Limit spectral measures of matrix distributions of metric triples

TL;DR

This paper introduces the concept of limit spectral measures for metric triples, showing conditions under which they are deterministic and providing an example where they are not, advancing understanding of spectral properties in metric measure spaces.

Contribution

It defines limit spectral measures for metric triples and explores their determinism, including an example demonstrating non-deterministic cases, thus extending spectral analysis in metric measure spaces.

Findings

Limit spectral measure coincides with the spectrum of an integral operator when the metric is square integrable.

Under certain conditions, the limit spectral measure is deterministic.

An example is constructed where the spectral measure is not deterministic.

Abstract

A notion of the limit spectral measure of a metric triple (i.e., a metric measure space) is defined. If the metric is square integrable, then the limit spectral measure is deterministic and coinsides with the spectrum of the integral operator in $L^2(\mu)$ with kernel $\rho$. We construct an example in which there is no deterministic spectral measure.