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\title{A note on the compact uniform integrability in metric spaces}
\alttitle{Note sur l'intégrabilité uniforme compacte dans les espaces métriques}

\author{\firstname{Dinh} \middlename{Thanh} \lastname{Giang}\CDRorcid{0000-0003-1569-5864}}
\address{Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam}
\email{giang.dt.dinh@gmail.com}

\author{\firstname{Duong} \middlename{Xuan} \lastname{Giap}\CDRorcid{0009-0009-3645-9642}}
\address{Department of Mathematics, Vinh University, Vinh, Nghe An, Vietnam}
\email{dxgiap@gmail.com}

\author{\firstname{Nguyen} \middlename{Tran} \lastname{Thuan}\CDRorcid{0000-0003-1354-3066}}
\address[2]{Department of Mathematics, Vinh University, Vinh, Nghe An, Vietnam}
\email{thuan.tr.nguyen@gmail.com}

%\thanks{This research was funded by the Ministry of Education and Training of Vietnam, under Grant~Number B2025-TDV-03}
%\CDRGrant[Ministry of Education and Training of Vietnam]{B2025-TDV-03}
\thanks{The authors were supported by the National Foundation for Science and Technology Development (NAFOSTED), grant no.~101.03-2021.32}
\CDRGrant[National Foundation for Science and Technology Development (NAFOSTED)]{101.03-2021.32}

\subjclass{60A10, 28A25}

\keywords{\kwd{Compact uniform integrability} \kwd{de la Vall\'ee Poussin criterion} \kwd{Kolmogorov extension theorem} \kwd{metric space} \kwd{uniform tightness}}
\altkeywords{\kwd{Intégrabilité uniforme compacte} \kwd{critère de la Vallée Poussin} \kwd{théorème d'extension de Kolmogorov} \kwd{espace métrique} \kwd{étanchéité uniforme}}

\begin{abstract} 
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 Vall\'ee Poussin criterion are discussed. Our approach to the proofs is different from the relevant works.
\end{abstract}

\begin{altabstract}
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.
\end{altabstract}

\dateposted{2025-07-22}
\begin{document}

\maketitle

\section{Introduction}

The uniform integrability (UI) is a fundamental notion in probability theory which describes a certain type of boundedness for a family of random variables. It is in particular a very useful tool for deriving limit theorems, for example, the celebrated Vitali theorem asserts that, under the UI condition, the convergence in probability is equivalent to the $L^1$-convergence for a sequence of random variables.

A collection of real-valued random variables $(X_i)_{i \in I}$, which are defined on a probability space $(\Omega, \cF, \bP)$, is called UI if $\sup_{ i \in I} \bE\bracks[\big]{\abs{X_i} \1_{\braces*{\abs{X_i} >a}}} \to 0$ when $a \to \infty$. To make this condition more applicable in different contexts, several equivalent formulations of the UI are established. Among others, we state here some characterizations that are frequently used: a family $(X_i)_{i \in I}$ is UI if and only if one (and hence, all) of the following two conditions holds true:
\begin{enumerate}\romanenumi
\item \label{condition_i} $\sup_{i \in I}\bE\bracks[\big]{\abs{X_i}} < \infty$, and in addition, for any $\ep >0$, there exists $\kappa >0$ such that whenever $A \in \cF$ with $\bP(A) \lee \kappa$ one has $\sup_{i \in I}\bE\bracks[\big]{\abs{X_i} \1_A} \lee \ep$;
\item \label{condition_ii} there is a non-decreasing function $G \colon [0, \infty) \to [0, \infty)$ with $G(0) = 0$ and $G(x)/x \to \infty$ as $x \to \infty$ such that $
\sup_{i \in I} \bE \bracks[\big]{G\parens[\big]{\abs{X_i}}} < \infty$.
\end{enumerate}
Condition~\eqref{condition_i} is the uniform boundedness in $L^1$ and the uniform absolute continuity, respectively. Condition~\eqref{condition_ii} is known as the celebrated \emph{de la Vall\'ee Poussin criterion} which gives an elegant description for UI. We refer to~\cite[Section~22]{Sc17} for a comprehensive summary about UI.

In the literature, the notion of UI and its characterizations have been extensively studied in various directions depending on which applications are of interest. For random variables taking real values, we can mention, among others, the weighted UI in the Ces\`aro sense which is then extended to $\{a_{n, k}\}$-UI in~\cite{Ca94}. Recently, the notions of $B$-statistical UI and $\mathcal J$-UI, together with their characterizations, have been introduced and discussed in~\cite{CRUV21} and~\cite{HCRUV23}, respectively. As the literature on UI in this line of research is vast, we refer the reader to, for example, the references in the aforementioned articles and their follow-up works.

In another direction, several authors consider random elements $(X_i)_i$ taking values in a separable Banach space which satisfy a compact uniform integrability (CUI) condition, see, e.g., \cite{HG76}. One can show that if $(X_i)_i$ is CUI, then there is a family of random elements $(Y_i)_i$ taking values in a finite set that approximates $(X_i)_i$, and moreover, the family $(Y_i)_i$ is constructed in a way that it inherits some good distributional properties of $(X_i)_i$. These features are crucial to derive limit theorems for $(X_i)_i$. Note that when the Banach space is especially a finite dimensional space so that every closed ball is compact, then CUI becomes the usual UI aforementioned above. We refer to, e.g., \cite{BCS14, Ca97, CRUV20, CV01, DT82, Ta78,WR87} for relevant discussions on CUI and its characterizations with applications in proving limit theorems.

However, in many applications it appears that random elements take values in metric spaces rather than in normed linear spaces. We can mention, for instance, the space of right-continuous with finite left limits functions on $[0, 1]$ with Skorokhod metric~\cite{Bi99}, the metric space of probability distributions with Wasserstein metric~\cite{Vi09}, and the space of compact subsets of a given Banach space with Hausdorff metric~\cite{Mo17}. Therefore, it might be necessary to consider the CUI condition in metric spaces. For this direction, the authors in~\cite{TQ20} have recently introduced the notion of CUI in the Ces\`aro sense for a sequence of random elements with values in a convex combination metric space and established some limit theorems under a CUI condition.

In this article, we study the notion of compact uniform $p$-th order integrability, abbreviated by $\CUI(p)$, for measurable functions or random elements in a separable and complete metric space, where $p \in (0, \infty)$. We first introduce in \defiref{def:CUIC} the $\CUI(p)$ condition for an arbitrary family of measurable functions, which are not necessarily defined on the same measure space, by means of its characterizations. It is shown in \theoref{thm:equivalence} that, under the uniform finiteness of the measure spaces and a uniform tightness condition, then $\CUI(p)$ is equivalent to an absolute continuity condition, and, also, to a de la Vall\'ee Poussin type condition.

Our approach to the proof is different from the relevant literature, even in case of UI random variables. Specifically, instead of directly proving the equivalences for the UI or CUI condition as in, e.g., \cite{Ca94, CV01}, we construct a suitable way to transform the (apparently) more general setting into a simplified setting where one can apply the known results in a straightforward way. To do that, the Kolmogorov extension theorem plays a key role.

We then discuss in \sectionref{sec:CUI-random} the $(\nu_\theta)_{\theta}$-$\CUI(p)$ condition for a collection of random elements $\braces[\big]{X_s, \: s \in \bigcup_{\theta} S_\theta}$ with values in a metric space. Here, $\nu_\theta$ is a finite measure on the (possibly uncountable) index set $S_\theta$. It is shown via examples that, when $\nu_\theta$ are in particular discrete measures, then $(\nu_\theta)_{\theta}$-$\CUI(p)$ boils down to several known notions of $\CUI(p)$ such as (classical) $\CUI(p)$, Ces\`aro $\CUI(p)$, and weighted $\CUI(p)$. Moreover, for certain choices of $(\nu_\theta)_\theta$, the $(\nu_\theta)_\theta$-$\CUI(p)$ can be seen as the usual $\CUI(p)$ for random elements with random indices, see \exemref{exam:CUI-random-indices}. We also provide in \theoref{thm:CUI-criterion} some standard equivalences for $(\nu_\theta)_{\theta}$-$\CUI(p)$ whose proof follows directly from \theoref{thm:equivalence} by realizing the measure spaces in a product form.

\subsection*{Notation}

Let $\bN \coloneqq \{1, 2, \dotsc\}$ and let $\bR^d$ be the $d$-dimensional Euclidean space equipped with the Borel $\sigma$-algebra $\cB(\bR^d)$. For a set $S$, we denote by $\cP(S)$ the collection of all subsets of $S$.

