2. Preliminaries

We review here the definitions of infinite countable discrete amenable group and in the process introduce some of the basic notation and terminology appearing throughout this paper.

Let G be an infinite countable discrete group and denote by e_G the identity of G. Denote by $\mathcal{F}_G$ the set of all non-empty finite subsets of G.

Definition 2.1. G is called amenable if for each $K \in \mathcal{F}_G$ and any δ > 0 there exists $F \in \mathcal{F}_G$ such that

\begin{equation*} |F \Delta K F| \lt \delta|F|, \end{equation*}

where $|\bullet|$ is the counting measure of the set $\bullet, K F=$ $\{k f: k \in K, f \in F\}$ and $F \Delta K F=(F \backslash K F) \cup(K F \backslash F)$.

Definition 2.2. A sequence $\left\{F_n: n \in \mathbb{N}\right\}$ in $\mathcal{F}_G$ is called a Følner sequence, if

\begin{equation*} \lim _{n \rightarrow \infty} \frac{\left|g F_n \Delta F_n\right|}{\left|F_n\right|}=0 \end{equation*}

for each $g \in G$.

It is not too hard to obtain the usual asymptotic invariance property from this, viz.: G is amenable if and only if G has a Følner sequence $\left\{F_n\right\}_{n \in \mathbb{N}}$. In the class of countable discrete groups, amenable groups include all solvable groups.

Throughout the current paper, we will assume that G is always an infinite countable discrete amenable group.

Let $(\Omega, \mathcal{F}, \mathbb{P},G)$ be a complete probability space together with a group G of invertible $\mathbb{P}$-preserving transformations of $(\Omega,\mathcal{F},\mathbb{P})$, where e_G acts as the identity transformation. Let X be a compact metric space with the distance function d and the Borel σ-algebra $\mathcal{B}(X)$. Let $\mathcal{E}$ be a measurable subset of $\Omega \times X$ with respect to the product σ-algebra $\mathcal{F} \otimes \mathcal{B}(X)$. The fibres $\mathcal{E}_\omega=\{x \in X:(\omega, x) \in \mathcal{E}\}, \omega \in \Omega$, are non-empty compact subsets of X.

A bundle RDS associated with $(\Omega, \mathcal{F}, \mathbb{P}, G)$ is a family

\begin{equation*} \mathbb{T}=\left\{T_{g,\omega}: \mathcal{E}_\omega \rightarrow \mathcal{E}_{g \omega}\mid g\in G, \omega\in \Omega \right\} \end{equation*}

satisfying:

(1) for each $\omega \in \Omega$, the transformation $T_{e_G, \omega}$ is the identity over $\mathcal{E}_\omega$,

(2) for each $g \in G$, the map $\left(\mathcal{E},(\mathcal{F} \times \mathcal{B}(X))_{\mathcal{E}}\right) \rightarrow(X, \mathcal{B}(X))$, given by $(\omega, x) \mapsto T_{g, \omega}(x)$, is measurable and

(3) for each $\omega \in \Omega$ and all $g_1, g_2 \in G, T_{g_2, g_1 \omega} \circ T_{g_1, \omega}=T_{g_2 g_1, \omega}$ (and so $T_{g^{-1}, \omega}=\left(T_{g, g^{-1} \omega}\right)^{-1}$ for each $\left.g \in G\right)$.

In this case, G has a natural measurable action on $\mathcal{E}$ with $\Theta_{g}(\omega, x) =\left(g \omega, T_{g, \omega} x\right)$ for each $g \in G$, called the corresponding skew product transformation.

If $T_{g, \omega}$ is a continuous map for each $g \in G$ (and so $T_{g, \omega}: \mathcal{E}_\omega \rightarrow \mathcal{E}_{g \omega}$ is a homeomorphism for $\mathbb{P}$-a.e. $\omega \in \Omega$ and each $g \in G$), then it is called a continuous bundle $R D S$.

By [Reference Castaing and Valadier11, Chapter III], the mapping $\omega \mapsto \mathcal{E}_\omega$ is measurable with respect to the Borel σ-algebra induced by the Hausdorff topology on the hyperspace 2^X of all non-empty compact subsets of X, and the distance function $d\left(x, \mathcal{E}_\omega\right)$ is measurable in $\omega \in \Omega$ for each $x \in X$.

These concepts generalize the classical concepts of dynamical systems as follows. We say that $(\Omega, \mathcal{F}, \mathbb{P}, G)$ is a trivial $M D S$, if Ω is a singleton.

For a compact metric space X, a continuous bundle RDS associated with a trivial measure dynamical system (MDS) means that there is a topological G-action (K, G) for some non-empty compact subset $K \subseteq X$, that is, the group G acts on K in the sense that there exists a family of homeomorphisms $\left\{F_g: g \in G\right\}$ of K such that $F_{g_2} \circ F_{g_1}=F_{g_2 g_1}$ for all $g_1, g_2 \in G$ and $F_{e_G}$ acts as the identity transformation over K. The pair (K, G) is also called a topological dynamical G-system.

Let $\mathrm{C}_{\mathcal{E}}^o$ be the set of all finite covers of $\mathcal{E}$ consisting of subsets of open random sets [Reference Arnold3, Definition 1.6.1]. Similarly, we denote by $\mathrm{C}_K^o$ the set of finite open covers of $K \subset X$. Denote by $\mathfrak{C}_{\mathcal{E}}^o$ the set of all countable open random covers α of $\mathcal{E}$ with $\alpha(\omega)=\{A(\omega): A \in$ $\alpha\} \in \mathrm{C}_{\mathcal{E}_\omega}^o$ for $\mathbb{P}$-a.a. ω, which have an increasing sequence $\left\{\Omega_1 \subset \Omega_2 \subset \cdots\right\} \subset \mathcal{F}$ such that $\mathbb{P}\left(\Omega_n\right) \rightarrow 1$ and $\alpha \cap\left(\Omega_n \times X\right)$ is a finite family for each $n \geq 1$ [Reference Dooley and Zhang19, Definition 11.1].

Suppose that $\alpha=\left\{A^{(i)}: i \in I\right\} \in \mathfrak{C}_{\mathcal{E}}^o$ and $F=\{g_{1},...,g_{n}\}\in \mathcal{F}_{G}$. Let

\begin{equation*} \alpha_{F}(\omega)=\bigvee_{i=1}^{n}\left(T_{g_i,\omega}\right)^{-1} \alpha\left(g_i \omega\right). \end{equation*}

Then $\alpha_{F}(\omega)$ is a finite open cover of $\mathcal{E}_\omega$ for $\mathbb{P}$-a.a. ω consisting of sets

\begin{equation*} A^{\left(j_1, j_2, \ldots, j_{n}\right)}(\omega)=\bigcap_{i=1}^{n}\left(T_{g_i,\omega}\right)^{-1} A^{\left(j_i\right)}\left(g_i \omega\right), \quad j_i \in I, \end{equation*}

(some of which may be empty).

By the measurability of $\Theta_{g_i}$, for every $g_i\in F$, it is easy to prove that, for any $j=\left(j_1, j_2, \ldots, j_{n}\right) \in I^n$, the graph of $\omega \mapsto A^j(\omega)$, denoted by $A^j=\{(\omega, x): x \in$ $\left.A^j(\omega)\right\}$, is measurable. In fact, $A^j=\bigcap_{i=1}^{n}\left(\Theta_{g_i}\right)^{-1} A^{\left(j_i\right)}$.

Let $K \subset X$ be compact and let α be a finite open cover of K. Write [Reference Lindenstrauss and Weiss40]

\begin{equation*} \operatorname{ord}(\alpha)=-1+\sup _{x \in K} \sum_{U \in \alpha} 1_U(x) \quad \text {and } \quad D(\alpha)=\min _{\beta \succ \alpha} \operatorname{ord}(\beta), \end{equation*}

where β runs over all finite open covers of K refining α.

For each $\omega \in \Omega$, write

\begin{equation*} D\left(\alpha_{F}(\omega)\right)=-1+\min _\beta \sup _{x \in \mathcal{E}_\omega} \sum_{U \in \beta} 1_U(x), \end{equation*}

where β runs over all finite open covers of $\mathcal{E}_\omega$ refining $\alpha_{F}(\omega)$. We will discuss the measurability and the subadditivity of the function $D\left(\alpha_{F}(\omega)\right)$ in the next section. We need the following standard facts, which will be used often in this paper.

These results can be found in [Reference Castaing and Valadier11, Chapter III]. Lemma 2.4 is a combination of Theorems III.22 and III.23. Lemma 2.5 is from Definition 10, Proposition III.13 and a trivial application of Lemma 2.4; see also [Reference Crauel14, Proposition 2.4]. We remark that the ω-section of G in Lemma 2.4 is not required to be a non-empty closed subset of X, which is a little different from the condition of the classical Kuratowski–Ryll–Nardzewski measurable selection theorem (see [Reference Castaing and Valadier11, Theorem III.8] or [Reference Kechris26, Theorem 12.13]).

We now give the definition of the isomorphism in the category of continuous bundle RDSs. For simplicity, we only consider the case of two continuous bundle RDSs over the same measure-preserving group G. For the general case, see Arnold [Reference Arnold3, 1.9].

In this definition, we do not require the measurability of $(\omega, y) \mapsto \psi_\omega^{-1}(y)$. In fact, Ψ is automatically bimeasurable by our assumption. Ψ is measurable, which can be easily obtained from the measurability of the map $(\omega, x) \mapsto \psi_\omega(x)$. In the other direction, set

\begin{equation*} H=\left\{\left(\omega, x, \psi_\omega(x)\right) \in \Omega \times X \times Y:(\omega, x) \in \mathcal{E}\right\} . \end{equation*}

Then H is $\mathcal{F} \otimes \mathcal{B}(X) \otimes \mathcal{B}(Y)$ measurable by Lemma 2.5 as the graph of $(\omega, x) \mapsto \psi_\omega(x)$. Let $\pi_{\Omega}$ be the natural projection from $\Omega \times X \times Y$ onto Ω. For any $y \in Y, E \in \mathcal{B}(X)$,

\begin{equation*} \left\{\omega \in \Omega: \psi_\omega^{-1}(y) \in E\right\}=\pi_{\Omega}(H \cap(\Omega \times E \times\{y\})) \end{equation*}

is $\mathcal{F}$ measurable by Lemma 2.4. So $\psi_\omega^{-1}(y)$ is measurable in ω. Note that ψ_ω is a homeomorphism, so $\psi_\omega^{-1}(y)$ is continuous in Y. It follows that $(\omega, y) \mapsto \psi_\omega^{-1}(y)$ is a Carath $\acute{e}$odory map and hence is measurable [Reference Aubin and Frankowska4, Definition 8.2.7, Lemma 8.2.6].

3. The function $D(\alpha_{F}(\omega))$

Recall that two families $\left(E_i\right)_{i \in I}$ and $\left(F_i\right)_{i \in I}$, with common index set I, are called combinatorially equivalent if

\begin{equation*} \bigcap_{i \in J} E_i \neq \emptyset \Longleftrightarrow \bigcap_{i \in J} F_i \neq \emptyset \end{equation*}

for every subset $J \subset I$.

The proof of the measurability of $D\left(\alpha_{F}(\omega)\right)$ relies on the following lemmas [Reference Coornaert13, § 1.6].

Lemma 3.3. Let K be a normal space and let α be a finite open cover of K. Then

\begin{equation*} D(\alpha)=\min _\gamma \operatorname{ord}(\gamma), \end{equation*}

where γ runs over all finite closed covers of K that are finer than α.

The proof of the following theorem comes from [Reference Ma, Yang and Chen44], but with some additions here.

