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Source. Theorem 5, p. 229, 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 definition of the double difference property (p. 218) and the proof (p. 229) were read clause by clause on the page images. Nothing here is independently reviewed.

Statement

Setting (p. 218). For a class $F_1$ of real functions on $\mathbb R$ and a class $F_2$ of real functions on $\mathbb R^2$, the pair $(F_1,F_2)$ has the double difference property if whenever $f(x+y)-f(x)-f(y)\in F_2$ for a function $f:\mathbb R\to\mathbb R$, $f=g+H$ with $g\in F_1$ and $H$ additive. $L$ is the class of Lebesgue measurable functions on $\mathbb R$ and $L^{(2)}$ that of Lebesgue measurable functions on $\mathbb R^2$ (p. 229).

Theorem 5 (p. 229, quoted). "If a function $f:\mathbf R\to\mathbf R$ is such that $f(x+y)-f(x)-f(y)$ is Lebesgue measurable (as a function of two variables), then $f$ is of the form $g+H$ where $g\in L$ and $H:\mathbf R\to\mathbf R$ is additive."

That is, the pair $(L,L^{(2)})$ has the double difference property. In contrast with Theorem 3, no third summand $S$ is needed under this two-variable hypothesis.

Proof pointer

P. 229. There is a set $Y$ of full measure such that $f(x+y)-f(x)-f(y)$ is measurable in $x$ for every $y\in Y$, and writing any $h$ as $y_1+y_2$ with $y_1,y_2\in Y$ shows that every difference $f(x+h)-f(x)$ is measurable. Theorem 3 then gives $f=g+H+S$. The function $S(x+y)-S(x)-S(y)$ is measurable on the plane and, for each fixed $x$, equals $-S(x)$ for almost every $y$, so integrating over $y\in[0,1]$ shows that $S$ is measurable and $f=(g+S)+H$.

Dependencies

Theorem 3 of the paper and Fubini's theorem.

Bears on

No Erdős problem page in this corpus. The paper derives from it Theorem 7 (p. 232: a Baire $\alpha$ Cauchy difference gives $f=g+H$ with $g$ Baire $\alpha$), Theorem 8 (p. 233: the approximately continuous functions have the difference property) and Theorem 9 (p. 233: a bounded function whose every difference is a derivative is a derivative).