Let $(M, \cM, \mu)$ be a measure space. The indicator function of an $A \in \cM$ is denoted by $\1_A$. For $p \in (0, \infty)$, let $L^p(\mu)\coloneqq L^p(M, \cM, \mu)$ be the (equivalent) class of $\cM/\cB(\bR)$-measurable functions $f \colon M \to \bR$ with $\int_M \abs[\big]{f(x)}^p \mu(\od x) <\infty$. For a measurable space $(U, \cU)$, the pushforward measure of $\mu$ via an $\cM/\cU$-measurable function $f\colon M \to U$ is denoted by $\mu \circ f^{-1}$, i.e.\ $\mu \circ f^{-1}(A) = \mu\parens[\big]{\{f \in A\}}$ for $A \in \cU$. The Dirac measure $\delta_x$ concentrated at a point $x \in M$ is defined by $\delta_x (A) = 1$ if $x \in A$ and $\delta_x(A) = 0$ if $x \notin A$ for $A \in \cM$. The notation $\int_M f \od \mu$ stands for $\int_M f(x) \mu(\od x)$ when the integration variable $x$ is not important to indicate.

Assume throughout this article that $(E, d_E)$ is a separable and complete metric space which is equipped with the Borel $\sigma$-algebra $\cB(E)$ induced by the metric $d_E$. Let $\cK(E)$ be the collection of all nonempty compact subsets in $E$.

For a probability space $(\Omega, \cF, \bP)$, an $\cF/\cB(E)$-measurable function $X\colon \Omega \to E$ is called an ($E$-valued) random element. In particular, if $\parens[\big]{E, \cB(E)} = \parens[\big]{\bR, \cB(\bR)}$ then $X$ is called a random variable. The expectation of a random variable $X$, if it exists, is denoted by $\bE[X]$.

\section{A result on the compact uniform integrability}

Let us fix an element $\bme \in E$ and denote
\[
\norm{x}_{\bme} \coloneqq d_E(x, \bme).
\]
In particular, if $E$ is a separable Banach space then we typically choose $\bme = 0$.

\begin{defi}\label{def:CUIC}
Let\/ $\Theta \neq \emptyset$ be a (possibly uncountable) index set and $\braces[\big]{(M_\theta, \cM_\theta, \mu_\theta), \: \theta \in \Theta}$ a family of finite measure spaces. Suppose $p \in (0, \infty)$. We say that the \emph{compact uniform $p$-th order integrability criterion (the $\CUI(p)$ criterion, for short) holds true on $\parens[\big]{E, \cB(E)}$ for $\braces[\big]{(M_\theta, \cM_\theta, \mu_\theta), \: \theta \in \Theta}$} if for any family $\braces[\big]{f_\theta, \: \theta \in \Theta}$ of $\cM_\theta/\cB(E)$-measurable functions $f_\theta \colon M_\theta \to E$, the following four statements~\ref{item:1:statement}--\ref{item:4:statement} are equivalent:
\begin{enumerate}[(S1)]
\item\label{item:1:statement}
$\braces[\big]{f_\theta, \: \theta \in \Theta}$ is \emph{$\CUI(p)$}, which means:
\[
\forall\ep >0, \ \exists K_{p,\ep} \in \cK(E)\quad \text{such that} \quad \sup_{\theta \in \Theta} \int_{M_\theta} \norm{f_\theta}_{\bme}^p \, \1_{\braces*{f_\theta \notin K_{p,\ep}}} \od \mu_\theta \lee \ep;
\]
\item \label{item:2:statement}
\begin{enumerate}\alphenumii
\item\label{item:2:statement-a}
$\braces[\big]{f_\theta, \: \theta \in \Theta}$ is \emph{uniformly tight (UT)}, which means:
\[
\forall \delta >0, \ \exists K'_\delta \in \cK(E) \quad \text{such that} \quad \sup_{\theta \in \Theta} \mu_\theta\parens[\big]{\braces[\big]{f_\theta \notin K'_\delta}} \lee \delta;
\]
\item\label{item:2:statement-b}
$\displaystyle \lim_{a \to \infty} \; \sup_{\theta \in \Theta} \int_{M_\theta} \norm{f_\theta}_{\bme}^p \, \1_{\braces*{\norm{f_\theta}_{\bme} > a}} \od \mu_\theta = 0$;
\end{enumerate}
\item \label{item:3:statement}
\begin{enumerate}\alphenumii
\item \label{item:3:statement-a} $\braces[\big]{f_\theta, \: \theta \in \Theta}$ is UT;
\item \label{item:3:statement-b} $\displaystyle R\coloneqq \sup_{\theta \in \Theta} \int_{M_\theta} \norm{f_\theta}^p_{\bme} \, \od \mu_\theta < \infty$, and for any $\ep >0$ there exists $\kappa = \kappa(R, p, \ep) >0$ such that
\[
\forall \theta \in \Theta, \ A_\theta \in \cM_\theta \; : \quad \mu_\theta(A_\theta) \lee \kappa \ \Longrightarrow \ \int_{M_\theta} \norm{f_\theta}_{\bme}^p \, \1_{A_\theta} \od \mu_\theta \lee \ep;
\]
\end{enumerate}
\item \label{item:4:statement}
\begin{enumerate}\alphenumii
\item \label{item:4:statement-a} $\braces[\big]{f_\theta, \: \theta \in \Theta}$ is UT;
\item \label{item:4:statement-b} there is a non-decreasing function $G \colon [0, \infty) \to [0, \infty)$ with $G(0) = 0$ and $G(x)/x^p \to \infty$ as $x \to \infty$ such that
\[
\sup_{\theta \in \Theta} \int_{M_\theta} G\parens[\big]{\norm{f_\theta}_{\bme}} \od \mu_\theta < \infty.
\]
\end{enumerate}
\end{enumerate}
\end{defi}

We will show in \propref{prop:independence-choice-e} below that our main result does not depend on the choice of the reference element $\bme$.

\begin{rema}\label{rema:CUIC}
\mbox{}
\begin{enumerate}
\item \label{rema:CUIC:item:1}
The tightness conditions~\ref{item:2:statement}-\eqref{item:2:statement-a}, \ref{item:3:statement}-\eqref{item:3:statement-a} and~\ref{item:4:statement}-\eqref{item:4:statement-a} coincide and they are a variant of UT in the literature, see, e.g., \cite[p.~9]{Bi99} or~\cite[Definition~5.2.1]{Ta78} (note that the above measure spaces indexed by $\Theta$ are not necessarily identical).
\item \label{rema:CUIC:item:2}
When $(M_\theta, \cM_\theta, \mu_\theta)$ does not depend on $\theta$, i.e.\ $(M_\theta, \cM_\theta, \mu_\theta) = (M, \cM, \mu)$ for all {$\theta \in \Theta$}, then the equivalences~\ref{item:2:statement}-\eqref{item:2:statement-b} $\Leftrightarrow$~\ref{item:3:statement}-\eqref{item:3:statement-b} $\Leftrightarrow$~\ref{item:4:statement}-\eqref{item:4:statement-b} hold true and they are a UI characterization of $\braces[\big]{\norm{f_\theta}_{\bme}^p, \: \theta \in \Theta}$, see, e.g., \cite[Theorem~22.9, (ii), (ix), (x)]{Sc17}.
\item \label{rema:CUIC:item:3}
For~\ref{item:2:statement}, it is easy to check that conditions~\eqref{item:2:statement-a} and~\eqref{item:2:statement-b} do not imply each other. Indeed, \eqref{item:2:statement-a} is not sufficient to obtain~\eqref{item:2:statement-b} as UT does not imply UI, even in finite dimensional spaces. Conversely, let $E = \ell^1$ be the Banach space of summable real sequences and $\braces[\big]{e_n, \: n \in \bN}$ the canonical basis of $\ell^1$. Set $\Theta = \bN$ and let $(M_n, \cM_n, \mu_n) = (\Omega, \cF, \bP)$ be a fixed probability space. For $f_n(\omega) \coloneqq e_n$, the family $\braces[\big]{f_n, \: n \in \bN}$ is uniformly bounded, and thus, \eqref{item:2:statement-b} holds. However, \eqref{item:2:statement-a} fails to hold as $\braces[\big]{e_n, \: n \in \bN}$ is not compact.
\end{enumerate}
\end{rema}

We are now in a position to formulate the main result.

\begin{theo}\label{thm:equivalence}
Let\/ $\Theta \neq \emptyset$ and $p \in (0, \infty)$. Then, one has the following:
\begin{enumerate}
\item \label{thm:equivalence:item:1}
the\/ $\CUI(p)$ criterion holds true on $\parens[\big]{E, \cB(E)}$ for all families $\braces[\big]{(M_\theta, \cM_\theta, \mu_{\theta}), \: \theta \in \Theta}$ satisfying $\sup_{\theta \in \Theta} \mu_\theta(M_\theta) < \infty$;
\item \label{thm:equivalence:item:2}
if $E$ is not a singleton and if the $\CUI(p)$ criterion holds true on $\parens[\big]{E, \cB(E)}$ for some family $\braces[\big]{(M_\theta, \cM_\theta, \mu_{\theta}), \: \theta \in \Theta}$, then\/ $\sup_{\theta \in \Theta} \mu_\theta(M_\theta) < \infty$.
\end{enumerate}
\end{theo}

