Limit spectral measures of matrix distributions of metric triples

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.