Topology and convergence on the space of measure-valued functions

TL;DR

This paper studies uniform convergence on compacta for measure-valued functions and explores related limit theorems, including Lévy's continuity theorem and functional limit theorems for additive processes, both classical and non-commutative.

Contribution

It provides new insights into convergence properties and limit theorems for functions valued in finite Borel measures, extending classical results to non-commutative settings.

Findings

Uniform convergence on compacta for measure-valued functions analyzed

Limit theorems like Lévy's continuity theorem extended to this setting

Functional limit theorems established for classical and non-commutative additive processes

Abstract

In these notes, uniform convergence on compacta is studied on the space of functions taking values in the set of finite Borel measures. Related limit theorems, including L\'evy's continuity theorem and functional limit theorems for (classical and non-commutative) additive processes, are also described. N.B.: the contents of this manuscript have been incorporated into another manuscript (arXiv:2412.18742).