We need the following lemma.

\begin{lemm}\label{lemm:equivalence-absolute}
Let $I \neq \emptyset$ and let $\braces[\big]{(\Omega_i, \cF_i, \bP_i), i \in I}$ be a family of probability spaces. Suppose that $X_i \colon \Omega_i \to \bR$ is $\cF_i/\cB(\bR)$-measurable with $R\coloneqq \sup_{i \in I} \bE_i\bracks[\big]{\abs{X_i}} <\infty$, where\/ $\bE_i$ is the expectation with respect to $\bP_i$. Then, the following assertions are equivalent:
\begin{enumerate}
\item \label{lemm:equivalence-absolute-item:1}
for any $\ep>0$, there exists $\kappa = \kappa(R, \ep)>0$ such that
\[
\forall i \in I, \ A_i \in \cF_i\;:\quad \bP_i(A_i) \lee \kappa \ \Longrightarrow \ \bE_i\bracks[\big]{\abs{X_i}\1_{A_i}} \lee \ep;
\]
\item \label{lemm:equivalence-absolute-item:2}
for any $\ep >0$, there exists $\kappa = \kappa(R, \ep) >0$ such that
\[
\forall i \in I, \ A_i \in \sigma\braces[\big]{\abs{X_i}}\; : \quad \bP_i(A_i) \lee \kappa \ \Longrightarrow \ \bE_i\bracks[\big]{\abs{X_i}\1_{A_i}} \lee \ep.
\]
\end{enumerate}
\end{lemm}

\begin{proof}
The implication~\eqref{lemm:equivalence-absolute-item:1} $\Rightarrow$~\eqref{lemm:equivalence-absolute-item:2} is obvious. Now, we assume that~\eqref{lemm:equivalence-absolute-item:2} is satisfied. Then, for any $\ep>0$, there exists $\kappa >0$ such that, for any $i \in I$, $A_i \in \sigma\braces[\big]{\abs{X_i}}$, $\bP_i(A_i) \lee \kappa$ implies $\bE_i\bracks[\big]{\abs{X_i}\1_{A_i}} \lee \ep/2$. Let $a>0$ be sufficiently large such that $R/a \lee \kappa$. Then,
\[
\sup_{i \in I}\bP_i\parens[\big]{\braces[\big]{\abs{X_i} >a}} \lee \frac{1}{a} \sup_{i \in I} \bE_i\bracks[\big]{\abs{X_i}} = \frac{R}{a} \lee \kappa.
\]
For $\kappa'\coloneqq \ep/(2a)$ and for any $A_i\in \cF_i$ with $\bP_i(A_i) \lee \kappa'$, one has
\[
\begin{split}
\bE_i\bracks[\big]{\abs{X_i}\1_{A_i}} & = \bE_i\bracks[\big]{\abs{X_i} \1_{A_i \cap \braces*{\vphantom{f}\abs{X_i} \lee a}}} + \bE_i\bracks[\big]{\abs{X_i} \1_{A_i \cap \braces*{\vphantom{f}\abs{X_i} > a}}} \\
& \lee a \, \bP_i(A_i) + \bE_i\bracks[\big]{\abs{X_i} \1_{\braces*{\vphantom{f}\abs{X_i} > a}}}\\
& \lee a \kappa' + \ep/2 = \ep.
\end{split}
\]
Hence, statement~\eqref{lemm:equivalence-absolute-item:1} holds true.
\end{proof}

\vspace{.5\baselineskip}

\begin{proof}[Proof of \theoref{thm:equivalence}]
\mbox{}\\*[-3ex]
\begin{proof}[(\ref{thm:equivalence:item:1})]
The proof is proceeded in three steps.
\begin{claim}\label{claim_1}
The\/ $\CUI(p)$ criterion holds true on $\parens[\big]{E, \cB(E)}$ for all families $\braces[\big]{(M_\theta, \cM_\theta, \mu_{\theta}), \: \theta \in \Theta}$ with
\[
(M_\theta, \cM_\theta, \mu_\theta) =(M_{\theta'}, \cM_{\theta'}, \mu_{\theta'}) \quad \text{and} \quad \mu_\theta (M_\theta) = 1, \quad \forall \theta, \theta' \in \Theta.
\]
\end{claim}
In this case, we let $(M_\theta, \cM_\theta, \mu_{\theta}) = (M, \cM, \mu)$ be the common probability space. The statements~\ref{item:2:statement}-\eqref{item:2:statement-a}, \ref{item:3:statement}-\eqref{item:3:statement-a} and~\ref{item:4:statement}-\eqref{item:4:statement-a} coincide, and, by \remaref{rema:CUIC}\eqref{rema:CUIC:item:2}, the relation~\ref{item:2:statement}-\eqref{item:2:statement-b} $\Leftrightarrow$~\ref{item:3:statement}-\eqref{item:3:statement-b} $\Leftrightarrow$~\ref{item:4:statement}-\eqref{item:4:statement-b} holds true. The remaining equivalence~\ref{item:1:statement} $\Leftrightarrow$~\ref{item:2:statement} also holds due to~\cite[Lemma~2.2]{WR87} where we note that the argument in the proof of~\cite[Lemma~2.2]{WR87} is also valid for any $\Theta$-indexed family of $E$-valued random elements and that the linearity of the underlying space is not necessary there.

