Limit Spectral Measures of Matrix Distributions of Metric Triples

Abstract

The 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 coincides with the spectrum of the integral operator on \(L^2(\mu)\) with kernel \(\rho\). An example in which there is no deterministic spectral measure is constructed.