Measures and fibers

TL;DR

This paper investigates measures on compact spaces via fibers of continuous maps into 2^omega, revealing conditions under which fibers carry uncountable Maharam type measures and implications for separability.

Contribution

It establishes new connections between measure properties on compact spaces and the structure of fibers under continuous mappings, including conditions for non-scattered fibers and separability.

Findings

Spaces with uncountable Maharam type have fibers with non-scattered measures.

Under Martin's Axiom, fibers can carry measures of uncountable Maharam type.

Spaces supporting positive measures and finite-to-one maps into 2^omega are separable.

Abstract

We study measures on compact spaces by analyzing the properties of fibers of continuous mappings into 2^omega. We show that if a compact zerodimensional space K carries a measure of uncountable Maharam type, then such a mapping has a non-scattered fiber and, if we assume additionally a weak version of Martin's Axiom, such a mapping has a fiber carrying a measure of uncountable Maharam type. Also, we prove that every compact zerodimensional space which supports a strictly positive measure and which can be mapped into 2^omega by a finite-to-one function is separable.