\begin{claim}\label{claim_2}
The\/ $\CUI(p)$ criterion holds true on $\parens[\big]{E, \cB(E)}$ for all families $\braces[\big]{(M_\theta, \cM_\theta, \mu_{\theta}), \: \theta \in \Theta}$ with
\[
\mu_\theta(M_\theta) =1, \quad \forall \theta \in \Theta.
\]
\end{claim}
In this case, $\braces[\big]{(M_\theta, \cM_\theta, \mu_{\theta}), \: \theta \in \Theta}$ is a family of probability spaces. Let $f_\theta\colon M_\theta \to E$ be $\cM_\theta/\cB(E)$-measurable for any $\theta \in \Theta$. According to the Kolmogorov extension theorem, which is recalled in \propref{prop:Kolmogorov-extension} below, there exist a probability space $(\bfM, \frM, \bmu)$ and $\frM/\cM_\theta$-measurable functions $\pi_\theta \colon \bfM \to M_\theta$, $\theta \in \Theta$, such that
\begin{equation}\label{eq:probability-measure}
\bmu \circ \pi_\theta^{-1} = \mu_\theta \quad \text{on } \cM_\theta.
\end{equation}
Then, for any non-negative and $\cM_\theta/\cB(\bR)$-measurable function $h \colon M_\theta\to \bR$, applying the change of variables formula (see, e.g., \cite[Theorem~15.1]{Sc17}) we get
\begin{equation}\label{eq:change-variable}
\int_{M_\theta} h(x) \mu_\theta (\od x) = \int_{M_\theta} h(x) (\bmu\circ \pi_\theta^{-1})(\od x) = \int_{\pi^{-1}_\theta(M_\theta)} h\parens[\big]{\pi_\theta(\bmx)} \bmu(\od \bmx) = \int_{\bfM} h\parens[\big]{\pi_\theta(\bmx)} \bmu(\od \bmx).
\end{equation}
Define $F_\theta \colon \bfM \to E$ by
\[
F_\theta (\bmx) \coloneqq f_\theta\parens[\big]{\pi_\theta(\bmx)}, \quad \bmx \in \bfM.
\]
We observe that:
\begin{itemize}
\item For~\ref{item:1:statement}, using~\eqref{eq:change-variable} with $h(x) = \norm{f_\theta(x)}_{\bme}^p\, \1_{\braces*{f_\theta(x) \notin K_{p,\ep}}}$ we obtain
\[
\int_{M_\theta} \norm{f_\theta}_{\bme}^p \, \1_{\braces*{f_\theta \notin K_{p,\ep}}} \od \mu_\theta = \int_{\bfM} \norm{F_\theta}_{\bme}^p \, \1_{\braces*{F_\theta \notin K_{p,\ep}}} \od \bmu.
\]
\item For~\ref{item:2:statement}-\eqref{item:2:statement-a}, using~\eqref{eq:change-variable} with $h(x) = \1_{\braces{f_\theta(x) \notin K'_\delta}}$ we get
\[
\mu_\theta\parens[\big]{\braces[\big]{f_\theta \notin K'_\delta}} = \int_{M_\theta} \1_{\braces*{\vphantom{f}\smash{f_\theta \notin K'_\delta}}} \od \mu_\theta = \int_{\bfM} \1_{\braces*{\vphantom{f}\smash{F_\theta \notin K'_\delta}}} \od \bmu = \bmu\parens[\big]{\braces[\big]{F_\theta \notin K'_\delta}}.
\]
For~\ref{item:2:statement}-\eqref{item:2:statement-b}, applying~\eqref{eq:change-variable} with $h(x) = \norm{f_\theta(x)}_{\bme}^p \, \1_{\braces*{\norm{f_\theta(x)}_{\bme} >a}}$ yields to
\[
\int_{M_\theta} \norm{f_\theta}_{\bme}^p \, \1_{\braces*{\norm{f_\theta}_{\bme} >a}} \od \mu_\theta = \int_{\bfM} \norm{F_\theta}_{\bme}^p \, \1_{\braces*{\norm{F_\theta}_{\bme} >a}} \od \bmu.
\]
\item~\ref{item:3:statement}-\eqref{item:3:statement-a} coincides with~\ref{item:2:statement}-\eqref{item:2:statement-a}. For~\ref{item:3:statement}-\eqref{item:3:statement-b}, applying~\eqref{eq:change-variable} for $h(x) = \norm{f_\theta(x)}_{\bme}^p$ we get
\[
\int_{M_\theta} \norm{f_\theta}_{\bme}^p \od\mu_\theta = \int_{\bfM} \norm[\big]{f_\theta\parens[\big]{\pi_\theta(\bmx)}}_{\bme}^p \, \bmu (\od \bmx) = \int_{\bfM} \norm{F_\theta}_{\bme}^p \od \bmu,
\]
which are finite. For the absolute continuity, we let $\phi \colon E \to \bR$ be defined by $\phi(x) \coloneqq \norm{x}_\bme^p$ and have the following equivalences:
\begin{align*}
& \text{``}\forall \ep >0, \ \exists \kappa >0 \; : \ \forall A_\theta \in \cM_\theta, \ \mu_\theta(A_\theta) \lee \kappa \ \Longrightarrow \ \int_{M_\theta} \norm{f_\theta}^p_\bme \1_{A_\theta} \od \mu_\theta \lee \ep\text{''}\\
\Longleftrightarrow \; & \text{``}\forall \ep >0, \ \exists \kappa >0 \; : \ \forall A_\theta \in \sigma\{\phi \circ f_\theta\}, \ \mu_\theta(A_\theta) \lee \kappa \ \Longrightarrow \ \int_{M_\theta} \phi \circ f_\theta\, \1_{A_\theta} \od \mu_\theta \lee \ep\text{''}\\
\Longleftrightarrow \; & \text{``}\forall \ep >0, \ \exists \kappa >0 \; : \ \forall B_\theta \in \cB(\bR), \ \mu_\theta \circ f_\theta^{-1} \circ \phi^{-1} (B_\theta) \lee \kappa \ \Longrightarrow \ \int_{M_\theta} \phi \circ f_\theta \, \1_{\braces*{\phi \circ f_\theta \in B_\theta}} \od \mu_\theta \lee \ep\text{''}\\
\Longleftrightarrow\; & \text{``}\forall \ep >0, \ \exists \kappa >0 \; : \ \forall \tilde A_\theta \in \sigma\{\phi \circ F_\theta\}, \ \bmu(\tilde A_\theta) \lee \kappa \ \Longrightarrow \ \int_{\bfM} \phi \circ F_\theta \, \1_{\tilde A_\theta} \od \bmu \lee \ep \text{''} \\
\Longleftrightarrow \; & \text{``}\forall \ep >0, \ \exists \kappa >0 \; : \ \forall \tilde A \in \frM, \ \bmu(\tilde A) \lee \kappa \ \Longrightarrow \ \int_{\bfM} \norm{F_\theta}^p_\bme \1_{\tilde A} \, \od \bmu \lee \ep \text{''}.
\end{align*}
Here, obtaining the first and the last equivalences is due to \lemmref{lemm:equivalence-absolute}, the third is due to $\mu_\theta \circ f_\theta^{-1} = \bmu \circ F_\theta^{-1}$ by~\eqref{eq:probability-measure}.
\item~\ref{item:4:statement}-\eqref{item:4:statement-a} coincides with~\ref{item:2:statement}-\eqref{item:2:statement-a}. For~\ref{item:4:statement}-\eqref{item:4:statement-b}, applying~\eqref{eq:change-variable} with $h(x) = G\parens[\big]{\norm{f_\theta(x)}_{\bme}}$ we obtain
\[
\int_{M_\theta} G\parens[\big]{\norm{f_\theta}_{\bme}} \od \mu_\theta = \int_{\bfM} G\parens[\big]{\norm{F_\theta}_{\bme}} \od \bmu.
\]
\end{itemize}
Hence, equivalences~\ref{item:1:statement} $\Leftrightarrow$~\ref{item:2:statement} $\Leftrightarrow$~\ref{item:3:statement} $\Leftrightarrow$~\ref{item:4:statement} hold for $\braces[\big]{F_\theta, \: \theta \in \Theta}$ on $(\bfM, \frM, \bmu)$ if and only if they hold for $\braces[\big]{f_\theta, \: \theta \in \Theta}$ in relation to $\braces[\big]{(M_\theta, \cM_\theta, \mu_\theta), \: \theta \in \Theta}$. Since the former holds true by \claimref{claim_1}, the latter, i.e.\ \claimref{claim_2}, follows.

\begin{claim}\label{claim_3}
The\/ $\CUI(p)$ criterion holds true on $\parens[\big]{E, \cB(E)}$ for all families $\braces[\big]{(M_\theta, \cM_\theta, \mu_{\theta}), \: \theta \in \Theta}$ with
\[
\sup_{\theta \in \Theta} \mu_\theta(M_\theta) < \infty.
\]
\end{claim}
We let $\braces[\big]{(M_\theta, \cM_\theta, \mu_{\theta}), \: \theta \in \Theta}$ be a family of finite measure spaces with $C\coloneqq \sup_{\theta \in \Theta} \mu_\theta(M_\theta) < \infty$. Without loss of generality, we may assume $C>0$, otherwise the equivalences~\ref{item:1:statement} $\Leftrightarrow$~\ref{item:2:statement} $\Leftrightarrow$~\ref{item:3:statement} $\Leftrightarrow$~\ref{item:4:statement} in \defiref{def:CUIC} are trivial. For $\theta \in \Theta$, denote
\[
C_\theta\coloneqq \mu_\theta(M_\theta) \in [0, C].
\]
Let $\bfo$ be an element such that $\bfo \notin \bigcup_{\theta \in \Theta} M_\theta$. For $\theta \in \Theta$ we set
\[
\tilde M_\theta\coloneqq M_\theta \cup \{\bfo\}, \quad \tilde\cM_\theta\coloneqq \sigma\braces[\big]{\cM_\theta, \{\bfo\}},
\]
and for $A \in \tilde\cM_\theta$, we define
\[
\tilde\mu_\theta(A) \coloneqq \frac{1}{C} \mu_\theta\parens[\big]{A\backslash\{\bfo\}} + \frac{C- C_\theta}{C} \delta_{\bfo}\parens[\big]{A \cap \{\bfo\}}.
\]
Note that
\begin{equation}\label{eq:relation-2}
\tilde\mu_\theta(A) = \frac{1}{C} \mu_\theta(A), \quad A \in \cM_\theta.
\end{equation}
Since
\[
\tilde\mu_\theta(\tilde M_\theta) = \frac{1}{C} \mu_\theta(M_\theta) + \frac{C-C_\theta}{C} \delta_\bfo\parens[\big]{\{\bfo\}} = \frac{C_\theta}{C} + \frac{C-C_\theta}{C} = 1,
\]
it implies that $\braces[\big]{(\tilde M_\theta, \tilde\cM_\theta, \tilde\mu_\theta), \: \theta \in \Theta}$ is a family of probability spaces. Define $\tilde f_\theta\colon \tilde M_\theta \to E$ by
\[
\tilde f_\theta(x) \coloneqq 
\begin{cases}
f_\theta(x) & \text{if } x\in M_\theta,\\
\bme & \text{if } x = \bfo.
\end{cases}
\]
Then, for $B \in \cB(E)$,
\[
\braces[\big]{x \in \tilde M_\theta : \tilde f_\theta(x) \in B} = \braces[\big]{x \in M_\theta : \tilde f_\theta(x) \in B} \cup \braces[\big]{x \in \{\bfo\} : \tilde f_\theta(x) \in B} = \braces[\big]{x \in M_\theta : f_\theta(x) \in B} \cup B_\bfo,
\]
where $B_\bfo \in \tilde \cM_\theta$ is given by
\[
B_\bfo\coloneqq 
\begin{cases}
\{\bfo\} & \text{if } \bme \in B,\\
\emptyset & \text{if } \bme \notin B.
\end{cases}
\]
Since $\braces[\big]{x \in M_\theta : f_\theta(x) \in B} \in \cM_\theta$, it implies that $\braces[\big]{x \in \tilde M_\theta : \tilde f_\theta(x) \in B} \in \tilde \cM_\theta$. Hence, $\tilde f_\theta$ is $\tilde\cM_\theta/\cB(E)$-measurable.