Proof. Let $\alpha=\{A^{(i)}\}\in \mathrm{C}_{\mathcal{E}}^o$, $F=\{g_1,g_2,...,g_n\}$. J is an index set with  $|J|=\left|\alpha\right|^n$. Define $f: 2^X \times\left(2^X\right)^{|J|} \rightarrow\{-1\} \cup \mathbb{N}$ by

\begin{equation*} f\left(K,\left(F^j\right)_{j \in J}\right)=-1+\min _\beta \sup _{x \in K} \sum_{B \in \beta} 1_B(x), \end{equation*}

where β runs over all families of finite open covers of K such that

\begin{equation*} \beta \succ\left(K \backslash F^j\right)_{j \in J} \quad \text {and } \quad K=\bigcup_{B \in \beta} B. \end{equation*}

For $K=\emptyset$ or $\beta \in \emptyset$, we set $f=-1$. Obviously,

\begin{equation*} D\left(\alpha_F(\omega)\right)=f\left(\mathcal{E}_\omega,\left(\mathcal{E}_\omega \backslash A^j(\omega)\right)_{j \in J}\right) \geq 0 . \end{equation*}

Note that $\omega \mapsto \mathcal{E}_\omega$ is measurable and, for every $j \in J,\left\{(\omega, x): x \in \mathcal{E} \backslash A^j(\omega)\right\}=$ $\mathcal{E} \backslash A^j$, the graph of the multifunction $\omega \mapsto \mathcal{E}_\omega \backslash A^j(\omega)$ is measurable. In fact,

\begin{equation*} \mathcal{E}\backslash A^{j}=\mathcal{E}\backslash \left(\bigcap_{i=1}^{n}\left(\Theta g_{i}\right)^{-1}A^{(j_i)} \right) =\bigcup_{i=1}^{n}\left(\mathcal{E}\backslash\left(\left(\Theta g_{i}\right)^{-1}A^{(j_i)}\right)\right). \end{equation*}

By the measurability of $\Theta_{g_{i}}$ and $A^{(j_i)}\in\alpha $, the graph of $\omega\rightarrow\mathcal{E}_{\omega}\backslash A^{j_i}(\omega)$, denoted by $\mathcal{E}\backslash A^{(j_i)}$, is measurable.

By Lemma 2.5, $\omega \mapsto \mathcal{E}_\omega \backslash A^j(\omega)$ is measurable for any $j \in J$. Recalling that the composition of two measurable functions is measurable, it suffices to prove that

\begin{equation*} \mathscr{Q}_q=\left\{\left(K,\left(F^j\right)_{j \in J}\right): 0 \leq f\left(K,\left(F^j\right)_{j \in J}\right) \leq q\right\} \end{equation*}

is measurable with respect to the product σ-algebra $\mathcal{B}\left(2^X\right) \otimes \mathcal{B}\left(2^X\right)^{\otimes|J|}$ for any $q \in \mathbb{N}$. In fact, $\mathscr{Q}_q$ is open in $2^X \times\left(2^X\right)^{|J|}$.

For any $\left(K_{0},\left(F_{0}^{j}\right)_{j \in J}\right) \in \mathscr{Q}_{q}$, then $K_0 \ne \emptyset $, and there exists $\beta\in \mathrm{C}_{K_0}^o$, such that

\begin{equation*}\beta \succ \left(K_{0}\backslash F_{0}^{j}\right)_{j \in J} .\end{equation*}

This implies that $\left(K_{0}\backslash F_{0}^{j}\right)_{j \in J}$ itself is a finite open covering on K ₀ and has $D\left(\left(K_{0}\backslash F_{0}^{j}\right)_{j \in J}\right)\leq q.$ By Lemma 3.3, there exists a family of finite closed subsets $\left(E^{i}\right)_{i \in I}$ of K ₀ and a map $\varphi: I \rightarrow J$ such that

\begin{equation*} \begin{cases}K_{0}=\bigcup_{i \in I} E^{i}; & \\ E^{i} \subset K_{0} \backslash F_{0}^{\varphi(i)}, & \text {for all } i \in I; \\ \sum_{i \in I} 1_{E^{i}}(x) \leq q+1, & \text {for all } x \in K_{0} .\end{cases} \end{equation*}

Applying Lemma 3.2 to $E^{i} \subset K_{0} \backslash F_{0}^{\varphi(i)}, i \in I$, there exists a family of open subsets $\left(V^{i}\right)_{i \in I}$ of K ₀ such that

(3.1)\begin{equation} E^{i} \subset V^{i} \subset \overline{V^{i}} \subset K_{0} \backslash F_{0}^{\varphi(i)} \quad \text {for all } i \in I \end{equation}

and $\left(E^{i}\right)_{i \in I},\left(V^{i}\right)_{i \in I}$ and $\left(\overline{V^{i}}\right)_{i \in I}$ are combinatorially equivalent.

(3.2)\begin{equation} \sup _{x \in K_{0}} \sum_{i \in I} 1_{\overline{V}^{i}}(x)=\sup _{x \in K_{0}} \sum_{i \in I} 1_{V^{i}}(x)=\sup _{x \in K_{0}} \sum_{i \in I} 1_{E^{i}}(x) \leq q+1. \end{equation}

Recall that, by Kuratowski [Reference Kuratowski37, §17], [Reference Kuratowski36, §42], the family of all sets $\{F: F \subset G\}$ and of all sets $\{F: F \cap G \neq \emptyset\}$, where G runs over all open subsets of X, is an open subbase for the Hausdorff metric topology, which coincides with the Vietoris topology on 2^X [Reference Klein and Thompson35, Corollary 4.2.3]. Set

\begin{align*} \mathscr{K}_{1}&=\left\{K \in 2^{X}: K \subset \bigcup_{i \in I} V^{i}\right\}, \\ \mathscr{K}_{2}&=\bigcap_{\substack{\Gamma \subset I \\ |\Gamma| \gt q+1}}\left\{K \in 2^{X}: K \cap \bigcap_{i \in \Gamma} \overline{V^{i}}=\emptyset\right\} \end{align*}

and, for $j \in J$, set

\begin{equation*} \mathscr{F}_{j}=\left\{F \in 2^{X}: F \subset \bigcap_{i \in \varphi^{-1}\{j\}}\left(X \backslash \overline{V^{i}}\right)\right\}. \end{equation*}

Hence $\mathscr{K}_{1}, \mathscr{K}_{2}$ and $\mathscr{F}_{j}$ are all open in 2^X.

\begin{equation*} \mathscr{O}=\left(\mathscr{K}_{1} \cap \mathscr{K}_{2}\right) \times \prod_{j \in J} \mathscr{F}_{j} \end{equation*}

is open in $2^{X} \times\left(2^{X}\right)^{|J|}$, so that $\left(K_{0},\left(F_{0}^{j}\right)_{j \in J}\right) \in \mathscr{O} \subset \mathscr{Q}_{q}$.

Indeed, $\left(K_{0},\left(F_{0}^{j}\right)_{j \in J}\right) \in \mathscr{O}$. For any $\Gamma \subset I$ with $|\Gamma| \gt q+1$, if $ K_0 \cap \bigcap_{i \in \Gamma} \overline{V^{i}}\ne \emptyset, $ then there exists $y\in K_{0}$ and $y\in \bigcap_{i \in \Gamma}\overline{V^{i}}.$ By $y \in K_{0}=\bigcup_{i \in I} E^{i}$ and (3.2) then $ \sum_{i \in I} 1_{\overline{V}^{i}}(y)\leq q+1. $ By $y\in \bigcap_{i \in \Gamma}\overline{V^{i}}$, then

\begin{equation*} \sum_{i \in I} 1_{\overline{V^{i}}^{i}}(y)\geq \sum_{i \in \Gamma} 1_{\overline{V^{i}}^{i}}(y)\geq |\Gamma| \gt q+1, \end{equation*}

a contradiction.

Combining $K_{0}=\bigcup_{i \in I} E^{i}\subset \bigcup_{i \in I} V^{i}$, we know that $K_{0}\in \mathscr{K}_1\cap\mathscr{K}_2$.

From (3.1), for any $i\in I$, $F_{0}^{\varphi(i)}\subset X\backslash \overline{V}^{i}$. Then for $j=\varphi(i),$

\begin{equation*} F_{0}^{j}\subset\bigcap_{i\in {\varphi}^{-1}\{j\}}( X\backslash \overline{V}^{i}), \quad \text{i.e.}\quad \left(F_{0}^{j}\right)_{j \in J}\in \prod_{j \in J} \mathscr{F}_{j}. \end{equation*}

Hence, $\left(K_{0},\left(F_{0}^{j}\right)_{j \in J}\right) \in \mathscr{O}$.

Next, for any $\left(K,\left(F^{j}\right)_{j \in J}\right)$ $\in \mathscr{O}$, we want to get $\left(K,\left(F^{j}\right)_{j \in J}\right)$ $\in \mathscr{Q}_{q}.$ By $K\in \mathscr{K}_1\bigcap \mathscr{K}_2$, then

\begin{equation*} K\subset \bigcup_{i\in I} V^{i}\subset \bigcup_{i\in I}\overline{V}^{i}\quad \text{and}\quad \sup_{x\in K}\sum_{i\in I}\overline{V}^i\leq q+1. \end{equation*}

It follows from $(F_{j})_{j\in J}\in \prod_{j \in J} \mathscr{F}_{j}$ that for any $j\in J$,

\begin{equation*} F^{j}\subset \bigcap_{i\in \varphi^{-1}\{j\}}\left(X\backslash \overline{V}^i\right), \end{equation*}

so we get for any $j\in J$, there exists $i\in \varphi^{-1}\{j\}$ such that $K\bigcap \overline{V}^i\subset K\backslash F^j$, that is,

\begin{equation*} \left(K\cap \overline{V}^i\right)_{i\in I} \succ (K\backslash F^{j})_{j\in J}. \end{equation*}

By Lemma 3.3 again, this implies that

\begin{equation*} f\left(K,\left(F^j\right)_{j \in J}\right)=-1+\min _\beta \sup _{x \in K} \sum_{B \in \beta} 1_B(x)\leq -1+ \sup _{x \in K} \sum_{i \in I} 1_{\overline{V}^{i}}(x)\leq q. \end{equation*}

Hence, $\left(K,\left(F^{j}\right)_{j \in J}\right)$ $\in \mathscr{Q}_{q}.$

Since $\left(K_{0},\left(F_{0}^{j}\right)_{j \in J}\right)$ is arbitrary, $\mathscr{Q}_{q}$ is open in $2^{X} \times\left(2^{X}\right)^{|J|}$.

To prove that the open set $\mathscr{Q}_{q}$ is measurable, we simply observe that (X, d) is a compact metric space, so [Reference Kuratowski36, § 42, V], [Reference Michael45] 2^X is also a compact metric space (and hence separable) with the Hausdorff metric $\left(\emptyset\right.$ is an isolated point in 2^X).

Hence [Reference Kallenberg25, Lemma 1.2] the product σ-algebra coincides with the σ-algebra generated by the product topology

\begin{equation*} \mathcal{B}\left(2^{X}\right) \otimes \mathcal{B}\left(2^{X}\right)^{\otimes|J|}=\mathcal{B}\left(2^{X} \times\left(2^{X}\right)^{|J|}\right) . \end{equation*}

It follows that the open set $\mathscr{Q}_{q}$ is $\mathcal{B}\left(2^{X}\right) \otimes \mathcal{B}\left(2^{X}\right)^{\otimes|J|}$ measurable.

