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arxiv: 2105.09804 · v1 · submitted 2021-05-20 · 🧮 math.CA · math.PR

Non-existence of measurable solutions of certain functional equations via probabilistic approaches

Kazuki Okamura This is my paper

Pith reviewed 2026-05-24 13:41 UTC · model grok-4.3

classification 🧮 math.CA math.PR
keywords functional equationsBorel measurabilityDirac measureuniform distributionCauchy distributionnon-existenceprobability characterizations
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The pith

No measurable solutions exist for f(x) + g(y) = h(x,y) when h is Borel measurable and tied to uniform or Cauchy distribution characterizations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines functional equations of the form f(x) + g(y) = h(x,y) with h given and f, g unknown. It establishes that if h is Borel measurable and arises from characterizations of the uniform or Cauchy distributions, then no measurable f and g satisfy the equation. The argument assumes measurable solutions exist and derives a contradiction via a property of the Dirac measure. The same technique shows non-existence for the arctan equation. A reader would care because these equations appear naturally when encoding distribution properties, so the result limits what can be expressed with measurable functions.

Core claim

If h is a Borel measurable function associated with characterizations of the uniform or Cauchy distributions, then there is no measurable solutions of the equation f(x) + g(y) = h(x,y). The proof uses a characterization of the Dirac measure and it is also applicable to the arctan equation.

What carries the argument

Characterization of the Dirac measure, used to obtain a contradiction from the assumption of measurable f and g.

If this is right

  • No measurable solutions exist when h comes from uniform distribution characterizations.
  • No measurable solutions exist when h comes from Cauchy distribution characterizations.
  • The arctan functional equation likewise admits no measurable solutions.
  • Probabilistic characterizations can be turned into non-existence proofs for additive equations under measurability constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar contradictions may appear for other distribution characterizations that produce Dirac-like concentration properties.
  • The result separates the measurable and non-measurable regimes for additive representations of probability identities.
  • Any search for solutions would need to invoke non-measurable functions, which typically require the axiom of choice.

Load-bearing premise

The given h must be Borel measurable and must arise specifically from the characterizations of the uniform or Cauchy distributions.

What would settle it

An explicit construction of Borel measurable f and g satisfying f(x) + g(y) = h(x,y) for any such h would falsify the non-existence result.

read the original abstract

This paper deals with functional equations in the form of $f(x) + g(y) = h(x,y)$ where $h$ is given and $f$ and $g$ are unknown. We will show that if $h$ is a Borel measurable function associated with characterizations of the uniform or Cauchy distributions, then there is no measurable solutions of the equation. Our proof uses a characterization of the Dirac measure and it is also applicable to the arctan equation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims that for the functional equation f(x) + g(y) = h(x,y) with h Borel measurable and associated with characterizations of the uniform or Cauchy distributions, there are no measurable solutions f,g. The argument relies on a characterization of the Dirac measure to derive a contradiction and is stated to apply also to the arctan equation.

Significance. If the central link between the given h and the distributional characterizations can be made rigorous, the probabilistic approach via Dirac-measure characterization would offer a compact non-existence proof for a class of functional equations arising in probability characterizations; this could be of interest in the intersection of functional equations and probability theory.

major comments (2)
  1. [Abstract / Introduction] The central claim requires that the supplied Borel h 'arises specifically from' the uniform or Cauchy characterizing equations so that the functional equation forces the measure to be Dirac. The manuscript provides no explicit construction, theorem reference, or reduction showing how those characterizations produce the particular h that triggers the Dirac step; without this link the contradiction does not follow.
  2. [Abstract] The arctan case is asserted to be covered by the same argument, yet no reduction or verification is supplied showing that the corresponding h satisfies the required association with a distributional characterization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. The points raised correctly identify gaps in the presentation of the link between the distributional characterizations and the specific h, as well as the arctan case. We address each comment below and will revise the manuscript to supply the missing explicit constructions and reductions.

read point-by-point responses

  1. Referee: [Abstract / Introduction] The central claim requires that the supplied Borel h 'arises specifically from' the uniform or Cauchy characterizing equations so that the functional equation forces the measure to be Dirac. The manuscript provides no explicit construction, theorem reference, or reduction showing how those characterizations produce the particular h that triggers the Dirac step; without this link the contradiction does not follow.

    Authors: We agree that the manuscript does not contain an explicit construction, theorem reference, or reduction showing how the uniform and Cauchy characterizations produce the Borel measurable h that triggers the Dirac-measure step. In the revised version we will add a short preliminary section (or subsection of the introduction) that supplies this link: we will state the relevant characterizing equations, derive the associated h(x,y), and verify that the functional equation then forces the measure to be Dirac, yielding the desired contradiction. revision: yes

  2. Referee: [Abstract] The arctan case is asserted to be covered by the same argument, yet no reduction or verification is supplied showing that the corresponding h satisfies the required association with a distributional characterization.

    Authors: The referee correctly notes that the abstract asserts coverage of the arctan equation without supplying a reduction or verification. We will revise the manuscript to include a brief paragraph (or appendix note) that reduces the arctan equation to the same framework, confirming that its h satisfies the association with a distributional characterization and is therefore covered by the Dirac-measure argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; proof applies external Dirac characterization independently

full rationale

The derivation applies an external characterization of the Dirac measure to the functional equation f(x) + g(y) = h(x,y) where h is given as Borel measurable and associated with uniform/Cauchy characterizations. No self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain appears; the Dirac step is invoked as an independent probabilistic tool rather than derived from the paper's own inputs or prior author work. The arctan applicability is likewise presented as an extension of the same external method. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the claim is a non-existence result resting on standard measure-theoretic notions and an external Dirac measure characterization.

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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    work page 2019