For any non-negative and $\cB(E)/\cB(\bR)$-measurable function $H \colon E \to \bR$ with $H(\bme) = 0$ one has
\begin{equation}\label{eq:transform-f-g}
\begin{split}
\int_{\tilde M_\theta} H\parens[\big]{\tilde f_\theta(x)} \tilde \mu_\theta(\od x) & = \int_{M_\theta} H\parens[\big]{\tilde f_\theta(x)} \tilde \mu_\theta(\od x) + \int_{\braces*{\bfo}} H\parens[\big]{\tilde f_\theta(x)} \tilde\mu_\theta(\od x) \\
& = \frac{1}{C} \int_{M_\theta} H\parens[\big]{f_\theta(x)} \mu_\theta(\od x) + \frac{C - C_\theta}{C} H\parens[\big]{\tilde f_\theta(\bfo)} \delta_\bfo\parens[\big]{\{\bfo\}} \\
& = \frac{1}{C} \int_{M_\theta} H\parens[\big]{f_\theta(x)} \mu_\theta(\od x).
\end{split}
\end{equation}
We observe that:
\begin{itemize}
\item For~\ref{item:1:statement}, we use~\eqref{eq:transform-f-g} with $H(x) = \norm{x}_{\bme}^p \, \1_{\braces*{x \notin K_{p,\ep}}}$ to get
\[
\int_{\tilde M_\theta} \norm{\tilde f_\theta}_{\bme}^p \, \1_{\braces*{\tilde f_\theta \notin K_{p,\ep}}} \od \tilde \mu_\theta = \int_{\tilde M_\theta} H\parens[\big]{\tilde f_\theta(x)}\tilde \mu_{\theta}(\od x) = \frac{1}{C}\int_{M_\theta} \norm{f_\theta}_{\bme}^p \, \1_{\braces*{f_\theta \notin K_{p,\ep}}} \od \mu_\theta.
\]
\item For~\ref{item:2:statement}-\eqref{item:2:statement-a}, we may assume without loss of generality that $\bme \in K'_\delta$, otherwise we can let $\tilde K'_\delta \coloneqq K'_\delta \cup\{\bme\}$ which is also a compact set in $E$, and moreover, \ref{item:2:statement}-\eqref{item:2:statement-a} holds for $\tilde K'_\delta$ provided that~\ref{item:2:statement}-\eqref{item:2:statement-a} holds for $K'_\delta$. Note that
\[
\begin{split}
\braces[\big]{x \in \tilde M_\theta : \tilde f_\theta(x) \in K'_\delta} 
	& = \braces[\big]{x \in M_\theta : \tilde f_\theta(x) \in K'_\delta} \cup \braces[\big]{x \in \{\bfo\} : \tilde f_\theta(x) \in K'_\delta}
\\	& = \braces[\big]{x \in M_\theta : f_\theta(x) \in K'_\delta} \cup \{\bfo\},
\end{split}
\]
which implies
\[
\braces[\big]{x \in \tilde M_\theta : \tilde f_\theta(x) \notin K'_\delta} = \parens[\big]{\tilde M_\theta \backslash \braces[\big]{x \in M_\theta : f_\theta(x) \in K'_\delta}} \cap M_\theta = \braces[\big]{x \in M_\theta : f_\theta(x) \notin K'_\delta}.
\]
Hence, by~\eqref{eq:relation-2},
\[
\tilde \mu_\theta\parens[\big]{\braces[\big]{\tilde f_\theta \notin K'_{\delta}}} = \frac{1}{C} \mu_{\theta} \parens[\big]{\braces[\big]{f_\theta \notin K'_{\delta}}}.
\]
For~\ref{item:2:statement}-\eqref{item:2:statement-b}, we apply~\eqref{eq:transform-f-g} with $H(x) = \norm{x}_{\bme}^p \, \1_{\braces*{\vphantom{f}\norm{x}_{\bme} >a}}$ to get
\[
\int_{\tilde M_\theta} \norm{\tilde f_\theta}_{\bme}^p \, \1_{\braces*{\vphantom{f}\norm{\smash{\tilde f_\theta}} > a}} \od \tilde \mu_\theta = \frac{1}{C}\int_{M_\theta} \norm{f_\theta}_{\bme}^p \, \1_{\braces*{\norm{f_\theta}_{\bme} >a}} \od \mu_\theta.
\]
\item~\ref{item:3:statement}-\eqref{item:3:statement-a} coincides with~\ref{item:2:statement}-\eqref{item:2:statement-a}. For~\ref{item:3:statement}-\eqref{item:3:statement-b}, we have $\tilde\mu_\theta(A_\theta) = \frac{1}{C}\mu_\theta(A_\theta)$ by~\eqref{eq:relation-2} and $\restr{\tilde{f}}{A} = f$ by the definition of $\tilde f$ for any $A \in \cM_\theta \subseteq \tilde \cM_\theta$, and hence,
\[
\int_{\tilde M_\theta} \norm{\tilde f_\theta}_{\bme}^p \, \1_{A_\theta} \od \tilde\mu_\theta = \frac{1}{C}\int_{M_\theta} \norm{f_\theta}_{\bme}^p \, \1_{A_\theta} \od\mu_\theta.
\]
\item~\ref{item:4:statement}-\eqref{item:4:statement-a} coincides with~\ref{item:2:statement}-\eqref{item:2:statement-a}. For~\ref{item:4:statement}-\eqref{item:4:statement-b}, it follows from~\eqref{eq:transform-f-g} with $H(x) \coloneqq G\parens[\big]{\norm{x}_{\bme}}$ that
\[
\int_{\tilde M_\theta} G\parens[\big]{\norm{\tilde f_\theta}_{\bme}} \od \tilde \mu_\theta = \int_{\tilde M_\theta} H(\tilde f_\theta) \od \tilde \mu_\theta = \frac{1}{C} \int_{M_\theta} H(f_\theta) \od \mu_\theta = \frac{1}{C} \int_{M_\theta} G\parens[\big]{\norm{f_\theta}_{\bme}} \od \mu_\theta.
\]
\end{itemize}
According to those observations, as $C \in (0, \infty)$, the equivalences~\ref{item:1:statement} $\Leftrightarrow$~\ref{item:2:statement} $\Leftrightarrow$~\ref{item:3:statement} $\Leftrightarrow$~\ref{item:4:statement} in \defiref{def:CUIC} for $\braces[\big]{f_\theta, \: \theta \in \Theta}$ in relation to $\braces[\big]{(M_\theta, \cM_\theta, \mu_\theta), \: \theta \in \Theta}$ can be derived from the correspondent equivalences for $\braces[\big]{\tilde f_\theta, \: \theta \in \Theta}$ in relation to $\braces[\big]{(\tilde M_\theta, \tilde \cM_\theta, \tilde \mu_\theta), \: \theta \in \Theta}$. Since the latter equivalences are satisfied by \claimref{claim_2}, we obtain \claimref{claim_3}. The proof of item~\eqref{thm:equivalence:item:1} is completed.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{thm:equivalence:item:2})]
Since $E$ is not a singleton, there exists $\tilde \bme \in E$ and $\tilde \bme \neq \bme$. We define $f_\theta(x) \coloneqq \tilde \bme$ for all $x\in M_\theta$ and $\theta \in \Theta$. Then, $f_\theta$ is clearly $\cM_\theta/\cB(E)$-measurable. For any $\ep >0$, we choose $K_{p, \ep} \coloneqq \{\tilde\bme\} \in \cK(E)$ to get $\1_{\braces*{f_\theta \notin K_{p, \ep}}} = 0$, and hence, \ref{item:1:statement} is satisfied for $\braces[\big]{f_\theta, \: \theta \in \Theta}$. Since, by assumption, the $\CUI(p)$ criterion holds true on $\parens[\big]{E, \cB(E)}$ for $\braces[\big]{(M_\theta, \cM_\theta, \mu_\theta), \: \theta \in \Theta}$, the condition $\sup_{\theta \in \Theta} \int_{M_\theta} \norm{f_\theta}^p_{\bme} \od \mu_\theta <\infty$ in~\ref{item:3:statement}-\eqref{item:3:statement-b} has to be satisfied, which then yields to $\sup_{\theta \in \Theta} \mu_\theta(M_\theta) < \infty$ as $\norm{f_\theta}_\bme = \norm{\tilde \bme}_\bme >0$.
\end{proof}
\let\qed\relax
\end{proof}

It is straightforward to check that the conclusion of \theoref{thm:equivalence}\eqref{thm:equivalence:item:2} does not depend on the reference element $\bme$. We now show that this observation is also true for \theoref{thm:equivalence}\eqref{thm:equivalence:item:1}.

