Almost additive functions

DefinitionErdős Problem 1126 on almost additive functions#✓fully proved

This is the statement that for every function \(f : \mathbb{R} \to \mathbb{R}\) with \(f(x + y) = f(x) + f(y)\) for almost every \((x, y) \in \mathbb{R}^2\) with respect to Lebesgue measure on \(\mathbb{R}^2\), there is a function \(h : \mathbb{R} \to \mathbb{R}\) with \(h(x + y) = h(x) + h(y)\) for all \(x, y \in \mathbb{R}\) and \(f(x) = h(x)\) for almost every \(x \in \mathbb{R}\).

def conjecture : Prop :=
  ∀ (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, h (x + y) = h x + h y) ∧ (∀ᵐ x ∂MeasureTheory.volume, f x = h x)

Sources

• erdosproblems.com/1126 (record FC-ErdosProblem1126).

Used by 1

Used byAlmost additive functions agree almost everywhere with additive functions

TheoremAlmost additive functions agree almost everywhere with additive functions#●ready to prove

Let \(f : \mathbb{R} \to \mathbb{R}\) satisfy \(f(x + y) = f(x) + f(y)\) for almost every \((x, y) \in \mathbb{R}^2\) with respect to Lebesgue measure on \(\mathbb{R}^2\). Then there is a function \(h : \mathbb{R} \to \mathbb{R}\) with \(h(x + y) = h(x) + h(y)\) for all \(x, y \in \mathbb{R}\) and \(f(x) = h(x)\) for almost every \(x \in \mathbb{R}\).

public theorem_wanted conjecture_holds : conjecture

Sources

• de Bruijn [dB66] and Jurkat [Ju65], independently, per erdosproblems.com/1126, https://www.erdosproblems.com/1126.

Uses 1

Statement usesErdős Problem 1126 on almost additive functions

1 fully proved1 ready to prove

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