FormalConjectures.ErdosProblems.«1126»

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import FormalConjecturesUtil

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.


open MeasureTheory


namespace Erdos1126

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].

@[category research solved, AMS 26 28, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1126.lean"]
theorem declaration uses 'sorry'erdos_1126 : answer(True) ↔ ∀
    (f : ℝ → ℝ)
    (h :
      ∀ᵐ (p : ℝ × ℝ) ∂(volume.prod volume),
        f (p.1 + p.2) = f p.1 + f p.2),
    ∃ h : ℝ → ℝ,
      (∀ x y, h (x + y) = h x + h y) ∧ (∀ᵐ x ∂volume, f x = h x) := by⊢ True ↔
  ∀ (f : ℝ → ℝ),
    (∀ᵐ (p : ℝ × ℝ) ∂volume.prod volume, f (p.1 + p.2) = f p.1 + f p.2) →
      ∃ h, (∀ (x y : ℝ), h (x + y) = h x + h y) ∧ ∀ᵐ (x : ℝ), f x = h x
  sorryAll goals completed! 🐙


end Erdos1126