\begin{prop}\label{prop:independence-choice-e}
Let $\bme' \in E$. Then, the conclusion in \theoref{thm:equivalence}(\ref{thm:equivalence:item:1}) holds true with respect to $\bme$ if and only if it holds true with respect to $\bme'$.
\end{prop}

\begin{proof}
The proof of \theoref{thm:equivalence}\eqref{thm:equivalence:item:1} reveals that \claimref{claim_1}, \claimref{claim_2} and \claimref{claim_3} (stated in relation to~$\bme$) are equivalent. Hence, it is sufficient to verify that \claimref{claim_1} stated in relation to $\bme$ is equivalent to \claimref{claim_1} stated in relation to $\bme'$. Indeed, let us consider $(M_\theta, \cM_\theta, \mu_\theta) \eqqcolon (M, \cM, \mu)$ a probability space and any $\cM/\cB(E)$-measurable functions $f_\theta \colon M \to E$, $\theta \in \Theta$. By writing~\ref{item:2:statement}$_{\bme}$, we mean assertion~\ref{item:2:statement} stated in relation to $\bme$, and analogously for~\ref{item:2:statement}$_{\bme'}$. Note that~\ref{item:2:statement}-\eqref{item:2:statement-a} does not depend on the choice of $\bme$. In addition, \ref{item:2:statement}$_{\bme}$-\eqref{item:2:statement-b} $\Leftrightarrow$~\ref{item:2:statement}$_{\bme'}$-\eqref{item:2:statement-b} due to $\abs[\big]{d_E(f_\theta, \bme) - d_E(f_\theta, \bme')} \lee d_E(\bme, \bme')$. Hence, we get~\ref{item:2:statement}$_{\bme}$ $\Leftrightarrow$~\ref{item:2:statement}$_{\bme'}$, which completes the proof.
\end{proof}

\section{\texorpdfstring{$(\nu_\theta)_{\theta \in \Theta}$}{(nu theta) theta in Theta}-compact uniform integrability and characterizations} \label{sec:CUI-random}

In this section, we assume that $(\Omega, \cF, \bP)$ is a probability space, $(E, d_E)$ is a separable and complete metric space equipped with the Borel $\sigma$-algebra $\cB(E)$.

\begin{assumption}\label{assumption}
Assume that $\Theta \neq \emptyset$ is an arbitrary index set and $\braces[\big]{(S_\theta, \cS_\theta, \nu_\theta), \: \theta \in \Theta}$ is a family of finite measure spaces satisfying
\[
\sup_{\theta \in \Theta} \nu_\theta(S_\theta) < \infty.
\]
Let $\braces[\big]{X_s, \: s \in \bigcup_{\theta \in \Theta} S_\theta}$ be a family of $E$-valued random elements. We assume that for all $\theta \in \Theta$ the map $\Omega \times S_\theta \ni (\omega, s) \mapsto X_s(\omega)$ is $\cF\otimes \cS_\theta/\cB(E)$-measurable, i.e.
\[
\braces[\big]{(\omega, s) \in \Omega \times S_\theta : X_s(\omega) \in B} \in \cF \otimes \cS_\theta, \quad \forall B \in \cB(E).
\]
\end{assumption}

\begin{rema}\label{rema:measurability}
We give some sufficient conditions for the joint measurability in Assumption~\ref{assumption}.
\begin{enumerate}
\item \label{item:1:rema:measurability}
If $S_\theta$ is countable and $\cS_\theta = \cP(S_\theta)$ for all $\theta \in \Theta$, then Assumption~\ref{assumption} is satisfied. Indeed, we show that $\Omega \times S_\theta \ni (\omega, s) \mapsto X_s(\omega)$ is $\cF\otimes \cS_\theta/\cB(E)$-measurable for all $\theta \in \Theta$. For any $B \in \cB(E)$, one has
\[
\braces[\big]{(\omega, s) \in \Omega \times S_\theta : X_s(\omega) \in B} = \bigcup_{s \in S_\theta} \parens[\Big]{\braces[\big]{\omega \in \Omega : X_s(\omega) \in B} \times \{s\}}.
\]
Since $\{s\} \in \cS_\theta$ and $\{X_s\in B\} \in \cF$, we conclude that the countable union $\bigcup_{s\in S_\theta}$ on the right-hand side belongs to $\cF\otimes \cS_\theta$. Hence, $\braces[\big]{(\omega, s) \in \Omega \times S_\theta : X_s(\omega) \in B} \in \cF\otimes \cS_\theta$.
\item Let $S_\theta = \bI$ and $\cS_\theta = \cB(\bI)$ for all $\theta \in \Theta$, where $\bI = [0, \infty)$ or $\bI = [0, T]$ for some fixed $T \in (0, \infty)$. Assume that $X_s \colon \Omega \to E$ is $\cF/\cB(E)$-measurable for every $s \in \bI$ and that the mapping $\bI \ni s \mapsto X_s(\omega)$ is right-continuous for all $\omega \in \Omega$, i.e.\ $\lim_{s + \ep \in \bI,\; 0 \le \ep \downarrow 0} d_E\parens[\big]{X_{s + \ep}(\omega), X_s(\omega)} = 0$ for all $(\omega, s) \in \Omega \times \bI$. Then, $(\omega, s) \mapsto X_s(\omega)$ satisfies the measurability condition in Assumption~\ref{assumption}. This assertion can be proven by using a standard approximation argument. Note that the joint measurability of $X$ also holds when the right-continuity is replaced by the left-continuity.
\end{enumerate}
\end{rema}

\begin{defi}\label{def:UI-mu}
Let $p \in (0, \infty)$. A collection $\braces[\big]{X_s, \: s \in \bigcup_{\theta \in \Theta} S_\theta}$ satisfying Assumption~\ref{assumption} is said to be \emph{$(\nu_\theta)_{\theta \in \Theta}$-compactly uniformly $p$-th order integrable}, or $(\nu_\theta)_{\theta \in \Theta}$-$\CUI(p)$ for short, if for any $\ep >0$ there exists $K_{p,\ep} \in \cK(E)$ such that
\begin{equation}\label{eq:def:UI-mu}
\sup_{\theta \in \Theta} \bE \bracks[\Bigg]{\int_{S_\theta} \norm{X_s}_{\bme}^p \, \1_{\braces*{X_s \notin K_{p,\ep}}} \nu_\theta (\od s)} \lee \ep.
\end{equation}
\end{defi}

The notion of $(\nu_\theta)_{\theta \in \Theta}$-$\CUI(p)$ generalizes several well-known notions of $\CUI(p)$ as shown in Examples~\ref{exam:classical-UI}--\ref{exam:weighted-UI} below.

\begin{exam}[Classical $\CUI(p)$]\label{exam:classical-UI}
Let $\braces[\big]{X_\theta, \: \theta \in \Theta}$ be a collection of $E$-valued random elements. We show that the $\CUI(p)$ condition for $\braces[\big]{X_\theta, \: \theta \in \Theta}$ can be regarded as the $(\nu_\theta)_{\theta \in \Theta}$-$\CUI(p)$ for a suitable choice of $\braces[\big]{(S_\theta, \cS_\theta, \nu_\theta), \: \theta \in \Theta}$. Indeed, for every $\theta \in \Theta$ we let $S_\theta \coloneqq \{\theta\}$, $\cS_\theta \coloneqq \cP(S_\theta) = \braces[\big]{\emptyset, \{\theta\}}$ and $\nu_\theta \coloneqq \delta_\theta$, where $\delta_\theta$ is the Dirac delta measure at $\theta \in S_\theta$.
Then, we apply \remaref{rema:measurability}\eqref{item:1:rema:measurability} to find that Assumption~\ref{assumption} is satisfied. Hence,
\[
\int_{S_\theta} \norm{X_s}_{\bme}^p \, \1_{\braces*{X_s \notin K_{p,\ep}}} \nu_\theta (\od s) = \int_{\braces*{\theta}} \norm{X_s}_{\bme}^p \, \1_{\braces*{X_s \notin K_{p,\ep}}} \delta_\theta (\od s)= \norm{X_\theta}_{\bme}^p \, \1_{\braces*{X_\theta \notin K_{p,\ep}}},
\]
and then condition~\eqref{eq:def:UI-mu} becomes
\[
\sup_{\theta \in \Theta} \bE\bracks[\big]{\norm{X_\theta}_{\bme}^p \, \1_{\braces*{X_\theta \notin K_{p,\ep}}}} \lee \ep,
\]
which is the classical $\CUI(p)$ of the family $\braces[\big]{X_\theta, \: \theta \in \Theta}$, see, e.g., \cite{HG76, Ta78, DT82} when $E$ is a separable Banach space (with $\bme = 0$).
\end{exam}

