On measure estimates arising from Hausdorff distance convergence

TL;DR

This paper presents a method to estimate the measure of compact sets approximated by Hausdorff distance, with applications to spectral measures of operators with periodic approximations.

Contribution

It introduces a technique using fattenings of compact sets to ensure measure convergence and applies this to spectral analysis of operators.

Findings

Measure estimates converge via fattenings of compact sets

Application to spectral measures of operators with periodic approximations

Provides a new approach for measure estimation in Hausdorff convergence

Abstract

We discuss a method to estimate the measure of a compact set which is approximated using the Hausdorff distance by a sequence of compact sets. We do this by considering corresponding fattenings of the sequence of compact sets and showing their measures converge. We further review applications of this result to study the measure of a spectrum of an operator which has a sequence of periodic approximations.