On Mazurkiewicz's sets, thin {\sigma}-ideals of compact sets and the space of probability measures on the rationals

TL;DR

This paper investigates properties of thin -ideals of compact sets in metric spaces and refines Preiss's theorem on non-uniformly tight probability measures on the rationals, using Mazurkiewicz's construction.

Contribution

It introduces new properties of -ideals and provides a refined version of Preiss's theorem utilizing Mazurkiewicz's 1927 construction.

Findings

Characterization of thin -ideals of compact sets

Refinement of Preiss's theorem on probability measures

Application of Mazurkiewicz's construction in measure theory

Abstract

We shall establish some properties of thin $\sigma$-ideals of compact sets in compact metric spaces (in particular, the $\sigma$-ideals of compact null-sets for thin subadditive capacities), and we shall refine the celebrated theorem of David Preiss that there exist compact non-uniformly tight sets of probability measures on the rationals. Both topics will be based on a construction of Stefan Mazurkiewicz from his 1927 paper containing a solution of a Urysohn's problem in dimension theory.