Let $\alpha\in \mathfrak{C}_{\mathcal{E}}^o$ and $\left\{\Omega_1\subset\Omega_2\subset\cdots\right\}\subset\mathcal{F}$ be an increasing sequence in the definition of $\mathfrak{C}_{\mathcal{E}}^o$. Set

\begin{equation*} \alpha^r=\left[\alpha \cap\left(\Omega_{r} \times X\right)\right] \bigcup\left[\left(\Omega_{r}^{c} \times X\right) \cap \mathcal{E}\right]. \end{equation*}

It is obvious that $D\left(\alpha_F(\omega)\right)=D\left(\alpha_F^r(\omega)\right)$ for any $\omega\in \bigcap_{i=1}^{n}\left(g_i\right)^{-1}\Omega_r$. Note that

\begin{equation*} \bigcap_{i=1}^n \left(g_i\right)^{-1}\Omega_r=\Omega\setminus \bigcup_{i=1}^{n}\{\omega:g_i\omega\notin \Omega_r\}, \end{equation*}

while $\mathbb{P}\{\omega: g_i\omega\notin \Omega_r\}=1-\mathbb{P}(\Omega_r)\rightarrow 0$. Hence for large enough r, $\mathbb{P}$-a.e. ω,

\begin{equation*} D(\alpha_{F}(\omega))=D(\alpha^r_{F}(\omega)). \end{equation*}

The proof of this theorem is complete.

Proof. Assume that $|\alpha|=l$.

\begin{equation*} D(\alpha(\omega)) \leq \operatorname{ord}(\alpha(\omega)) \leq l-1 \Longrightarrow \int D(\alpha(\omega)) d \mathbb{P}(\omega) \leq l-1. \end{equation*}

4. Mean topological dimension

Before introducing the notion of mean topological dimension for a continuous bundle RDS, we first review some properties of $D(\alpha)$. These lemmas can be found in [Reference Lindenstrauss and Weiss40] and [Reference Coornaert13]. In this chapter, we will assume that G is always an infinite countable discrete amenable group admitting a tiling Fø lner sequence.

Lemma 4.2. Let X and Y be topological spaces and let $f: X \rightarrow Y$ be a continuous map. Let β be a finite open cover of Y. Then

\begin{equation*} D\left(f^{-1}(\beta)\right) \leq D(\beta) \text {. } \end{equation*}

Lemma 4.3. Let X be a normal space. Let α and β be finite open covers of X. Then

\begin{equation*} D(\alpha \vee \beta) \leq D(\alpha)+D(\beta). \end{equation*}

Let X and Y be topological spaces. Let α be a finite open cover of X. A continuous map $f: X \rightarrow Y$ is said to be α-compatible if there exists a finite open cover β of Y such that $f^{-1}(\beta) \succ \alpha$. We will use the notation $f \succ \alpha$ to denote that f is α-compatible.

The following lemma [Reference Arnold3, Theorem 3.3.2] can be seen as a random version of Fekete’s subadditive lemma.

Lemma 4.6. (Kingman’s subadditive ergodic theorem)

Let ϑ be a measure-preserving transformation on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and let $\left(f_{n}\right)_{n \geq 1}$ be a sequence of non-negative random variables such that

\begin{equation*} f_{n+m}(\omega) \leq f_{n}(\omega)+f_{m}\left(\vartheta^{n} \omega\right). \end{equation*}

If $f_{n} \in \mathbb{L}^{1}(\Omega, \mathcal{F}, \mathbb{P})$ for all n, then there exists an integrable function $f: \Omega \rightarrow \mathbb{R}$ such that, for $\mathbb{P}$-a.a. $\omega, n^{-1} f_{n}(\omega) \rightarrow f(\omega)$. Moreover, the convergence also holds in $\mathbb{L}^{1}$ and satisfies

\begin{equation*} \int_{\Omega} f(\omega) d \mathbb{P}(\omega)=\lim _{n \rightarrow \infty} \frac{1}{n} \int_{\Omega} f_{n}(\omega) d \mathbb{P}(\omega)=\inf _{n \geq 1} \frac{1}{n} \int_{\Omega} f_{n}(\omega) d \mathbb{P}(\omega) \lt \infty. \end{equation*}

If, in addition, ϑ is ergodic, then $f(\omega)=\int f d \mathbb{P}$ for $\mathbb{P}$-a.a. ω.

In contrast to the amenable group $\mathbb{Z}$, a general infinite countable discrete amenable group may have a complicated combinatorial structure.

Let $f: \mathcal{F}_G \rightarrow \mathbb{R}$ be a function. Following [Reference Huang, Ye and Zhang23], we say that f is:

1. (1) monotone, if $f(E) \leq f(F)$ for any $E, F \in \mathcal{F}_G$ satisfying $E \subseteq F$;

2. (2) non-negative, if $f(F) \geq 0$ for any $F \in \mathcal{F}_G$;

3. (3) G-invariant, if $f(F g)=f(F)$ for any $F \in \mathcal{F}_G$ and $g \in G$;

4. (4) sub-additive, if $f(E \cup F) \leq f(E)+f(F)$ for any $E, F \in \mathcal{F}_G$ satisfying $E\cap F=\emptyset$.

The following convergence property is well known (see, for example, [Reference Huang, Ye and Zhang23, Lemma 2.4] or [Reference Lindenstrauss and Weiss40, Theorem 6.1]).

For a function f as in Proposition 4.7, in general, we cannot conclude that the limit of the sequence $\left\{\frac{f\left(F_n\right)}{\left|F_n\right|}: n \in \mathbb{N}\right\}$ is its infimum. This is shown by [Reference Dooley and Zhang19, Example 2.7].

Let $\mathbf{D}=\left\{d_F: F \in \mathcal{F}_G\right\}$ be a family of functions in $\mathbb{L}^{1}(Y, \mathcal{D}, \nu)$. We make some natural assumptions on the family D which will enable us to state analogues of Theorem 4.6.

We say that D is:

1. (1) non-negative if: each element from D is non-negative;

2. (2) G-invariant if: $d_{F g}(y)=d_F(g y)$ for ν-a.e. $y \in Y$ and all $F \in \mathcal{F}_G, g \in G$;

3. (3) G-bi-invariant if: it is G-invariant, and in addition $d_{g F}(y)=d_{F g}(y)$ for ν-a.e. $y \in Y$ and all $F \in \mathcal{F}_G, g \in G$;

4. (4) monotone if: $d_E(y) \leq d_F(y)$ for ν-a.e. $y \in Y$ and all $\emptyset \neq E \subseteq F \in \mathcal{F}_G$;

5. (5) sub-additive (sup-additive, respectively) if: $d_{E \cup F}(y) \leq d_E(y)+d_F(y)$ for ν-a.e. $y \in Y$ and all disjoint $E, F \in \mathcal{F}_G$ (-D is sub-additive, respectively);

6. (6) strongly sub-additive (strongly sup-additive, respectively) if $d_{E \cap F}(y)+d_{E \cup F}(y) \leq$ $d_E(y)+d_F(y)$ for ν-a.e. $y \in Y$ and all $E, F \in \mathcal{F}_G$, here by convention we set $d_{\emptyset}(y)=0$ for each $y \in Y$ (-D is strongly sub-additive, respectively).

Proposition 4.8. Let $\mathbf{D}=\left\{d_F: F \in \mathcal{F}_G\right\} \subseteq \mathbb{L}^{1}(\Omega,\mathcal{F},\mathbb{P})$ be a sub-additive G-invariant family and $\mathbb{P}$ be a G-invariant probability measure on Ω. Then, for the function

\begin{equation*} f: \mathcal{F}_G \rightarrow \mathbb{R}, F \mapsto \int_{\Omega} d_F(\omega) d \mathbb{P}(\omega), \end{equation*}

$f(E g)=f(E)$ and $f(E \cup F) \leq f(E)+f(F)$ whenever $g \in G$ and $E, F \in \mathcal{F}_G$ satisfy $E \cap F=\emptyset$. Moreover, if D is monotone then D is non-negative, and so f is a monotone non-negative sub-additive G-invariant function.

Let $\alpha \in \mathfrak{C}_{\mathcal{E}}^{o}$ and $\mathbf{D}=\left\{D(\alpha_{F}(\omega)): F \in \mathcal{F}_G\right\}$ be a family functions on $(\Omega,\mathcal{F},\mathbb{P})$. By Lemmas 4.2 and 4.3, for $\mathbb{P}$- $a.e.$  $\omega\in \Omega$ and $E,F\in \mathcal{F}_{G}$ with $E\cap F=\emptyset$,

\begin{equation*} \begin{aligned} D\left(\alpha_{E\cup F}(\omega)\right) & = D\left(\bigvee_{g\in E\cup F}\left(T_{g, \omega}\right)^{-1} \alpha\left(g \omega\right)\right) \\ & = D\left(\bigvee_{g\in E}\left(T_{g, \omega}\right)^{-1} \alpha\left(g \omega\right)\vee \bigvee_{g\in F}\left(T_{g, \omega}\right)^{-1} \alpha\left(g \omega\right)\right)\\ & \leq D\left(\bigvee_{g\in E}\left(T_{g, \omega}\right)^{-1} \alpha\left(g \omega\right)\right) + D\left(\bigvee_{g\in F}\left(T_{g, \omega}\right)^{-1} \alpha\left(g \omega\right)\right)\\ & = D\left(\alpha_{E}(\omega)\right)+D\left(\alpha_{F}(\omega)\right). \end{aligned} \end{equation*}

For $\mathbb{P}$- $a.e.$  $\omega\in\Omega$ and all $F\in \mathcal{F}_{G}$, $g\in G$,

\begin{equation*} \begin{aligned} D\left(\alpha_{Eg}(\omega)\right) & = D\left(\bigvee_{h\in Eg}\left(T_{h, \omega}\right)^{-1} \alpha\left(h \omega\right)\right) = D\left(\bigvee_{h\in E}\left(T_{hg, \omega}\right)^{-1} \alpha\left(hg \omega\right)\right) \\ &= D\left(\bigvee_{h\in E}\left(T_{g, \omega}\right)^{-1}\circ \left(T_{h, g\omega}\right)^{-1} \alpha\left(h(g \omega)\right)\right) \\ & \leq D\left(\bigvee_{h\in E}\left(T_{h, g\omega}\right)^{-1} \alpha\left(h(g \omega)\right)\right) \\ &=D\left(\alpha_{E}(g\omega)\right). \end{aligned} \end{equation*}

Similarly,

\begin{equation*} \begin{aligned} D\left(\alpha_{E}(g\omega)\right) & = D\left(\bigvee_{h\in E}\left(T_{h, g\omega}\right)^{-1} \alpha\left(h (g\omega)\right)\right) = D\left(\bigvee_{hg^{-1}\in E}\left(T_{hg^{-1}, g\omega}\right)^{-1} \alpha\left(h\omega\right)\right) \\ &= D\left(\bigvee_{hg^{-1}\in E}\left(T_{h,\omega}\circ T_{g^{-1}, g\omega}\right)^{-1} \alpha\left(h\omega\right)\right) \\ &\leq D\left(\alpha_{Eg}(\omega)\right). \end{aligned} \end{equation*}

Thus,

\begin{equation*}D(\alpha_{Eg}(\omega))=D(\alpha_{E}(g\omega)).\end{equation*}

For all $\emptyset \neq E \subseteq F \in \mathcal{F}_G$, then $D\left(\alpha_{E}(\omega)\right) \leq D\left(\alpha_{F}(\omega)\right) $ by $\alpha_{F}(\omega) \succ \alpha_{E}(\omega)$ for $\mathbb{P}$-a.e. $\omega \in \Omega$.

Therefore, $\mathbf{D}=\left\{D(\alpha_{F}(\omega)): F \in \mathcal{F}_G\right\}$ is monotone non-negative subadditive and G-invariant function.

Set

\begin{equation*} \mathcal{D}=\left\{\alpha \in \mathfrak{C}_{\mathcal{E}}^{o}: \int D(\alpha(\omega)) d \mathbb{P}(\omega) \lt \infty\right\}. \end{equation*}

By Corollary 3.5, $\mathrm{C}_{\mathcal{E}}^{o} \subset \mathcal{D}$. Note also that by Lemma 4.3, if $\alpha, \beta \in \mathcal{D}$, then $\alpha \vee \beta \in \mathcal{D}$. Thus $\mathcal{D}$ is a directed set.

