Erdős Problem 1126

References:

• erdosproblems.com/1126
• [Er60c] Erdős, P., Problem 310. Colloq. Math., 311.
• [dB66] de Bruijn, N. G., On almost additive functions. Colloq. Math. (1966), 59-63.
• [Ju65] Jurkat, Wolfgang B., On Cauchy's functional equation. Proc. Amer. Math. Soc. (1965), 683-686.

theorem Erdos1126.erdos_1126 : True ↔ ∀ (f : ℝ → ℝ), (∀ᵐ (p : ℝ × ℝ) ∂MeasureTheory.volume.prod MeasureTheory.volume, f (p.1 + p.2) = f p.1 + f p.2) → ∃ (h : ℝ → ℝ), (∀ (x y : ℝ), h (x + y) = h x + h y) ∧ ∀ᵐ (x : ℝ), f x = h x

If [f(x+y)=f(x)+f(y)] for almost all $x,y\in \mathbb{R}$ then there exists a function $g$ such that [g(x+y)=g(x)+g(y)] for all $x,y\in\mathbb{R}$ such that $f(x)=g(x)$ for almost all $x$.

Proved independently by de Bruijn [dB66] and Jurkat [Ju65].