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Statement

Setting (p. 195, Section 1): a class $C$ of real functions on $-\infty<x<\infty$ has the difference property if every real $f$ with $\Delta_hf=f(x+h)-f(x)\in C$ for each $h$ has the form $f=g+H$ with $g\in C$ and $H$ additive.

Erdős's remark (p. 195; a footnote says "This remark is due to P. Erdös."). The difference property cannot be proved for the class of measurable functions, nor for the class of bounded measurable functions. Assuming the continuum hypothesis, Sierpiński constructed a non-measurable $S$ on the line taking only the values 0 and 1 such that, for each $h$, $S(x+h)-S(x)=0$ except for at most countably many $x$. Each $\Delta_hS$ is then measurable, but $S$ is not a measurable function plus an additive one: the additive part would be bounded on a set of positive measure, hence measurable by Ostrowski's theorem. Since the continuum hypothesis cannot be disproved (Gödel), the difference property of the measurable functions cannot be proved.

Conjecture (p. 195, quoted). "ERDÖS conjectured furthermore, that if any function $f(x)$ has the property that, for each $h$, $\Delta_h\,f(x)$ is measurable, then it can be written in the form $f(x)=g(x)+H(x)+S(x)$, where $g(x)$ is measurable, $H(x)$ is additive and $S(x)$ has, for each $h$, the property that $S(x+h)=S(x)$ for almost all $x$."

Sierpiński's function satisfies the condition on the third summand, so the remark does not refute the conjecture (an observation of this page). The paper continues: "We can prove this (§ 5) for the class of functions integrable ($L_2$) (over any finite interval) instead of measurable functions", which is Theorem 5.1.

Source. N. G. de Bruijn, Functions whose differences belong to a given class, Nieuw Arch. Wiskunde (2) 23 (1951), 194--218: the difference property, Erdős's remark and the conjecture on printed p. 195.

Read depth. Claims checked: the definition, the remark and the conjecture were read clause by clause on the page image. It is a conjecture; the paper gives no proof of it.

Proof pointer

None in the paper beyond the $L_2$ case, Theorem 5.1. The corpus records the measurable case as Laczkovich's Theorem 3; see Laczkovich 1980.

Bears on

• Problem 908: the conjecture is the problem's corrected Statement, with a measurable summand, as printed in 1951, except that the paper assumes measurable differences for each $h$ where the Statement says every $h>0$; the two hypotheses are equivalent, since a difference with a negative shift is minus a translate of one with a positive shift. The problem page cites this page of the paper, printed p. 195, for the word "measurable" against the continuous summand of the site's wording.