Suppose $\alpha \in \mathcal{D}$, for any $F\in\mathcal{F}_{G}$, let $F=\{g_1,g_2,...,g_n\}$.

\begin{equation*} \begin{aligned} D\left(\alpha_{F}(\omega)\right) & = D\left(\bigvee_{g\in F}\left(T_{g, \omega}\right)^{-1} \alpha\left(g \omega\right)\right) \\ &= D\left(\left(T_{g_1, \omega}\right)^{-1}\alpha\left(g_1\omega\right)\vee \left(T_{g_2, \omega}\right)^{-1}\alpha\left(g_2\omega\right)\vee \cdots \vee \left(T_{g_n, \omega}\right)^{-1}\alpha\left(g_n\omega\right)\right) \\ & \leq D\left(\alpha(g_1\omega)\right)+D\left(\alpha(g_2\omega)\right)+\cdots+D\left(\alpha(g_n\omega)\right). \end{aligned} \end{equation*}

Since for $i=1,2,...,n$,

\begin{equation*} \int D\left(\alpha(g_i\omega)\right)d \mathbb{P}(\omega)=\int D(\alpha(\omega))d\mathbb{P}(g_i^{-1}\omega)=\int D\left(\alpha(\omega)\right)d \mathbb{P}(\omega), \end{equation*}

that is,

\begin{equation*} \int D\left(\alpha_{F}(\omega)\right)d\mathbb{P}(\omega) \leq n\int D(\alpha(\omega))d\mathbb{P}(\omega). \end{equation*}

Then $ D\left(\alpha_{F}(\omega)\right)$ is integrable for every $F\in \mathcal{F}_{G}$.

Now write $D(\alpha, T, G)$ for the $\mathbb{P}$-almost surely (a.s.) limit of $ \frac{1}{\left|F_n\right|} \int_{\Omega} D\left(\alpha_{F_{n}}(\omega)\right) d \mathbb{P}(\omega).$ Then, by Propositions 4.7 and 4.8,

\begin{equation*} D(\alpha, T, G)=\lim _{n \rightarrow \infty}\frac{1}{\left|F_n\right|} \int_{\Omega} D\left(\alpha_{F_{n}}(\omega)\right) d \mathbb{P}(\omega), \end{equation*}

for any Fø lner sequence $\left\{F_n: n \in \mathbb{N}\right\}$ of G and the value of the limit is independent of the choice of the Fø lner sequence $\left\{F_n: n \in \mathbb{N}\right\}$.

Proposition 4.11. Let $\beta \in \mathcal{D}$ with an increasing sequence $\left(\Omega_{r}\right)_{r \geq 1} \subset \mathcal{F}$ satisfying $\mathbb{P}\left(\Omega_{r}\right) \rightarrow 1$, where each $\beta \cap\left(\Omega_{r} \times X\right), r \geq 1$, is a finite family.

Let

\begin{equation*} \beta^{r}=\left[\beta \cap\left(\Omega_{r} \times X\right)\right] \bigcup\left[\left(\Omega_{r}^{c} \times X\right) \cap \mathcal{E}\right]\in \mathrm{C}_{\mathcal{E}}^{o}, \end{equation*}

then

\begin{equation*} D\left(\beta^{r}, T, G\right) \rightarrow D(\beta, T, G). \end{equation*}

Proof. It is not hard to check that $D\left(\beta^{r}, T, G\right)$ is increasing in r. In fact, if $r_1 \lt r_2$, by the definition of β ^r,

\begin{equation*} \beta^{r_1}(\omega)= \begin{cases} \beta(\omega), & \text{if } \omega\in \Omega_{r_1}; \\ \mathcal{E}_{\omega}, & \text{if } \omega\in \Omega_{r_2}\backslash \Omega_{r_1}; \\ \mathcal{E}_{\omega}, & \text{if } \omega\notin \Omega_{r_2} \end{cases} \quad \text{and} \quad \beta^{r_2}(\omega)= \begin{cases} \beta(\omega), & \text{if } \omega\in \Omega_{r_1}; \\ \beta(\omega), & \text{if } \omega\in \Omega_{r_2}\backslash \Omega_{r_1}; \\ \mathcal{E}_{\omega}, & \text{if } \omega\notin \Omega_{r_2}, \end{cases} \end{equation*}

we have: for $\mathbb{P}$-a.e. $\omega\in \Omega$, $\beta^{r_2}(\omega)\succ\beta^{r_1}(\omega)$.

Recall Remark 4.10, so

\begin{equation*} D\left(\beta^{r_1}, T, G\right)\leq D\left(\beta^{r_2}, T, G\right). \end{equation*}

And that each member is pointwise less than $D(\beta, T, \omega)$ in ω by Lemma 4.1. It follows that

(4.1)\begin{equation} \lim _{r \rightarrow \infty} D\left(\beta^{r}, T, G\right) \leq D(\beta, T, G). \end{equation}

In the other direction, for any Fø lner sequences $F_{n}=\left\{g_1,g_2,...,g_{k_n}\right\}$, set

\begin{equation*} \begin{aligned} & I_{r, F_n}(\omega)=\left\{1 \leq i\leq k_n: g_i \omega \in \Omega_{r}\right\}, \\ & I^{r, F_n}(\omega)=\left\{1 \leq i\leq k_n: g_i \omega \notin \Omega_{r}\right\} \end{aligned} \end{equation*}

and note that, by Lemmas 4.1 and 4.3,

\begin{equation*} \begin{aligned} D\left(\beta_{F_{n}}(\omega)\right) & = D\left(\bigvee_{i=1}^{k_n}\left(T_{g_i,\omega}\right)^{-1} \beta\left(g_i \omega\right)\right) \\ & \leq D\left(\bigvee_{i \in I_{r, F_n}(\omega)}\left(T_{g_i,\omega}\right)^{-1} \beta\left(g_i \omega\right)\right)+D\left(\bigvee_{i \in I^{r, F_n}(\omega)}\left(T_{g_i,\omega}\right)^{-1} \beta\left(g_i \omega\right)\right) \\ & =D\left(\beta^{r}_{F_n}(\omega)\right)+D\left(\bigvee_{i \in I^{r, F_n}(\omega)}\left(T_{g_i,\omega}\right)^{-1} \beta\left(g_i \omega\right)\right) \\ & \leq D\left(\beta^{r}_{F_n}(\omega)\right)+\sum_{i \in I^{r, F_n}(\omega)} D\left(\beta\left(g_i \omega\right)\right). \end{aligned} \end{equation*}

It follows that

\begin{equation*} \begin{aligned} D(\beta, T, G) & =\lim _{n \rightarrow \infty}\frac{1}{\left|F_n\right|} \int_{\Omega} D\left(\beta_{F_{n}}(\omega)\right) d \mathbb{P}(\omega)\\ & \leq \lim _{n \rightarrow \infty}\frac{1}{\left|F_n\right|} \int_{\Omega} D\left(\beta_{F_{n}}^r(\omega)\right) d \mathbb{P}(\omega)\\ &\quad +\limsup _{n \rightarrow \infty} \frac{1}{\left|F_n\right|} \int \sum_{i=1}^{k_n} 1_{\Omega \backslash \Omega_{r}}\left(g_i \omega\right) D\left(\beta\left(g_i \omega\right)\right) d \mathbb{P}(\omega) \\ & \leq D(\beta^r, T, G)+\int 1_{\Omega \backslash \Omega_{r}}(\omega) D(\beta(\omega)) d \mathbb{P}(\omega). \end{aligned} \end{equation*}

By the assumption that $\mathbb{P}\left(\Omega_{r}\right) \rightarrow 1$ and $\beta \in \mathcal{D}$, according to Lebesgue’s dominated convergence theorem, we have

\begin{equation*} \lim _{r \rightarrow \infty} \int 1_{\Omega \backslash \Omega_{r}}(\omega) D(\beta(\omega)) d \mathbb{P}(\omega)=0 \end{equation*}

and hence

(4.2)\begin{equation} D(\beta, T, G) \leq \lim _{r \rightarrow \infty} D(\beta^r, T, G). \end{equation}

Combining (4.1) with (4.2) we complete the proof.

Proposition 4.12. For every $\beta \in \mathcal{D}$ and ϵ > 0, there exists a finite open cover $\mathcal{V}$ of X such that

\begin{equation*} D(\mathcal{V}, T, G) \geq D(\beta, T, G) -\epsilon, \end{equation*}

where

\begin{equation*} D(\mathcal{V}, T, G)=\lim _{n \rightarrow \infty}\frac{1}{\left|F_n\right|} \int_{\Omega} D\left(\mathcal{V}_{F_{n}}(\omega)\right) d \mathbb{P}(\omega) \end{equation*}

with $\mathcal{V}(\omega)=\mathcal{V} \cap \mathcal{E}_{\omega}$ for any Fø lner sequence $\{F_{n}:n\in\mathbb{N}\}$ of G.

Proof. Recall that the Lebesgue number $\lambda(\beta(\omega))$ of an open cover $\beta(\omega)$ on the fibre $\mathcal{E}_{\omega}$ is the largest λ such that, for every $x \in \mathcal{E}_{\omega}$, there is a $B(\omega) \in \beta(\omega)$ such that $d\left(x, \mathcal{E}_{\omega} \backslash B(\omega)\right) \geq$ λ. Suppose that $\beta=\left\{B^{(1)}, B^{(2)}, \ldots, B^{\left(l_{q}\right)}\right\}$ on $\Omega_{q}$, where $\left(\Omega_{q}\right)_{q \geq 1} \subset \mathcal{F}$ is the family of increasing sets in the definition of $\mathfrak{C}_{\mathcal{E}}^{o}$. When $\omega\in \Omega_q$, $\beta(\omega)\in \mathrm{C}_{\mathcal{E}_{\omega}}^o$ and its Lebesgue number is

\begin{equation*} \lambda(\beta(\omega))=\frac{1}{l_{q}} \min _{x \in \mathcal{E}_{\omega}} \sum_{i=1}^{l_{q}} d\left(x, \mathcal{E}_{\omega} \backslash B^{(i)}(\omega)\right) \quad \text {on } \Omega_{q}. \end{equation*}

Set $d(x, \emptyset)=\infty$, for convenience.

Note that $(\omega, x) \mapsto d\left(x, \mathcal{E}_{\omega} \backslash B^{(i)}(\omega)\right)$ is a Carath $\acute{e}$odory function and hence $\mathcal{F} \otimes \mathcal{B}(X)$ measurable [Reference Castaing and Valadier11, Lemma III.14]. By Lemma III.39 from [Reference Castaing and Valadier11], $\lambda(\beta(\omega))$ is measurable.

By the compactness of $\mathcal{E}_{\omega}$, we see that $\lambda(\beta(\omega)) \gt 0$ for all ω. Set

\begin{equation*} \Omega_{p, q}=\left\{\omega \in \Omega_{q}: \lambda(\beta(\omega)) \geq \frac{1}{p}\right\}. \end{equation*}

Then $\bigcup_{p=1}^{\infty} \Omega_{p, q}=\Omega_{q}$. Let $\Omega_{0, q}=\emptyset, \widetilde{\Omega}_{p, q}=\Omega_{p, q} \backslash \Omega_{p-1, q}$ and $\widetilde{\Omega}_{p}=\bigcup_{q \geq 1} \widetilde{\Omega}_{p, q}$.

