Uniform retractions onto a probability measure algebra and an application to topological groups
Abstract
We prove that the indicator functions form a uniformly continuous retract of the set of [0,1]-valued functions in L2([0,1]). Consequently, the Lebesgue probability measure algebra, equipped with the symmetric-difference metric, is an injective uniform space. The construction uses dyadic conditional expectations and a common exponential coupling to obtain Lipschitz maps into the indicator functions that approximate the identity uniformly on that set. As an application, we construct a Polish topological group whose left uniform space is injective but which admits a bounded left uniformly continuous real-valued function that is not right uniformly continuous. This answers the injective part of a question of Bouziad and Troallic in the negative. The group is a semidirect product of two copies of the measure algebra and is also bounded, locally finite, and of exponent at most four.
Related
- An event is a set, not a number: building the algebra of events before the probability measure in a school progression
- Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions
- Best proximity point results of operator type condensing mappings via measure of noncompactness and application to an infinite system of differential equations in c0 and ℓ1
- A Hybrid Spatial Statistical Learning Framework for Individualized Probability Estimation: Application to Multisite Autism Neuroimaging
- Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality
BibTeX
@misc{peng2026uniform,
title = {{Uniform retractions onto a probability measure algebra and an application to topological groups}},
author = {Dekui Peng and Li-Hong Xie and Jiang Yang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01802},
eprint = {2611.01802},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01802},
}
RIS
TY - UNPB
TI - Uniform retractions onto a probability measure algebra and an application to topological groups
AU - Dekui Peng
AU - Li-Hong Xie
AU - Jiang Yang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01802
UR - https://arxiv.org/abs/2611.01802
AB - We prove that the indicator functions form a uniformly continuous retract of the set of [0,1]-valued functions in L^2([0,1]). Consequently, the Lebesgue probability measure algebra, equipped with the symmetric-difference metric, is an injective uniform space. The construction uses dyadic conditional expectations and a common exponential coupling to obtain Lipschitz maps into the indicator functions that approximate the identity uniformly on that set. As an application, we construct a Polish topological group whose left uniform space is injective but which admits a bounded left uniformly continuous real-valued function that is not right uniformly continuous. This answers the injective part of a question of Bouziad and Troallic in the negative. The group is a semidirect product of two copies of the measure algebra and is also bounded, locally finite, and of exponent at most four.
ER -