\begin{exam}[Ces\`aro $\CUI(p)$ for multi-indexed random elements]\label{exam:Cesaro-CUI}
Let $d \in \bN$ and let $(X_{\bfn})_{\bfn \in \bN^d}$ be a family of $E$-valued random elements.
We set $\Theta \coloneqq \bN^d$, $S_\bfn \coloneqq \braces[\big]{\bfk \in \bN^d : \bfk \lec \bfn}$ and $\cS_\bfn \coloneqq \cP(S_\bfn)$ for all $\bfn \in \bN^d$. Here, $\bfk = (k_1, \dotsc, k_d) \lec \bfn = (n_1, \dotsc, n_d)$ means that $k_i \lee n_i$ for all $i = 1, \dotsc, d$. For any $\bfn = (n_1, n_2, \dotsc, n_d) \in \bN^d$ we set $\abs{\bfn} \coloneqq n_1 \cdot n_2 \dotsm n_d$ and define
\[
\nu_\bfn\coloneqq \frac{1}{\abs{\bfn}} \sum_{\bN^d \ni \bfk \lec \bfn} \delta_\bfk
\]
with $\delta_{\bfk}$ the Dirac measure at $\bfk$. Then, $\nu_{\bfn}(S_\bfn) = 1$. Due to \remaref{rema:measurability}\eqref{item:1:rema:measurability}, one has that Assumption~\ref{assumption} is satisfied. Then,
\[
\int_{\bN^d} \norm{X_s}_{\bme}^p \, \1_{\braces*{X_s \notin K_{p,\ep}}} \nu_{\bfn} (\od s) = \frac{1}{\abs{\bfn}} \sum_{\bN^d \ni \bfk \lec \bfn} \int_{\bN^d} \norm{X_s}_{\bme}^p \, \1_{\braces*{X_s \notin K_{p, \ep}}} \delta_{\bfk} (\od s) = \frac{1}{\abs{\bfn}} \sum_{\bN^d \ni \bfk \lec \bfn} \norm{X_{\bfk}}_{\bme}^p \, \1_{\braces*{X_{\bfk} \notin K_{p, \ep}}}.
\]
Hence, condition~\eqref{eq:def:UI-mu} becomes
\[
\sup_{\bfn \in \bN^d} \frac{1}{\abs{\bfn}} \sum_{\bN^d \ni \bfk \lec \bfn} \bE\bracks[\big]{\norm{X_\bfk}_{\bme}^p \, \1_{\braces*{X_\bfk \notin K_{p, \ep}}}} \lee \ep,
\]
which is the Ces\`aro $\CUI(p)$ condition for $(X_{\bfn})_{\bfn \in \bN^d}$, see~\cite{TQ20}, or see~\cite{BCS14, WR87} when $E$ is a separable Banach space (with $\bme = 0$).
\end{exam}

\begin{exam}[Weighted $\CUI(p)$]\label{exam:weighted-UI}
Let $d, d' \in \bN$ and let $(b_{\bfn, \bfn'})_{\bfn \in \bN^d, \: \bfn' \in \bN^{d'}}$ be an array of real numbers such that
\begin{equation}\label{eq:exam:weighted-UI}
0< B_\infty\coloneqq \sup_{\bfn \in \bN^d} B_{\bfn} < \infty, \quad \text{where } B_{\bfn}\coloneqq \sum_{\bfn' \in \bN^{d'}} \abs{b_{\bfn, \bfn'}}.
\end{equation}
For $\bfn \in \Theta \coloneqq \bN^d$, we let $S_{\bfn} \coloneqq \bN^{d'}$, $\cS_{\bfn} \coloneqq \cP(\bN^{d'})$ and define the measure $\nu_\bfn$ on $\cS_{\bfn}$ by
\[
\nu_{\bfn} \coloneqq \sum_{\bfn' \in \bN^{d'}} \abs{b_{\bfn, \bfn'}} \,\delta_{\bfn'}.
\]
Then, $\nu_{\bfn}(\bN^{d'}) = B_{\bfn}$, and hence, $\sup_{\bfn \in \bN^d} \nu_{\bfn}(\bN^{d'}) = B_\infty < \infty$. In this case, Assumption~\ref{assumption} is satisfied and condition~\eqref{eq:def:UI-mu} becomes
\begin{equation}\label{eq:weighted-UI}
\sup_{\bfn \in \bN^d} \sum_{\bfn' \in \bN^{d'}} \abs{b_{\bfn, \bfn'}} \, \bE\bracks[\big]{\norm{X_{\bfn, \bfn'}}_{\bme}^p \, \1_{\braces*{X_{\bfn, \bfn'} \notin K_{p,\ep}}}} \lee \ep.
\end{equation}
In particular, if $d = d' = 1$ and $b_{n, j} = 0$ for all $j \notin [u_n, v_n]$ and for some given sequences $(u_n)_{n \in \bN}$, $(v_n)_{n \in \bN}$ of integers with $u_n \lee v_n$, then condition~\eqref{eq:exam:weighted-UI} simply boils down to $\sup_{n \gee 1} \sum_{j =u_n}^{v_n} \abs{b_{n, j}} \in (0, \infty)$ and~\eqref{eq:weighted-UI} becomes
\[
\sup_{n \gee 1} \sum_{j = u_n}^{v_n} \abs{b_{n, j}} \, \bE\bracks[\big]{\norm{X_{n, j}}_{\bme}^p\, \1_{\braces*{ X_{n, j} \notin K_{p,\ep}}}} \lee \ep.
\]
Condition~\eqref{eq:weighted-UI} is called $\{b_{\bfn, \bfn'}\}$-$\CUI(p)$, see~\cite{Ca97} when $E$ is a separable Banach space.
\end{exam}

\begin{exam}[$\CUI(p)$ with random indices]\label{exam:CUI-random-indices}
For $\theta \in \Theta$, let $\tau_\theta \colon \Omega \to S_\theta$ be $\cF/\cS_\theta$-measurable and assume that $\sigma\braces[\big]{X_s, \: s\in S_\theta}$ is independent of $\sigma\{\tau_\theta\}$. Then, $\braces[\big]{X_s, \: s \in \bigcup_{\theta \in \Theta} S_\theta}$ is $(\bP\circ \tau_\theta^{-1})_{\theta \in \Theta}$-$\CUI(p)$ if and only if $\braces[\big]{X_{\tau_\theta}, \: \theta\in \Theta}$ is $\CUI(p)$, where $X_{\tau_\theta}(\omega) \coloneqq X_{\tau_\theta(\omega)}(\omega)$.

Indeed, let us fix $\theta \in \Theta$ and define $T_\theta \colon \Omega \to \Omega \times S_\theta$ by setting $T_\theta(\omega) \coloneqq \parens[\big]{\omega, \tau_\theta(\omega)}$. It is easy to check that $T_\theta$ is $\cF/\cF\otimes \cS_\theta$-measurable. Then, $X_{\tau_\theta} = X \circ T_\theta$. Now, for any non-negative and $\cB(E)/\cB(\bR)$-measurable function $h \colon E \to \bR$, one has
\[
\bE\bracks[\big]{h(X_{\tau_\theta})} = \int_{\Omega} h(X\circ T_\theta) \od \bP = \int_{\Omega \times S_\theta} h\parens[\big]{X_s(\omega)} (\bP \circ T_{\theta}^{-1})(\od \omega, \od s).
\]
Since $\sigma\braces[\big]{X_s, \: s\in S_\theta}$ is independent of $\sigma\{\tau_\theta\}$, the measure $\bP \circ T_{\theta}^{-1}$ restricted on $\sigma\braces[\big]{X_s, \: s\in S_\theta} \otimes \cS_\theta$ is equal to $\bP \otimes (\bP\circ \tau_\theta^{-1})$. To see this, letting any $A \in \sigma\braces[\big]{X_s, \: s\in S_\theta}$ and $B \in \cS_\theta$ we get
\[
\bP \circ T_\theta^{-1}(A \times B) = \bP\parens[\big]{T^{-1}_\theta(A \times B)} = \bP\parens[\big]{A \cap \{\tau_\theta \in B\}} = \bP(A) \, \bP\circ\tau^{-1}_\theta(B).
\]
Hence, using Tonelli--Fubini's theorem (see, e.g., \cite[Theorem~14.8]{Sc17}) we obtain
\[
\bE\bracks[\big]{h(X_{\tau_\theta})} = \int_{\Omega \times S_\theta} h\parens[\big]{X_s(\omega)}\, \bP \otimes (\bP \circ \tau^{-1}_\theta)(\od \omega, \od s) = \bE \bracks[\Bigg]{\int_{S_\theta} h(X_s) \, \bP\circ \tau^{-1}_\theta(\od s)},
\]
which then implies the desired assertion by choosing $h(x) = \norm{x}_{\bme}^p\,\1_{\braces*{x \notin K_{p, \ep}}}$.
\end{exam}

\begin{rema}
In view of \exemref{exam:CUI-random-indices}, the Ces\`aro $\CUI(p)$ condition for $(X_{\bfn})_{\bfn \in \bN^d}$ in \exemref{exam:Cesaro-CUI} can be regarded as the $\CUI(p)$ condition for $(X_{\tau_\bfn})_{\bfn \in \bN^d}$, where each $\tau_\bfn$ is a random variable uniformly distributed on $\braces[\big]{\bfk \in \bN^d : \bfk \lec \bfn}$ and is independent of $\braces[\big]{X_\bfk, \: \bfk \lec \bfn}$.
\end{rema}