Now we can choose a maximal set $S_{p} \subset X$ of n_p points in X such that $d(x, y) \geq 1/(4p)$ for all $x, y \in S_{p}, x \neq y$, and let $\mathcal{U}_{p}$ be the cover of X by open balls of radius $1/(2p)$ centred at points of S_p. In fact, by the maximum of S_p, we have: for any $x\in X$, there exists a $s\in S_{p}$ such that $d(x,s) \lt 1/(4p) \lt 1/(2p)$. Define an open random cover α by

\begin{equation*} \alpha=\left\{\left(\widetilde{\Omega}_{p} \times U\right) \cap \mathcal{E}: U \in \mathcal{U}_{p}\right\} \quad \text {on } \widetilde{\Omega}_{p} \subset \Omega. \end{equation*}

Then, for $\omega \in \widetilde{\Omega}_{p}$,

\begin{equation*} \operatorname{diam} \alpha(\omega) \leq \operatorname{diam} \mathcal{U}_{p} \lt \frac{1}{p} \leq \lambda(\beta(\omega)), \end{equation*}

which implies that $\alpha(\omega) \succ$ $\beta(\omega)$. Now $\lambda(\alpha(\omega)) \geq 1/(2p)$ by the construction of S_p. To see this, we only need check that for any $x\in \mathcal{E}_{\omega}$, there is a $A(\omega) \in \alpha(\omega)$ such that $d\left(x, \mathcal{E}_{\omega} \backslash A(\omega)\right) \geq$ $1/(2p)$. By the maximum of S_p, for any $x\in \mathcal{E}_{\omega}$, there exists a $s_x\in S_{p}$, such that $d(x,s_x) \lt 1/(4p)$, that is $x\in B\left(s_x,1/(4p)\right)\subset B\left(s_x,1/(2p)\right)\in \alpha(\omega).$ Then $\lambda\left(\alpha(\omega)\right)\geq 1/(2p).$

Set $\Omega_{r}=\bigcup_{p=1}^{r} \widetilde{\Omega}_{p}$ and let $\alpha^{r}=\left[\alpha \cap\left(\Omega_{r} \times X\right)\right] \bigcup\left[\left(\Omega_{r}^{c} \times X\right) \cap \mathcal{E}\right]$. It is not hard to check that the sequence of non-negative random variables $D\left(\alpha^{r}_{F_{n}}(\omega)\right)$ increases to $D\left(\alpha_{F_{n}}(\omega)\right)$ for every Fø lner sequence $\{F_{n}: n\in \mathbb{N}\}$, $\mathbb{P}$-a.a. ω, as $r \rightarrow \infty$. Again, choose a sequence $\left(\mathcal{V}^{r}\right)_{r \geq 1}$ of finite open covers of X such that diam $\mathcal{V}^{r} \lt 1/(4r)$. Recall that $\mathcal{V}^{r}(\omega)=\mathcal{V}^{r} \cap \mathcal{E}_{\omega}$, so $\mathcal{V}^{r}(\omega) \succ \alpha^{r}(\omega)$ for $\mathbb{P}$-a.a. ω. In fact, if $\omega\in \Omega_r=\bigcup_{p=1}^{r}\widetilde{\Omega}_{p}$, then $\lambda\left(\alpha(\omega)\right)\geq1/(2p) \geq 1/(2r).$ By the definition of $\alpha^r(\omega)$, for $\omega\in \Omega_r$, we have $\alpha^r(\omega)=\alpha(\omega),$ then $\mathcal{V}^{r}(\omega) \succ \alpha^{r}(\omega)$ for $\omega\in \Omega_r$. If $\omega\in \Omega_r^{c}$, then $\alpha^r(\omega)=\left\{\mathcal{E}_{\omega}\right\}.$ so, $\mathcal{V}^{r}(\omega) \succ \alpha^{r}(\omega)$ for $\mathbb{P}$-a.a. ω. Hence,

\begin{equation*} D\left(\mathcal{V}^{r}_{F_{n}}(\omega)\right) \geq D\left(\alpha^{r}_{F_{n}}(\omega)\right) \rightarrow D\left(\alpha_{F_{n}}(\omega)\right) \geq D\left(\beta_{F_{n}}(\omega)\right). \end{equation*}

It follows from Proposition 4.11 that

\begin{equation*} \begin{aligned} D\left(\mathcal{V}^{r}, T, G\right) \geq D\left(\alpha^{r}, T, G\right) & \rightarrow D(\alpha, T, G) \geq D(\beta, T, G). \end{aligned} \end{equation*}

Hence, for a sufficiently large value of $r, \mathcal{V}^{r}$ will satisfy our requirements.

Definition 4.13. The mean topological dimension of a bundle RDS T on $\mathcal{E}$ for an amenable group G, denoted by $\mathbb{E}\mathrm{midm}(T,G)$, is defined by

\begin{equation*} \mathbb{E}\mathrm{mdim}(T,G)=\mathbb{E} \operatorname{mdim}(\mathcal{E}, T,G)=\sup _{\alpha} D(\alpha, T, G) \end{equation*}

where α runs over all countable random covers in $\mathcal{D}$.

Proof. Suppose T and S are isomorphic via ψ_ω and let $\Psi: \mathcal{E} \rightarrow \mathcal{G}$ be given by $\Psi(\omega, x)=\left(\omega, \psi_{\omega} x\right)$. Let $\alpha \in \mathrm{C}_{\mathcal{E}}^{o}$. Then $\beta=\Psi(\alpha) \in \mathrm{C}_{\mathcal{H}}^{o}$. As $S_{g,\omega} \circ \psi_{\omega}=\psi_{g \omega} \circ T_{g,\omega}$, the homeomorphism ψ_ω sends $D\left(\alpha_{F_{n}}(\omega)\right)$ to $D\left(\beta_{F_{n}}(\omega)\right)$ for any Fø lner sequence $\{F_{n}:n\in \mathbb{N}\}$ of G. In fact,

\begin{equation*} \begin{aligned} D\left(\alpha_{F_{n}}(\omega)\right) & = D\left(\bigvee_{g\in F_{n}}\left(T_{g, \omega}\right)^{-1} \alpha\left(g \omega\right)\right) = D\left(\bigvee_{g\in F_{n}}\left(T_{g, \omega}\right)^{-1} \circ \left(\psi_{g\omega}\right)^{-1} \beta\left(g \omega\right)\right) \\ &= D\left(\bigvee_{g\in F_{n}}\left(\psi_{\omega}\right)^{-1}\circ \left(S_{g,\omega}\right)^{-1} \beta\left(g \omega\right)\right) \\ & \leq D\left(\bigvee_{g\in F_{n}}\left(S_{g,\omega}\right)^{-1} \beta\left(g \omega\right)\right) = D\left(\beta_{F_{n}}(\omega)\right). \end{aligned} \end{equation*}

Similarly,

\begin{equation*} \begin{aligned} D\left(\beta_{F_{n}}(\omega)\right) & = D\left(\bigvee_{g\in F_{n}}\left(S_{g, \omega}\right)^{-1} \beta\left(g \omega\right)\right) = D\left(\bigvee_{g\in F_{n}}S_{g^{-1}, g\omega} \circ \psi_{g\omega} \alpha\left(g \omega\right)\right) \\ &= D\left(\bigvee_{g\in F_{n}}\psi_{\omega} \circ T_{g^{-1},g\omega} \alpha\left(g \omega\right)\right) \\ & \leq D\left(\bigvee_{g\in F_{n}}\left(T_{g,\omega}\right)^{-1} \alpha\left(g \omega\right)\right) = D\left(\alpha_{F_{n}}(\omega)\right). \end{aligned} \end{equation*}

It follows that $D\left(\alpha_{F_{n}}(\omega)\right)=$ $D\left(\beta_{F_{n}}(\omega)\right)$ and hence $D(\alpha, T, G)=D(\beta, S,G)$. Since $\alpha \mapsto \Psi(\alpha)$ provides a bijective correspondence between the finite open random covers of $\mathcal{E}$ and those of $\mathcal{H}$, by Remark 4.15 we deduce that $\mathbb{E} \operatorname{mdim}(T,G)=\mathbb{E} \operatorname{mdim}(S,G)$.

Proof. Let $\alpha\in \mathfrak{C}_{\mathcal{E}}^o.$ Then for any Fø lner sequence of infinite amenable group F_n,

\begin{equation*} D(\alpha_{F_{n}}(\omega))\leq \mathrm{dim} \mathcal{E}_{\omega}\leq \mathrm{dim} X \lt \infty \end{equation*}

and

\begin{equation*} D(\alpha,T,G)=\lim _{n \rightarrow \infty}\frac{1}{\left|F_n\right|} \int_{\Omega} D\left(\alpha_{F_{n}}(\omega)\right) d \mathbb{P}(\omega). \end{equation*}

Since $\left|F_{n}\right|\rightarrow \infty$ as $n\rightarrow \infty$ for any Fø lner sequence of infinite amenable group, then $\mathbb{E}\mathrm{mdim}(T,G)=0.$

Proposition 4.18. If a closed random set $\mathcal{C} \subset \mathcal{E}$ is forward invariant, i.e. for any $g\in G$

\begin{equation*} \mathcal{C}_\omega \subset\left(T_{g,\omega}\right)^{-1} \mathcal{C}_{g \omega} \quad \mathbb{P} \text {-a.a. } \omega, \end{equation*}

then $\operatorname{Emdim}\left(\mathcal{C},\left.T\right|_{\mathcal{C}},G\right) \leq \mathbb{E} \operatorname{mdim}(\mathcal{E}, T,G)$.

Proof. Since $\mathcal{C}$ is forward invariant, $\left(\mathcal{C},\left.T\right|_{\mathcal{C}}\right)$ is a continuous bundle RDS. For every $\alpha=$ $\left\{A^{(i)}: i \in I\right\} \in \mathfrak{C}_{\mathcal{C}}^o$, let $B^{(i)}=\mathcal{E} \backslash\left(\mathcal{C} \backslash A^{(i)}\right)$. Then $\beta=\left\{B^{(i)}: i \in I\right\} \in \mathfrak{C}_{\mathcal{E}}^o$. We claim that $D\left(\alpha_{F_{n}}(\omega)\right) \leq D\left(\beta_{F_{n}}(\omega)\right)$. It suffices to prove that, for every Fø lner sequence and $\mathbb{P}$-a.a. ω, if $\gamma \succ$ $\beta_{F_{n}}(\omega)$ is an open cover of $\mathcal{E}_\omega$, then there exists an open cover $\gamma^{\prime} \succ \alpha_{F_{n}}(\omega)$ of $\mathcal{C}_\omega$ such that $\operatorname{ord}\left(\gamma^{\prime}\right) \leq \operatorname{ord}(\gamma)$. In other words, it suffices to prove that

\begin{equation*} \sum_{C \in \gamma^{\prime}} 1_C(x) \leq \sum_{U \in \gamma} 1_U(x), \quad x \in \mathcal{C}_\omega \subset \mathcal{E}_\omega . \end{equation*}

Indeed, we can take $\gamma^{\prime}=\gamma \cap \mathcal{C}_\omega$.

