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Claim. Every Lebesgue measurable $f:\mathbb R\to\mathbb R$ with $2f(x)\le f(x+h)+f(x+2h)$ for all real $x$ and all $h>0$ is nondecreasing. The result is a case of Theorem 7.18(I) (p. 90) of J. H. B. Kemperman, On the regularity of generalized convex functions, Trans. Amer. Math. Soc. 135 (1969), 69--93. Under the paper's Assumption 7.16, that theorem makes monotonic every measurable $f$ on an interval that satisfies $\sum_j a_jf(x+T_jy)\ge0$ (the paper's (7.10)) with $\sum_j a_j=0$, when the index $k$ is $0$. Kemperman's inequality is the case $a=(-2,1,1)$, $T=(0,1,2)$, where $b_1=3\ne0$ gives $k=0$. Applied on each interval $(-N,N)$, the theorem makes $f$ monotonic on $\mathbb R$; a nonincreasing solution is constant, since the inequality then gives $f(x)\le f(x+h)$, so $f$ is nondecreasing. The question of Problem 1125 is Kemperman's; Laczkovich's 1984 paper, digested on its library card, cites it as Kemperman's Problem 60 in Aequationes Math. 4 (1970), 248--249, and opens by recording that Kemperman proved the affirmative answer for measurable functions in the 1969 paper; the site's commentary records the same under [Ke69].

Covers. The instances of Problem 1125 in which $f$ is Lebesgue measurable: for these the answer is yes, and the conclusion is nondecreasing monotonicity, since constants satisfy the inequality. It leaves the question for arbitrary $f$, which Laczkovich's theorem settles without any regularity assumption.

Depends on. No page of this wiki: the proof is the paper's own.

Acceptance. Refereed: the paper appeared in the Transactions of the American Mathematical Society, volume 135 (1969); the page is dated by the year of publication, the volume's nominal first day. The site's commentary credits the measurable case to Kemperman as [Ke69], but its label credits Laczkovich with the solution of the whole problem, so no reviewed evidence is listed for this partial claim. No Lean checks this statement separately, so no formalized evidence is listed. The proof is not compiled in this wiki.