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Topology and convergence on the space of measure-valued functions

Takahiro Hasebe, Ikkei Hotta, Takuya Murayama

TL;DR

The paper develops a topology on the space $C(T; \mathbf{M}(S))$ of measure-valued functions by enforcing uniform convergence on compacta against bounded continuous test functions, and shows this topology is equivalent to the compact-open topology via uniform-space methods. It proves Lévy-type continuity results for measure-valued processes by linking convergence of test-function integrals, uniform weak metrics, and convergence of characteristic functions, and extends these ideas to vague convergence and convergence of moments, with $p$-Wasserstein metrics providing concrete characterizations. The framework yields functional limit theorems for additive processes through the Lévy–Khintchine representation, establishing equivalences between convergence of generating triplets, marginal laws, and process convergence on compact time intervals. It also applies the topology to non-commutative probability via the Bercovici–Pata bijection, showing continuity/topological compatibility of the bijection across classical and non-classical convolutions, with implications for dynamical versions on both the circle and the real line. The results unify several convergence notions for measure-valued processes and underpin functional limit theorems and bijections in both classical and non-commutative settings; the work is integrated into a broader manuscript (arXiv:2412.18742).

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 Lévy'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).

Topology and convergence on the space of measure-valued functions

TL;DR

The paper develops a topology on the space of measure-valued functions by enforcing uniform convergence on compacta against bounded continuous test functions, and shows this topology is equivalent to the compact-open topology via uniform-space methods. It proves Lévy-type continuity results for measure-valued processes by linking convergence of test-function integrals, uniform weak metrics, and convergence of characteristic functions, and extends these ideas to vague convergence and convergence of moments, with -Wasserstein metrics providing concrete characterizations. The framework yields functional limit theorems for additive processes through the Lévy–Khintchine representation, establishing equivalences between convergence of generating triplets, marginal laws, and process convergence on compact time intervals. It also applies the topology to non-commutative probability via the Bercovici–Pata bijection, showing continuity/topological compatibility of the bijection across classical and non-classical convolutions, with implications for dynamical versions on both the circle and the real line. The results unify several convergence notions for measure-valued processes and underpin functional limit theorems and bijections in both classical and non-commutative settings; the work is integrated into a broader manuscript (arXiv:2412.18742).

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 Lévy'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).
Paper Structure (7 sections, 8 theorems, 17 equations)

This paper contains 7 sections, 8 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Topology on $C(T; \mathbf{M}(S))$
  3. Characteristic function and Lévy's continuity theorem
  4. Vague convergence of probability measures and convergence of moments
  5. Application I: convergence of additive processes
  6. Application II: Bercovici--Pata bijection for non-commutative additive processes
  7. Other related studies

Key Result

Proposition 2.1

Let $X$ be a topological space and $(Y, \mathcal{U})$ be a uniform space. On the set $C(X,Y)$ of continuous mappings from $X$ to $Y$, the topology of $\mathcal{U}$-uniform convergence on compacta is identical with the compact-open topology. In particular, the former topology depends only on the topo

Theorems & Definitions (17)

  • Proposition 2.1
  • Corollary 2.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Remark 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Remark 5.1
  • ...and 7 more