5. Metric mean dimension

For each $F\in \mathcal{F}_G $ and a positive ϵ > 0, define a family of metrics $d_{F}^{\omega}$ on $\mathcal{E}_{\omega}$ by the formula

\begin{equation*} d_{F}^{\omega}(x, y)=\max _{g\in F} d\left(T_{g,\omega} x, T_{g,\omega} y\right), \quad x, y \in \mathcal{E}_{\omega} . \end{equation*}

Set

\begin{equation*} B_{d_{F}^{\omega}}(x,\epsilon)=\left\{y\in \mathcal{E}_{\omega}: d_{F}^{\omega}(x, y) \lt \epsilon\right\}. \end{equation*}

Fix ω and let $\alpha=\left\{A_{i}: i=1,2, \ldots, l\right\}$ be a finite open cover of $\mathcal{E}_{\omega}$. Define the mesh of α with respect to the metric $d_{F}^{\omega}$ by

\begin{equation*} \operatorname{mesh}\left(\alpha, d_{F}^{\omega}\right)=\max _{1 \leq i \leq l} \operatorname{diam}\left(A_{i}, d_{F}^{\omega}\right), \end{equation*}

where

\begin{equation*} \operatorname{diam}\left(A_{i}, d_{F}^{\omega}\right)=\sup \left\{d_{F}^{\omega}(x, y): x, y \in A_{i}\right\}. \end{equation*}

Let $\operatorname{cov}(\omega, \epsilon, F)$ be the minimal cardinality of finite open covers of $\mathcal{E}_{\omega}$ by sets of $d_{F}^{\omega}$ diameter less than ϵ, that is,

\begin{equation*} \operatorname{cov}(\omega, \epsilon, F)=\inf \left\{|\alpha|: \alpha \in \mathrm{C}_{\mathcal{E}_{\omega}}^{o}, \operatorname{mesh}\left(\alpha, d_{F}^{\omega}\right) \lt \epsilon\right\}. \end{equation*}

We define

\begin{equation*} \operatorname{sep}(\omega, \epsilon, F)=\sup \left\{|H|: H \subset \mathcal{E}_{\omega} \text {is an }(\omega, \epsilon, F) \text {-separated set}\right\} \end{equation*}

as the maximum cardinality of $(\omega, \epsilon, F)$-separated sets of $\mathcal{E}_{\omega}$.

We define

\begin{equation*} \operatorname{span}(\omega, \epsilon, F)=\inf \left\{|H|: H \subset \mathcal{E}_{\omega} \text { is an }(\omega, \epsilon, F) \text {-spanning set}\right\} \end{equation*}

as the minimum cardinality of $(\omega, \epsilon, F)$-spanning sets of $\mathcal{E}_{\omega}$.

As in the deterministic case [Reference Brin and Stuck10, Lemma 2.5.1], we have the following facts.

Remark 5.3.

(5.1)\begin{equation} \operatorname{cov}(\omega, 2 \epsilon, F)\leq \operatorname{span}(\omega, \epsilon, F) \leq \operatorname{sep}(\omega, \epsilon, F) \leq \operatorname{cov}(\omega, \epsilon, F). \end{equation}

Proof. Let H be a spanning set of minimal cardinality, then

\begin{equation*} \mathcal{E}_{\omega}\subset \bigcup_{h\in H} \bar{B}_{d_{F}^{\omega}}(h,\epsilon). \end{equation*}

By compactness, then there exists $\epsilon_{1} \lt \epsilon$, such that, the finite balls of radius ϵ ₁ centred at the points of H over $\mathcal{E}_{\omega}$. Hence

\begin{equation*} \operatorname{cov}(\omega, 2 \epsilon, F)\leq \operatorname{span}(\omega, \epsilon, F). \end{equation*}

Let H be a separated set of maximal cardinality, then for any $x\in \mathcal{E}_{\omega}$, $H\bigcup \{x\} $ is not a $(\omega, \epsilon, F)$- separated set. That is, there exists $z\in H$ such that $d_{F}^{\omega}(x,z)\leq \epsilon$, then H be a $(\omega,\epsilon,F)$-spanning set. Hence,

\begin{equation*} \operatorname{span}(\omega, \epsilon, F) \leq \operatorname{sep}(\omega, \epsilon, F). \end{equation*}

Suppose $\alpha \in \mathrm{C}_{\mathcal{E}_{\omega}}^o$ of minimal cardinality with mesh $(\alpha, d_{F}^{\omega}) \lt \epsilon.$ Suppose $H\subset \mathcal{E}_{\omega}$ to be an $(\omega,\epsilon, F)$-separated set of maximal cardinality. If $|H| \gt |\alpha|$, then by the drawer principle, H must have at least two elements in α. It’s a contradiction.

Hence,

\begin{equation*} \operatorname{sep}(\omega, \epsilon, F) \leq \operatorname{cov}(\omega, \epsilon, F). \end{equation*}

Moreover, let $\alpha,\gamma \in \mathrm{C}_{\mathcal{E}}^o$, for any $\omega \in \Omega$, denote by $N(\alpha(\omega))$ the minimal cardinality of a subfamily of $\alpha(\omega)$ covering $\mathcal{E}_\omega$, equivalently,

\begin{equation*} N(\alpha(\omega))=\min_{\beta\subset\alpha(\omega)}\left\{|\beta|:\beta\in \mathrm{C}_{\mathcal{E}_{\omega}}^o\right\}. \end{equation*}

Let $\lambda(\omega)$ be the Lebesgue number of $\alpha(\omega)$, and assume that $\operatorname{mesh}(\gamma(\omega), d_F^{\omega})=\delta(\omega)$. In case similar to deterministic systems [Reference Walters54, Theorem 7.7], for $F\in \mathcal{F}_{G}$, we have the following inequality.

Remark 5.4.

\begin{equation*} \begin{gathered} N\left(\alpha_{F}(\omega)\right) \leq \operatorname{span}(\omega, \lambda / 2, {F}) \leq \operatorname{sep}(\omega, \lambda / 2, {F}) . \\ \operatorname{span}(\omega, \delta, {F}) \leq \operatorname{sep}(\omega, \delta, {F}) \leq N\left(\gamma_{F}(\omega)\right). \end{gathered} \end{equation*}

Proof. In fact, if A is an $(\omega, \lambda / 2, F)$-spanning set with minimal cardinality, then $\mathcal{E}_\omega \subset \bigcup_{a \in A} \bar{B}_{d_{F}^{\omega}}\left(a, \lambda / 2\right)$. And let $F=\{g_1,g_2,...,g_n\}$, then

\begin{equation*} \bar{B}_{d_{F}^{\omega}}\left(a, \lambda / 2\right)=\bigcap_{k=1}^{n}\left\{x: d\left(T_{g_k,\omega} x, T_{g_k,\omega} a\right) \leq \frac{1}{2} \lambda\right\}, \end{equation*}

and

\begin{equation*} \operatorname{diam}\bar{B}_{d_{F}^{\omega}}\left(a, \lambda / 2\right) \leq \lambda. \end{equation*}

Therefore, $\bar{B}_{d_{F}^{\omega}}\left(a, \lambda / 2\right)$ must lie in some element of $\alpha_F(\omega)$. Hence $N\left(\alpha_F(\omega)\right) \leq \operatorname{span}(\omega, \lambda / 2, F)$. The other side has been previously proven.

For the second inequality, it is only necessary to prove the inequality on the right-hand side. Let H be an $(\omega, \delta, F)$-separated set with maximal cardinality.

If $|H| \gt N\left(\gamma_F(\omega)\right)$, then by the drawer principle, at least two points in H fall into the same element of $\gamma_F(\omega)$. This contradicts our assumption about H.

Referring to [Reference Dooley and Zhang19, Proposition 5.4], it is not difficult to prove that $\omega \mapsto N\left(\alpha_F(\omega)\right)$ is measurable. Furthermore, $\left\{\text{log} N\left(\alpha_F(\omega)\right): F\in \mathcal{F_{G}}\right\}$ be a monotone sub-additive G-invariant family in $L^1(\Omega,\mathcal{F},\mathbb{P})$. Thus by Propositions 4.7 and 4.8, we have the following definitions.

Definition 5.5. Let $\omega\in \Omega$, and $F\in \mathcal{F}_{G}$, one has the fibre topological entropy of T with respect to α and the fibre topological entropy of T, respectively, by

\begin{equation*} h_{top}(T,\alpha,G)=\lim_{n\rightarrow \infty} \frac{1}{|F_{n}|}\int_{\Omega} logN(\alpha_{F_n}(\omega))d\mathbb{P}(\omega) \end{equation*}

and

\begin{equation*} h_{top}(T,G)=\sup_{\gamma \in \mathrm{C}_{\mathcal{E}}^o} h_{top}(T,\gamma,G). \end{equation*}

Remark that the values of these invariants are all independent of the selection of the Fø lner sequence $\{F_{n}: n\in\mathbb{N}\}$.

Notice that for any $F\in \mathcal{F}_{G}$ and a positive random variable $\epsilon=\epsilon(\omega)$ the function $\operatorname{sep}(\omega, \epsilon, F)$ is measurable in ω (even for random $\epsilon(\omega)$) [Reference Kifer32, Lemma 1.2].

Proof. For $q, n \in \mathbb{N}_{+}$, set

\begin{equation*} \begin{aligned} D_q & =\left\{\left(\omega, x_1, \ldots, x_q\right): \omega \in \Omega, x_i \in \mathcal{E}_\omega, \forall i\right\}, \\ E_q^{F} & =\left\{\left(\omega, x_1, \ldots, x_q\right) \in D_q: d_{{F}}^\omega\left(x_i, x_j\right) \gt \varepsilon, \forall i \neq j\right\}, \\ E_q^{{F}, k} & =\left\{\left(\omega, x_1, \ldots, x_q\right) \in D_q: d_{{F}}^\omega\left(x_i, x_j\right) \geqslant \varepsilon(1+k^{-1}), \forall i \neq j\right\}, \\ E_q^{F}(\omega) & =\left\{\left(x_1, \ldots, x_q\right):\left(\omega, x_1, \ldots, x_q\right) \in E_q^{F}\right\}, \\ E_q^{{F}, k}(\omega) & =\left\{\left(x_1, \ldots, x_q\right):\left(\omega, x_1, \ldots, x_q\right) \in E_q^{{F}, k}\right\} . \end{aligned} \end{equation*}

Observe that $D_q \in \mathcal{F} \times \left(\mathcal{B}(X)\right)^q$, where $\left(\mathcal{B}(X)\right)^q$ is the product σ-algebra on the product X^q of q copies of X. This follows from Theorem III.30 in [Reference Castaing and Valadier11] since

\begin{equation*} d_q\left(\left(x_1, \ldots, x_q\right),\left(y_1, \ldots, y_q\right)\right)=\sum_{i=1}^q d\left(x_i, y_i\right) \end{equation*}

is the distance function on X^q; and if $\mathcal{E}_\omega^q$ denotes the product of q copies of $\mathcal{E}_\omega$, then

\begin{equation*} d_q\left(\left(x_1, \ldots, x_q\right), \mathcal{E}_\omega^q\right)=\sum_{i=1}^q d\left(x_i, \mathcal{E}_\omega\right) \end{equation*}

is measurable in ω for each $\left(x_1, \ldots, x_q\right) \in X^q$. Next, define $q(q-1) / 2$ measurable functions $\psi_{i j}, 1 \leqslant i \lt j \leqslant q$, on D_q by $\psi_{i j}\left(\omega, x_1, \ldots, x_q\right)=d_{{F}}^\omega\left(x_i, x_j\right)$. Then

\begin{equation*} E_q^{F, k}=\bigcap_{1 \leqslant i \lt j \leqslant q} \psi_{i j}^{-1}\left[\varepsilon(1+k^{-1}), \infty\right) \in \mathcal{F} \times \left(\mathcal{B}(X)\right)^q . \end{equation*}

By continuity of the RDS T, each $E_q^{F, k}(\omega)$ is a closed subset of $\mathcal{E}_\omega^q$, and so it is compact. Clearly, $E_q^{F, k} \uparrow E_q^{F}$ and $E_q^{F, k}(\omega) \uparrow E_q^{F}(\omega)$ as $k \uparrow \infty$, in particular, $E_q^{F} \in \mathcal{F} \times \left(\mathcal{B}(X)\right)^q$.

Let, again, $\operatorname{sep}(\omega, \epsilon, F)$ denote the largest cardinality of an $(\omega, \varepsilon, F)$-separated set in $\mathcal{E}_\omega$ and set $t_{F, k}(\omega, \varepsilon)=\max \left\{q: E_q^{{F}, k}(\omega) \neq \emptyset\right\}$. By Theorem III.23 in [Reference Castaing and Valadier11], it follows that

\begin{equation*} \left\{\omega: t_{F, k}(\omega, \varepsilon) \geqslant q\right\}=\left\{\omega: E_q^{F, k}(\omega) \neq \emptyset\right\}=\operatorname{Pr}_{\Omega} E_q^{F, k} \in \mathcal{F}. \end{equation*}

Set

\begin{equation*} S(\omega,\epsilon,G)=\liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|} \log \operatorname{sep}(\omega, \epsilon, F_{n}), \end{equation*}

for any Fø lner sequence $\{F_{n}:n\in\mathbb{N}\}$ of G.

