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A note on a result of Saks and Zygmund on additive functions of rectangles

Julià Cufí, Juan J. Donaire

TL;DR

The article tightens Saks–Zygmund's lemma on additive functions of axis-aligned rectangles by weakening the nonnegativity hypotheses and providing a robust route to conclude $F(I_0)\ge 0$ from local nonnegativity, using localized dyadic arguments and square decompositions. It introduces Lemma 5.1' with $\underline{F}_\alpha(x)\ge 0$ off a $\Lambda_\alpha$-null set $\Sigma$ and $\underline{F}(x)\ge 0$ off a $B_\alpha$-set, and shows $F(Q)\ge 0$ for all squares $Q\subset I_0$, leading to weaker yet sufficient hypotheses for the main theorems. The work generalizes the original results to settings where exceptional sets have zero $\Lambda_\alpha$ measure and strengthens connections to Besicovitch-type removable singularities for analytic functions. It also clarifies when stronger continuity notions would suffice for the dyadic reduction to hold, and extends conclusions to the complex setting via $(\ell_1)$-type boundary behavior.

Abstract

We modify the proof of the basic lemma of a paper of Saks and Zygmund on additive functions of rectangles.

A note on a result of Saks and Zygmund on additive functions of rectangles

TL;DR

The article tightens Saks–Zygmund's lemma on additive functions of axis-aligned rectangles by weakening the nonnegativity hypotheses and providing a robust route to conclude from local nonnegativity, using localized dyadic arguments and square decompositions. It introduces Lemma 5.1' with off a -null set and off a -set, and shows for all squares , leading to weaker yet sufficient hypotheses for the main theorems. The work generalizes the original results to settings where exceptional sets have zero measure and strengthens connections to Besicovitch-type removable singularities for analytic functions. It also clarifies when stronger continuity notions would suffice for the dyadic reduction to hold, and extends conclusions to the complex setting via -type boundary behavior.

Abstract

We modify the proof of the basic lemma of a paper of Saks and Zygmund on additive functions of rectangles.
Paper Structure (4 sections, 2 theorems, 11 equations)

This paper contains 4 sections, 2 theorems, 11 equations.

Table of Contents

  1. Introduction.
  2. Discussion on the proof of Saks-Zygmund's lemma.
  3. Modification of the proof of the lemma.
  4. Modified results

Key Result

Proposition 1

There is an additive and continuous function of rectangles $F$ such that $F(Q_d)\geq 0$ for every dyadic square $Q_d$ but $F(I_0)<0$ for some rectangle $I_0$.

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2