a linear null set and adds that the additive $g$ is unique. The proof chooses, by Fubini's theorem, a conull set $M$ on which the equation can be composed, shows that $f(x)+f(y)$ depends only on $x+y$ for $x,y\in M$, and uses $\mathbb R=M+M$ to define $g$ on the whole line. The route differs from de Bruijn's, whose independent proof defines $g(x)$ as the almost-everywhere constant value of $f(x+y)-f(y)$; Jurkat's added-in-proof note records that de Bruijn sent him that manuscript in September 1964.

Acceptance. Refereed: the paper appeared in the Proceedings of the American Mathematical Society, received by the editors on 23 December 1963. Reviewed: Thomas Bloom, the curator of erdosproblems.com, labels the problem proved and credits it as proved independently by de Bruijn and Jurkat. The page is dated by the issue of August 1965; Jurkat's 1964 abstract in the Notices of the American Mathematical Society (11, p. 240, notice 64T-171) announced the result without a proof. No formalization of this proof is recorded; the Lean proof the site's label refers to formalizes de Bruijn's argument and is linked from his page.

Depends on. Nothing in this wiki: the argument is the paper's own.