A note on the compact uniform integrability in metric spaces

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.