Obviously, S is monotone non-decreasing as non-random $\epsilon \rightarrow 0$, and, for the purpose of measuring how fast it increases, we define the lower metric mean dimension of T on the fibre $\mathcal{E}_{\omega}$ as

\begin{equation*} \underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}}(T,G,\omega) =\liminf _{\epsilon \rightarrow 0} \frac{1}{|\log \epsilon|}S(\omega, \epsilon, G). \end{equation*}

It follows from Remark 5.3 and the fact that $\lim _{\epsilon \rightarrow 0}(\log \epsilon / \log 2 \epsilon)=1$ that

\begin{equation*} \underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}}(T,G,\omega) =\liminf _{\epsilon \rightarrow 0} \frac{1}{|\log \epsilon|} \liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|} \log \operatorname{cov}(\omega, \epsilon, F_{n}), \end{equation*}

for any Fø lner sequence $\{F_{n}:n\in\mathbb{N}\}$ of G.

To satisfy the measurability of the function, we define a slightly smaller-scale lower metric mean dimension. Set

\begin{align*} \mathbb{S}&=\left\{\left\{\epsilon_m\right\}_{m=1}^{+\infty}: \left\{\epsilon_m\right\}\ \text{is a sequence of positive numbers that is monotonically }\notag \right. \\ &\quad \times \left. \text{decreasing with} \lim_{m\rightarrow \infty} \epsilon_m=0 \right\}. \end{align*}

We define the $\mathbb{S}$-lower metric mean dimension of T on the fibre $\mathcal{E}_{\omega}$ as

\begin{equation*} \underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,G,\omega) =\liminf _{m \rightarrow \infty} \frac{1}{|\log \epsilon_m|}S(\omega, \epsilon_m, G). \end{equation*}

Since $\operatorname{sep}(\omega, \epsilon_m, F_{n})$ is measurable, it follows that $\underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,G,\omega)$ is measurable as well. We have the following definition.

Definition 5.7. The $\mathbb{S}$-lower metric mean topological dimension of bundle RDS T for a countable amenable group G, denoted by $\underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,d,G)$, is defined by

\begin{equation*} \underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,d,G)=\int \underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,G,\omega)d\mathbb{P}(\omega). \end{equation*}

Let $\alpha,\gamma \in \mathrm{C}_{\mathcal{E}}^o$. Let $2\epsilon(\omega)$ be the Lebesgue number of $\alpha(\omega)$, and assume that $\operatorname{mesh}(\gamma(\omega), d_F^{\omega})=\epsilon(\omega)$, for any $\omega \in \Omega$. Then by Remark 5.4,

\begin{equation*} N\left(\alpha^{\epsilon}_{F}(\omega)\right) \leq \operatorname{span}(\omega, \epsilon, {F}) \leq \operatorname{sep}(\omega, \epsilon, {F}) \leq N\left(\gamma^{\epsilon}_{F}(\omega)\right). \end{equation*}

Hence,

\begin{equation*} h_{\mathrm{top}}(T,G)= \lim _{\epsilon \rightarrow 0} \liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|}\int \log \operatorname{sep}(\omega, \epsilon, F_{n}) d \mathbb{P}(\omega). \end{equation*}

Compare

\begin{equation*} \underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,d,G)=\int \liminf _{m\rightarrow \infty} \frac{1}{|\log \epsilon_m|} \liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|} \log \operatorname{sep}\left(\omega, \epsilon_m, F_{n}\right) d\mathbb{P}(\omega) \end{equation*}

with

\begin{equation*} h_{\mathrm{top}}(T,G)= \lim _{\epsilon \rightarrow 0} \liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|}\int \log \operatorname{sep}(\omega, \epsilon, F_{n}) d \mathbb{P}(\omega). \end{equation*}

We can immediately state the relationship between the metric mean dimension and the fibre topological entropy of a system.

Proof. Set

\begin{equation*} S(\epsilon,G)=\liminf _{\epsilon \rightarrow 0} \frac{1}{|F_{n}|}\int \log \operatorname{sep}(\omega, \epsilon, F_{n})\mathbb{P}(\omega). \end{equation*}

Let $\epsilon=\frac{1}{m}$, then

\begin{equation*} \begin{aligned} \underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,G,\omega) &\leq \liminf _{m \rightarrow \infty} \frac{1}{|\log m|} \liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|} \log \operatorname{sep}\left(\omega, \frac{1}{m}, F_{n}\right)\\ &=\liminf _{m \rightarrow \infty} \frac{1}{|\log m|} \liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|} \log \operatorname{cov}\left(\omega, \frac{1}{m}, F_{n}\right). \end{aligned} \end{equation*}

From Fatou’s lemma, we have

\begin{equation*} \begin{aligned} \underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,d,G) & \leq \int \liminf _{m \rightarrow \infty} \frac{1}{|\log m|} \liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|} \log \operatorname{sep}\left(\omega, \frac{1}{m}, F_{n}\right) d\mathbb{P}(\omega) \\ &\leq \liminf _{m \rightarrow \infty} \frac{1}{|\log m|}\int \liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|} \log \operatorname{sep}\left(\omega, \frac{1}{m}, F_{n}\right) d\mathbb{P}(\omega) \\ & \leq \liminf _{m \rightarrow \infty} \frac{1}{|\log m|} \liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|}\int \log \operatorname{sep}\left(\omega, \frac{1}{m}, F_{n}\right) d\mathbb{P}(\omega)\\ & = \liminf_{m\rightarrow \infty} \frac{1}{|\log m|} S\left(\frac{1}{m}, G\right). \end{aligned} \end{equation*}

If $\underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,d,G) \gt 0$, then

\begin{equation*} \liminf_{m\rightarrow \infty} \frac{1}{|\log m|} S\left(\frac{1}{m}, G\right) \gt 0, \end{equation*}

that is,

\begin{equation*} \lim_{m\rightarrow \infty} S\left(\frac{1}{m}, G\right)=\infty. \end{equation*}

Since $S(\epsilon,G)$ is monotone non-decreasing as non-random $\epsilon \rightarrow 0$, then we have

\begin{equation*} h_{\mathrm{top}}(T,G)= \lim_{\epsilon\rightarrow 0} S\left(\epsilon, G\right)=\lim_{m\rightarrow \infty} S\left(\frac{1}{m}, G\right). \end{equation*}

Hence

\begin{equation*} h_{\mathrm{top}}(T,G)=\lim_{\epsilon\rightarrow 0} S\left(\epsilon, G\right)=\infty. \end{equation*}

We conclude this section by pointing out that the mean topological dimension is not larger than the $\mathbb{S}$-lower metric mean dimension for a continuous bundle RDS for amenable group as well as in the deterministic case. The main idea from Lindenstrauss and Weiss still works for the random transformations between different fibres [Reference Lindenstrauss and Weiss40, Theorem 4.2].

By Remark 4.15, it suffices to prove that, for any $\mathcal{A}=\left\{A_i\right\}_{i=1}^l \in \mathrm{C}_X^o$,

\begin{equation*} D(\mathcal{A},T,G)=\lim _{n \rightarrow \infty} \frac{1}{|F_{n}|} \int D\left(\mathcal{A}_{F_{n}}(\omega)\right)d\mathbb{P}(\omega) \leq \underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,d,G). \end{equation*}

By Lemma 4.1, without loss of generality, we can refine $\mathcal{A}$ to be of the form

\begin{equation*} \mathcal{A}=\bigvee_{i=1}^r\left\{U_i, V_i\right\} \end{equation*}

where, for every $1 \leq i \leq r,\left\{U_i, V_i\right\}$ is a two element open cover of X. Let $U_i(\omega)=U_i\cap \mathcal{E}_{\omega}$ and $V_i(\omega)=V_i\cap \mathcal{E}_{\omega}$.

Define $\varphi_i^\omega: \mathcal{E}_\omega \rightarrow[0,1]$ by

\begin{equation*} \varphi_i^\omega(x)=\frac{d\left(x, X \backslash V_i(\omega)\right)}{d\left(x, X \backslash V_i(\omega)\right)+d\left(x, X \backslash U_i(\omega)\right)} \end{equation*}

and let $\Phi^\omega(x)=\left(\varphi_1^\omega(x), \varphi_2^\omega(x), \ldots, \varphi_r^\omega(x)\right)$.

And

\begin{align*} U_{i}(\omega)&=\left(\varphi_i^\omega\right)^{-1}[0,1),\\ V_{i}(\omega)&=\left(\varphi_i^\omega\right)^{-1}(0,1]. \end{align*}

Notice that $\varphi_i^\omega$ is Lipschitz. In fact, let $A(i,\omega)=X \backslash V_i(\omega)$ and $B(i,\omega)=X \backslash U_i(\omega)$, then

\begin{equation*} \varphi_i^\omega(x)=\frac{d\left(x, A\right)}{d\left(x, A\right)+d\left(x, B\right)}. \end{equation*}

One has

\begin{equation*} |\varphi_i^\omega(x)-\varphi_i^\omega(y)|=\frac{|d\left(x, A\right)d\left(y, B\right)-d\left(x, B\right)d\left(y, A\right)|}{\left(d\left(x, A\right)+d\left(x, B\right)\right)\left(d\left(y, A\right)+d\left(y, B\right)\right)}. \end{equation*}

If

\begin{equation*} d(x, A) d(y, B)-d(x, B) d(y, A) \geq 0. \end{equation*}

As A and B are closed, then $d\left(x,A\right)\leq d\left(x,y\right)+d\left(y,A\right)$ and by the symmetry it is clear to see that $d\left(y,B\right)\leq d\left(x,y\right)+d\left(x,B\right)$. Then

\begin{equation*} \begin{aligned} |\varphi_i^\omega(x)-\varphi_i^\omega(y)| & \leq \frac{\left( d\left(x,y\right)+d\left(y,A\right)\right)d\left(y, B\right)+\left(d\left(x,y\right)-d\left(y,B\right)\right)d\left(y, A\right)}{\left(d\left(x, A\right)+d\left(x, B\right)\right)\left(d\left(y, A\right)+\left(y, B\right)\right)} \\ &= \frac{d\left(x, y\right)\left(d\left(y, A\right)+d\left(y, B\right)\right)}{\left(d\left(x, A\right)+d\left(x, B\right)\right)\left(d\left(y, A\right)+d\left(y, B\right)\right)} \\ & = \frac{d\left(x, y\right)}{d\left(x, A\right)+d\left(x, B\right)}. \end{aligned} \end{equation*}

Similarly, if

\begin{equation*} d(x, A) d(y, B)-d(x, B) d(y, A) \lt 0, \end{equation*}

one has

\begin{equation*} |\varphi_i^\omega(x)-\varphi_i^\omega(y)|\leq \frac{d\left(x, y\right)}{d\left(y, A\right)+d\left(y, B\right)}. \end{equation*}

That is,

\begin{equation*} \left|\varphi_i^\omega(x)-\varphi_i^\omega(y)\right| \leq \max \left\{\frac{d(x, y)}{d(x, A)+d(x, B)}, \frac{d(x, y)}{d(y, A)+d(y, B)}\right\}. \end{equation*}

Since

\begin{equation*} d(A, B) \leq \min \{d(x, A)+d(x, B), d(y, A)+d(y, B)\}, \end{equation*}

then

\begin{equation*} \left|\varphi_i^\omega(x)-\varphi_i^\omega(y)\right| \leq \frac{1}{d(A, B)} \cdot d(x, y). \end{equation*}

Therefore, $\varphi_i^\omega$ is Lipschitz and the Lipschitz constant is $1/\left(d\left(X \backslash V_i(\omega), X \backslash U_i(\omega)\right)\right)$.

