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Source. Theorem 3, p. 224, of M. Laczkovich, Functions with measurable differences, Acta Mathematica Academiae Scientiarum Hungaricae 35 (1980), 217--235, the edition named on the source card.

Read depth. Claims checked: the statement, the definitions of p. 217 it uses, and the reduction and final step of the proof were read clause by clause on the page images. The proof (pp. 224--227) and its preparatory results were read for structure only. Nothing here is independently reviewed beyond the bounded formulation review filed with the source card.

Statement

Setting (p. 217). For a class $F$ of real functions on $\mathbb R$, $F$ has the weak difference property when every $f:\mathbb R\to\mathbb R$ with $f(x+h)-f(x)\in F$ for every $h$ admits a decomposition $f=g+H+S$ with $g\in F$, $H$ additive ($H(x+y)=H(x)+H(y)$), and $S$ such that for every $h$, $S(x+h)-S(x)=0$ for almost every $x$. $L$ is the class of Lebesgue measurable functions on $\mathbb R$.

Theorem 3 (p. 224, quoted). "The class $L$ has the weak difference property."

Unwound: if $f:\mathbb R\to\mathbb R$ and $x\mapsto f(x+h)-f(x)$ is Lebesgue measurable for every real $h$, then $f=g+H+S$ pointwise on $\mathbb R$, with $g$ Lebesgue measurable, $H$ additive, and, for each fixed $h$, $S(x+h)-S(x)=0$ for almost every $x$. The exceptional null set may depend on $h$; no common null set is asserted, and $g$ is not claimed to be continuous.

The introduction (p. 217) states this as Erdős's conjecture, with $g$ measurable, and says that the main purpose of the paper is to prove it. It also recalls Erdős's example: under the continuum hypothesis there is a bounded non-measurable $S$ with $S(x+h)-S(x)=0$ for all but countably many $x$, for every $h$, which is not of the form $g+H$ with $g$ measurable and $H$ additive. So the term $S$ cannot be dropped in general, and $L$ does not have the (plain) difference property under that hypothesis.

Proof pointer

Pp. 224--227. Following de Bruijn, the proof first reduces to $f$ periodic mod 1, by comparing $f$ with its periodic extension from $[0,1)$. For such $f$ every difference lies in the class $S$ of measurable functions periodic mod 1, which carries Fréchet's pseudo-norm $\|f\|=\inf\{a+\lambda(\{x\in[0,1]:|f(x)|\ge a\}):a>0\}$ (p. 219), a metric for convergence in measure. For each $h$, Lemma 1 a) (p. 220) gives a constant $c(h)$ nearest to $f(x+h)-f(x)$ in that pseudo-norm. Theorem 2 (p. 221) shows that these distances tend to $0$ as $h\to0$, and from this $c$ satisfies the hypothesis of Theorem 1 (p. 218), so $c=H+u$ with $H$ additive and $u(x)\to u(0)=0$. The function $K(x,y)=f(x+y)-f(x)-H(y)$ is then continuous in $y$ for the pseudo-norm in $x$ (p. 225), which allows a measurable $G(x,y)$ on $\mathbb R^2$ with $G(\cdot,y)=K(\cdot,y)$ almost everywhere for every $y$ (p. 226). A Fubini argument on $S_1=K-G$ finds a point $x_0$ at which $S_1(x_0,\cdot)$ has almost everywhere vanishing differences, and $g(x)=G(x_0,x-x_0)+f(x_0)-H(x_0)$ is the measurable summand (p. 227).

Dependencies

Theorem 1 (p. 218), Lemma 1 (p. 220) and Theorem 2 (p. 221) of the paper, and de Bruijn's reduction (de Bruijn 1951, §1, cited by the paper as [1]).

Bears on

• Problem 908: the theorem answers the problem's corrected Statement, which asks for a measurable summand, in the affirmative. The problem's hypothesis is stated for $h>0$; the identity $\Delta_uf(x)=-\Delta_{-u}f(x+u)$ for $u<0$, where $\Delta_tf(x)=f(x+t)-f(x)$, recorded on the problem page, carries it to every real shift. The theorem does not give a continuous summand, the site's wording, which the problem page shows to be false.