Lorentz and Gale-Ryser theorems on general measure spaces

TL;DR

This paper extends Lorentz's classical characterization of plane measurable sets via cross sections to general measure spaces, building on Gale-Ryser's theorem about (0,1)-matrices and partitions.

Contribution

It generalizes Lorentz's theorem from the plane to arbitrary measure spaces using Gale-Ryser's matrix existence results.

Findings

Extended Lorentz's theorem to measure spaces

Connected matrix partition conditions with measure space properties

Provided new criteria for measurable set characterization

Abstract

Based on the Gale-Ryser theorem for the existence of suitable $(0,1)$-matrices for different partitions of a natural number, we revisit the classical result of G. G. Lorentz regarding the characterization of a plane measurable set, in terms of its cross sections, and extend it to general measure spaces.