\begin{theo}\label{thm:CUI-criterion}
Let $p \in (0, \infty)$. Under Assumption~\ref{assumption}, the following assertions are equivalent:
\begin{enumerate}
\item\label{item:1:CUI-criterion}
$\braces[\big]{X_s, \: s \in \bigcup_{\theta \in \Theta} S_\theta}$ is $(\nu_\theta)_{\theta \in \Theta}$-$\CUI(p)$;
\item \label{item:2:CUI-criterion}
one has
\begin{equation}\label{eq:UT}
\forall \delta > 0, \ \exists K'_\delta \in \cK(E) \;:\; \sup_{\theta \in \Theta} (\bP \otimes \nu_\theta)\parens[\Big]{\braces[\big]{(\omega, s) \in \Omega \times S_\theta: X_s(\omega) \notin K'_\delta}} \lee \delta,
\end{equation}
and
\begin{equation}\label{eq:UI}
\lim_{a \to \infty} \; \sup_{\theta \in \Theta} \bE \bracks[\Bigg]{\int_{S_\theta} \norm{X_s}_{\bme}^p\, \1_{\braces*{\vphantom{f}\norm{X_s}_{\bme} > a}} \nu_\theta(\od s)} = 0;
\end{equation}
\item \label{item:3:CUI-criterion}
condition~\eqref{eq:UT} holds, $\sup_{\theta \in \Theta} \bE\bracks[\big]{\int_{S_\theta} \norm{X_s}_{\bme}^p \, \nu_\theta(\od s)} < \infty$, and in addition, for any $\ep >0$ there exists $\kappa >0$ such that
\[
\forall A_\theta \in \cF \otimes \cS_\theta, \quad \sup_{\theta \in \Theta} (\bP \otimes \nu_\theta)(A_\theta) \lee \kappa \ \Longrightarrow \ \sup_{\theta \in \Theta} \bE \bracks[\Bigg]{\int_{S_\theta} \norm{X_s}_{\bme}^p\, \1_{A_\theta} \nu_\theta(\od s)} \lee \ep;
\]
\item \label{item:4:CUI-criterion}
condition~\eqref{eq:UT} holds, and in addition, there is a non-decreasing function $G \colon [0, \infty) \to [0, \infty)$ with $G(0) = 0$ and $G(x)/x^p \to \infty$ as $x \to \infty$ such that
\[
\sup_{\theta \in \Theta} \bE \bracks[\Bigg]{\int_{S_\theta} G\parens[\big]{\norm{X_s}_{\bme}} \nu_\theta(\od s)} < \infty.
\]
\end{enumerate}
\end{theo}

\begin{proof}
We let
$M_\theta \coloneqq \Omega \times S_\theta$, $\cM_\theta\coloneqq \cF\otimes \cS_\theta$, $\mu_\theta \coloneqq \bP \otimes \nu_\theta$. Then, under Assumption~\ref{assumption} one has
\[
\sup_{\theta \in \Theta} \mu_\theta(M_\theta) = \sup_{\theta \in \Theta} (\bP \otimes \nu_\theta)(\Omega \times S_\theta) = \sup_{\theta \in \Theta} \nu_\theta(S_\theta) < \infty.
\]
Using \theoref{thm:equivalence}\eqref{thm:equivalence:item:1} with $f_\theta \colon M_\theta \to E$ defined by $f_\theta(\omega, s) \coloneqq X_s(\omega)$, together with Tonelli--Fubini's theorem saying that $\int_{\Omega \times S_\theta} F\od \bP \otimes \nu_\theta = \bE\bracks[\big]{\int_{S_\theta} F \od \nu_\theta}$ for any $\cF \otimes \cS_\theta/\cB(\bR)$-measurable and non-negative $F$, we obtain the desired conclusion.
\end{proof}

In the particular case when $E \Mk=\Mk \bR^d$ equipped with the usual Euclidean norm (and let {$\bme \Mk=\Mk 0$}), then the $(\nu_\theta)_{\theta \in \Theta}$-$\CUI(p)$ condition of $\braces[\big]{X_s, \: s \in \bigcup_{\theta \in \Theta} S_\theta}$ is equivalent to the $(\nu_\theta)_{\theta \in \Theta}$-$\UI(p)$ condition~\eqref{eq:UI}. Moreover, \eqref{eq:UI} implies~\eqref{eq:UT} in this case, which can be argued by using Markov's inequality and Tonelli--Fubini's theorem
\[
\sup_{\theta \in \Theta}(\bP \otimes \nu_\theta)\parens[\Big]{\braces[\big]{(\omega, s) \in \Omega \times S_\theta : \norm[\big]{X_s(\omega)} > R }} \lee \frac{1}{R^p} \sup_{\theta \in \Theta} \bE \bracks[\Bigg]{\int_{S_\theta} \norm{X_s}^p \1_{\braces*{\vphantom{f}\norm{X_s} >R}} \nu_\theta(\od s)} \xrightarrow{R \to \infty} 0.
\]
Therefore, we derive from \theoref{thm:CUI-criterion} the following corollary.

\begin{coro}\label{coro:finite-dim}
Let $E =\bR^d$ be the $d$-dimensional Euclidean space with $\bme = 0$. Let $p \in (0, \infty)$. Then, under Assumption~\ref{assumption}, the following assertions are equivalent:
\begin{enumerate}
\item $\sup_{\theta \in \Theta} \bE\bracks[\big]{\int_{S_\theta} \norm{X_s}^p \1_{\braces*{\vphantom{f}\norm{X_s} > a}} \nu_\theta(\od s) } \to 0$ as $a \to \infty$;
\item $\sup_{\theta \in \Theta} \bE\bracks[\big]{\int_{S_\theta} \norm{X_s}^p \nu_\theta(\od s)} < \infty$, and for any $\ep >0$ there is $\kappa >0$ such that
\[
\forall A_\theta \in \cF \otimes \cS_\theta, \quad \sup_{\theta \in \Theta} (\bP \otimes \nu_\theta)(A_\theta) \lee \kappa \ \Longrightarrow \ \sup_{\theta \in \Theta} \bE \bracks[\Bigg]{\int_{S_\theta} \norm{X_s}^p \1_{A_\theta} \nu_\theta(\od s)} \lee \ep;
\]
\item there is a non-decreasing function $G \colon [0, \infty) \to [0, \infty)$ with $G(0) = 0$ and $G(x)/x^p \to \infty$ as $x \to \infty$ such that $\sup_{\theta \in \Theta} \bE\bracks[\big]{\int_{S_\theta} G\parens[\big]{\norm{X_s}} \nu_\theta(\od s) } < \infty$.
\end{enumerate}
\end{coro}

\begin{rema}
\mbox{}
\begin{enumerate}
\item If $E$ is a separable Banach space and $(\nu_\theta)_{\theta \in \Theta}$ is chosen as in \exemref{exam:weighted-UI}, then the equivalence~\eqref{item:1:CUI-criterion} $\Leftrightarrow$~\eqref{item:2:CUI-criterion} in \theoref{thm:CUI-criterion} is similar to~\cite[Corollary~1]{CV01}.
\item In \cororef{coro:finite-dim}, by letting $d = 1$ and choosing $(\nu_\theta)_{\theta \in \Theta}$ as in \exemref{exam:weighted-UI} we obtain~\cite[Theorems~2 and~3]{Ca94}.
\end{enumerate}
\end{rema}

\appendix

\section{Kolmogorov extension theorem}

We recall the Kolmogorov extension theorem presented in~\cite{Ba96}, where the index set is not necessarily countable.

\begin{prop}[{\cite[Theorem~9.2]{Ba96}}]\label{prop:Kolmogorov-extension}
Let\/ $\Theta \neq \emptyset$ be an arbitrary index set and $\braces[\big]{(M_\theta, \cM_\theta, \mu_\theta), \: \theta \in \Theta}$ a family of probability spaces. Then there exist a probability space $(\bfM, \frM, \bmu)$ and\/ $\frM/\cM_\theta$-measurable functions $\pi_\theta \colon \bfM \to M_\theta$, $\theta \in \Theta$, such that:
\begin{enumerate}\romanenumi
\item $\braces[\big]{\pi_\theta, \: \theta \in \Theta}$ is an independent family under $\bmu$;
\item $\bmu \circ \pi^{-1}_\theta = \mu_\theta$, i.e.\ $\bmu(\pi_\theta \in B) = \mu_\theta(B)$ for all $B\in \cM_\theta$ and $\theta \in \Theta$.
\end{enumerate}
\end{prop}

\section*{Acknowledgments}

A part of this article was written when the third author was affiliated with the Department of Mathematics, Saarland University, Germany.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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