# Equivalence between GLT sequences and measurable functions

## TL;DR

This paper establishes a theoretical link between GLT matrix sequences and measurable functions, proving the space of GLT sequences is complete and equivalent to the space of measurable functions.

## Contribution

It proves the space of GLT sequences is complete and identifies it with the space of measurable functions, advancing the theoretical understanding of spectral distribution analysis.

## Key findings

- The space of GLT sequences is complete under the a.c.s. metric.
- GLT sequences are shown to be equivalent to measurable functions.
- Theoretical results deepen the understanding of spectral distribution asymptotics.

## Abstract

The theory of Generalized Locally Toeplitz (GLT) sequences of matrices has been developed in order to study the asymptotic behaviour of particular spectral distributions when the dimension of the matrices tends to infinity. A key concepts in this theory are the notion of Approximating Classes of Sequences (a.c.s.), and spectral symbols, that lead to define a metric structure on the space of matrix sequences, and provide a link with the measurable functions. In this document we prove additional results regarding theoretical aspects, such as the completeness of the matrix sequences space with respect to the metric a.c.s., and the identification of the space of GLT sequences with the space of measurable functions.