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September 10, 2025Open Access

A note on the compact uniform integrability in metric spaces

DGDinh Thanh GiangNational Institute of Nutrition DGDuong Xuan GiapVinh University NTNguyen Tran ThuanUniversity of Economics Ho Chi Minh City

Key Points

The characterization of compact uniform integrability reveals new insights into measurable functions in metric spaces.
Based on uniform tightness, this work discusses uniform absolute continuity and de la Vallée Poussin criterion for specific metric space valued random elements.
Introducing a new notion of CUI for random elements extends existing literature while enhancing the interpretative framework.
This approach diverges from traditional proofs, potentially opening pathways for further research in uniform integrability conditions.

Abstract

In this note, we characterize the compact uniform p-th order integrability (CUI(p)) condition for measurable functions taking values in a metric space, where p∈(0,∞). Based on that, we then introduce the notion of (ν θ ) θ -CUI(p) for a family of metric space valued random elements which not only extends several known notions of CUI(p) in the literature but also provides insight into interpreting them. Under a uniform tightness condition, characterizations of (ν θ ) θ -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.

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