The Riesz representation in submetric spaces and its probabilistic aspects
Abstract
A submetric space is a topological space equipped with a continuous metric that generates a metric topology (usually) weaker than the original one (e.g., a~separable Hilbert space with the weak topology). We show that a positive linear functional on the space of bounded continuous functions on a submetric space is given as an integral with respect to a tight measure if, and only if, it is continuous in the strict topology. Neither local compactness nor complete regularity is required. We link the equicontinuity of a family of functionals in the strict topology to the uniform tightness of the underlying measures, and we provide a probabilistic representation (Skorokhod's parametrization) for such families. Finally, we identify the finest topology on the space of probability measures on a submetric space in which uniform tightness implies relative compactness. In non-metric spaces this topology is strictly stronger than the topology of weak convergence of probability measures.
Related
- C{à}dl{à}guity and compactness in submetric spaces
- H-Operator Approximation Spaces via Delayed Riesz Means and Quasi-Banach Moduli
- Riesz transform characterization of Hardy spaces associated with Dunkl--Schrödinger operators
- A General Framework for Operator Continuity between Riesz Spaces
- Truncation bands, unitality, and homomorphisms in truncated Riesz spaces: Boulabiar's problems and beyond
BibTeX
@misc{jakubowski2026riesz,
title = {{The Riesz representation in submetric spaces and its probabilistic aspects}},
author = {Adam Jakubowski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35661},
eprint = {2610.35661},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35661},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Riesz representation in submetric spaces and its probabilistic aspects
AU - Adam Jakubowski
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35661
UR - https://arxiv.org/abs/2610.35661
AB - A submetric space is a topological space equipped with a continuous metric that generates a metric topology (usually) weaker than the original one (e.g., a~separable Hilbert space with the weak topology). We show that a positive linear functional on the space of bounded continuous functions on a submetric space is given as an integral with respect to a tight measure if, and only if, it is continuous in the strict topology. Neither local compactness nor complete regularity is required. We link the equicontinuity of a family of functionals in the strict topology to the uniform tightness of the underlying measures, and we provide a probabilistic representation (Skorokhod's parametrization) for such families. Finally, we identify the finest topology on the space of probability measures on a submetric space in which uniform tightness implies relative compactness. In non-metric spaces this topology is strictly stronger than the topology of weak convergence of probability measures.
ER -