In this note, we characterize the compact uniform $p$-th order integrability ($\operatorname{CUI}(p)$) condition for measurable functions taking values in a metric space, where $p \in (0, \infty )$. Based on that, we then introduce the notion of $(\nu _\theta )_\theta $-$\operatorname{CUI}(p)$ for a family of metric space valued random elements which not only extends several known notions of $\operatorname{CUI}(p)$ in the literature but also provides insight into interpreting them. Under a uniform tightness condition, characterizations of $(\nu _\theta )_\theta $-$\operatorname{CUI}(p)$ in terms of the uniform absolute continuity and of the de la Vallée Poussin criterion are discussed. Our approach to the proofs is different from the relevant works.
Dans cette note, nous caractérisons la condition d’intégrabilité d’ordre $p$-uniforme compact ($\operatorname{CUI}(p)$) pour les fonctions mesurables prenant des valeurs dans un espace métrique, où $p \in (0, \infty )$. Sur cette base, nous introduisons la notion de $(\nu _\theta )_\theta $-$\operatorname{CUI}(p)$ pour une famille d’éléments aléatoires valués dans un espace métrique qui non seulement étend plusieurs notions connues de $\operatorname{CUI}(p)$ dans la littérature, mais fournit également un aperçu de leur interprétation. Sous une condition d’étanchéité uniforme, les caractérisations de $(\nu _\theta )_\theta $-$\operatorname{CUI}(p)$ en termes de continuité absolue uniforme et du critère de de la Vallée Poussin sont discutées. Notre approche des preuves est différente de celle des travaux existants.
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Keywords: Compact uniform integrability, de la Vallée Poussin criterion, Kolmogorov extension theorem, metric space, uniform tightness
Mots-clés : Intégrabilité uniforme compacte, critère de la Vallée Poussin, théorème d’extension de Kolmogorov, espace métrique, étanchéité uniforme
Dinh Thanh Giang 1 ; Duong Xuan Giap 2 ; Nguyen Tran Thuan 2
CC-BY 4.0
@article{CRMATH_2025__363_G9_905_0,
author = {Dinh Thanh Giang and Duong Xuan Giap and Nguyen Tran Thuan},
title = {A note on the compact uniform integrability in metric spaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {905--916},
year = {2025},
publisher = {Acad\'emie des sciences, Paris},
volume = {363},
doi = {10.5802/crmath.748},
language = {en},
}
TY - JOUR AU - Dinh Thanh Giang AU - Duong Xuan Giap AU - Nguyen Tran Thuan TI - A note on the compact uniform integrability in metric spaces JO - Comptes Rendus. Mathématique PY - 2025 SP - 905 EP - 916 VL - 363 PB - Académie des sciences, Paris DO - 10.5802/crmath.748 LA - en ID - CRMATH_2025__363_G9_905_0 ER -
Dinh Thanh Giang; Duong Xuan Giap; Nguyen Tran Thuan. A note on the compact uniform integrability in metric spaces. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 905-916. doi: 10.5802/crmath.748
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