Let L_ω be a bound on the Lipschitz constants of all the $\left(\varphi_i^\omega\right)_{i=1}^r$. For any Fø lner sequence F_n of G, $F_{n}=\left\{g_1,g_2,...,g_{b}\right\}$, where $b=b(n)$.

Define $R_{F_{n}}^\omega$: $\mathcal{E}_\omega \rightarrow[0,1]^{r b}$ by

\begin{equation*} \begin{aligned} R_{F_{n}}^\omega(x)& =\left(\Phi^{g_1 \omega}\left(T_{g_1, \omega} x\right), \ldots, \Phi^{g_{b}\omega}\left(T_{g_{b},\omega} x\right)\right)\\ &=\left(\varphi_1^{g_1 \omega}(T_{g_1, \omega} x), \varphi_2^{g_1 \omega}(T_{g_1, \omega} x), \ldots, \varphi_r^{g_1 \omega}(T_{g_1, \omega} x),\right. \\ &\quad \quad \quad \quad \vdots \\ & \quad \quad \quad \left.\varphi_1^{g_{b}\omega}\left(T_{g_{b},\omega} x\right), \varphi_2^{g_{b}\omega}\left(T_{g_{b},\omega} x\right), \ldots, \varphi_r^{g_{b}\omega}\left(T_{g_{b},\omega} x\right)\right) . \end{aligned} \end{equation*}

From Lemma 4.4, we see that $R_{F_{n}}^\omega \succ \mathcal{A}_{F_n}(\omega)$, and

\begin{equation*} \left\|R_{F_{n}}^\omega(x)-R_{F_{n}}^\omega(y)\right\|_{\infty} \leq L_\omega d_{F_{n}}^\omega(x, y) . \end{equation*}

As usual, if $S \subset\{1,2, \ldots, r b\}$, denote by $R_{F_{n}}^\omega(x)_S \in[0,1]^{|S|}$ the projection of $R_{F_{n}}^\omega(x)$ to the coordinates in the index set S. Let $D(\omega)=\underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,G,\omega)$. We need the following two fibrewise lemmas, which correspond to [Reference Lindenstrauss and Weiss40, Lemmas 4.3 and 4.4].

Lemma 5.10. Let $\delta \gt 0,$ $D(\omega)=\underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,G,\omega)$. For any Fø lner sequence F_n of G, $F_{n}=\left\{g_1,g_2,...,g_{b}\right\}$, where $b=b(n)$. If n is larger than some $N(\omega,\delta)$, there is a $ \xi(\omega) \in(0,1)^{r b}$ such that, for any $|S| \geq(D(\omega)+\delta) b$,

\begin{equation*} \xi(\omega)_S \notin R_{F_{n}}^\omega\left(\mathcal{E}_\omega\right)_S. \end{equation*}

Proof. Notice that

\begin{equation*} D(\omega)=\underline{\mathbb{E}\operatorname{mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,G,\omega) =\liminf _{m \rightarrow \infty} \frac{1}{|\log \epsilon_m|} \liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|} \log \operatorname{cov}\left(\omega, \epsilon_m, F_{n}\right). \end{equation*}

Fix ω and let m is large enough such that,

\begin{equation*} 0 \lt \epsilon_m \lt \left(2^r\left(2 L_{\omega}\right)^{D(\omega)}\right)^{-2 / \delta}, \end{equation*}

and

\begin{equation*} \frac{1}{|\log \epsilon_m|} \liminf _{n \rightarrow \infty} \frac{1}{|F_{n}|} \log \operatorname{cov}\left(\omega, \epsilon_m, F_{n}\right) \leq D(\omega)+\delta / 4. \end{equation*}

If n is large enough (depending on ϵ_m and δ), $\mathcal{E}_{\omega}$ can be covered by $\epsilon_m^{-(D(\omega)+\delta / 2) b}$ balls of the form

\begin{equation*} B_{d_{F_{n}}^{\omega}}\left(x, \epsilon_m\right)=\left\{x^{\prime} \in \mathcal{E}_{\omega}: d_{F_{n}}^{\omega}(x,x^{\prime}) \lt \epsilon_m\right\}. \end{equation*}

Since L_ω is a Lipschitz constant for all $\left(\varphi_i^\omega\right)_{i=1}^r$,

\begin{equation*} R_{F_{n}}^\omega\left(B_{d_{F_{n}}^{\omega}}\left(x, \epsilon_m\right)\right) \subset\left\{a \in[0,1]^{r b}:\| R_{F_{n}}^\omega(x)-a\|_{\infty} \lt L_{\omega}\epsilon_m\right\}, \end{equation*}

the set $R_{F_{n}}^\omega(\mathcal{E}_{\omega})$ can be covered by $\epsilon_m^{-(D(\omega)+\delta/ 2) b}$ balls in the $\|\cdot\|_{\infty}$ norm of radius $L_{\omega}\epsilon_m $. Enumerate these balls as $B(\omega, k), k=1,2 \ldots, K=\epsilon_m^{-(D(\omega)+\delta/ 2) b}$.

Choose ξ with uniform probability in $[0,1]^{r b}$. For any $S\subset \left\{1,2,...,r b\right\}$,

\begin{equation*} P\left(\xi_S \in R_{F_{n}}^\omega(\mathcal{E}_{\omega})_S\right) \leq \sum_{k=1}^K P\left(\left.\xi_S \in B(\omega, k)\right|_S\right) \leq \epsilon_m^{-(D(\omega)+\delta / 2) b}\left(2 L_{\omega} \epsilon_m \right)^{|S|}. \end{equation*}

Hence

\begin{equation*} \begin{aligned} P\left(\exists S:|S| \right. & \left.\geq(D(\omega)+\delta) b \text {and } \xi_S \in R_{F_{n}}^\omega(\mathcal{E}_{\omega})_S\right) \\ \leq & \sum_{|S| \geq(D(\omega)+\delta) b} P\left(\xi_S \in R_{F_{n}}^\omega(\mathcal{E}_{\omega})_S\right) \\ \leq & \# \left\{S:|S|\geq (D(\omega)+\delta)b\right\} \times \epsilon_m^{-(D(\omega)+\delta / 2) b}\left(2 L_{\omega}\epsilon_m\right)^{(D(\omega)+\delta) b} \\ \leq & 2^{r b}\left(\left(2 L_{\omega}\right)^{D(\omega)} \epsilon_m^{\delta / 2}\right)^b \ll 1, \end{aligned} \end{equation*}

and so, with high probability, a random ξ will satisfy the requirements.

Lemma 5.11. For any Fø lner sequence F_n of G, $F_{n}=\left\{g_1,g_2,...,g_{b}\right\}$, where $b=b(n)$. If $\pi: R_{F_{n}}^\omega\left(\mathcal{E}_\omega\right) \rightarrow[0,1]^{r b}$ satisfies for both a = 0 and 1, and, for all $\xi \in$ $[0,1]^{r b}$,

\begin{equation*} \left\{1 \leq k \leq r b: \xi_k=a\right\} \subset\left\{1 \leq k \leq r b: \pi(\xi)_k=a\right\}, \end{equation*}

then $\pi \circ R_{F_{n}}^\omega$ is compatible with $\mathcal{A}_{F_{n}}(\omega)$.

Proof. Indeed, if $\xi \in[0,1]^{r b}$, define for $0 \leq j \lt N$ and $1 \leq i \lt r$,

\begin{equation*} W_{i, j}= \begin{cases}\left(T_{g_{j},\omega}\right)^{-1} U_i, & \text {if } \xi_{j r+i}=0; \\ \left(T_{g_{j},\omega}\right)^{-1} V_i, & \text {otherwise. }\end{cases} \end{equation*}

It is easy to see that

\begin{equation*} \left(\pi\circ R_{F_{n}}^\omega\right)^{-1}(\xi) \subset \bigcap_{\substack{1 \leq i \lt r \\ 1 \leq j\leq b}} W_{i, j} \in \mathcal{A}_{F_{n}}(\omega). \end{equation*}

Proof. Fix ω and let δ > 0. For any Fø lner sequence F_n of G, $F_{n}=\left\{g_1,g_2,...,g_{b}\right\}$, where $b=b(n)$. Find $\bar{\xi}, b$ as in Lemma 5.10. Let

\begin{equation*} \Phi=\left\{\xi \in[0,1]^{r b}: \#\left\{k: \xi_k = \bar{\xi}_k \right\} \gt (D(\omega)+\delta) b \right\},\end{equation*}

thus $R_{F_{n}}^\omega(\mathcal{E}_{\omega}) \subset[0,1]^{r b} \backslash \Phi$. For brevity, we shall denote the latter set as $\Phi^c$,

\begin{equation*} \Phi^c=\left\{\xi \in[0,1]^{r b}: \#\left\{k: \xi_k\ne \bar{\xi}_k \right\}\geq rb-\lfloor(D(\omega)+\delta) b\rfloor \right\}. \end{equation*}

We shall construct a continuous retraction π of $\Phi^c$ onto the $\lfloor(D(\omega)+\delta) b\rfloor+1$ skeleton of the cube $I=[0,1]^{r b}$. For $k=1,2, \ldots$, we denote by J_k the set

\begin{equation*}J_k=\left\{\xi \in I: \#\left\{1 \leq i \leq r b: \xi_i \in\{0,1\}\right\}\geq k \right\}, \end{equation*}

and $\text{dim} J_{k}=rb-k$.

Since $\bar{\xi}$ is in the interior of I, one can define $\pi_1: I \backslash\{\bar{\xi}\} \rightarrow J_1$ by mapping each ξ to the intersection of the ray starting at $\bar{\xi}$ and passing through ξ and J ₁. For each of the $(rb-1)$-dimensional cubes I ^ʹ that comprise J ₁, we can define a retraction on I ^ʹ in a similar fashion using as a centre the projection of $\bar{\xi}$ onto I ^ʹ. This will define a continuous retraction π ₂ of $\Phi^c$ into J ₂. As long as there is some intersection of Φ with the cubes in J_k, this process can be continued, and thus we get finally a continuous projection π of $\Phi^c$ onto $J_{k_0}$, with

\begin{equation*} k_0=rb-\lfloor(D(\omega)+\delta) b\rfloor-1. \end{equation*}

Clearly, π satisfies the conditions of Lemma 5.11. Thus $\pi \circ R_{F_{n}}^{\omega} \succ \mathcal{A}_{F_{n}}(\omega)$.

Also, as $R_{F_{n}}^{\omega}(\mathcal{E}_{\omega}) \subset \Phi^c$,

\begin{equation*} \pi \circ R_{F_{n}}^{\omega}\subset J_{k_0}, \end{equation*}

the latter having topological dimension $\lfloor(D(\omega)+\delta) b\rfloor+1$. To summarize: we have constructed an $\mathcal{A}_{F_{n}}(\omega)$ compatible function from $\mathcal{E}_{\omega}$ to a space of topological dimension $\leq(D(\omega)+\delta) b+1$, and so by Lemma 4.5

\begin{equation*} \mathcal{D}\left(\mathcal{A}_{F_{n}}(\omega)\right) \leq \text{dim}J_{k_0}\leq (D(\omega)+\delta) b+1, \end{equation*}

then

\begin{equation*} \lim_{n\rightarrow \infty}\frac{1}{|F_{n}|}\int \mathcal{D}\left(\mathcal{A}_{F_{n}}(\omega)\right)d\mathbb{P}(\omega) \leq \underline{\operatorname{\mathbb{E}mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,d,G)+\delta. \end{equation*}

As δ > 0 can be made as small as we like, we see that indeed

\begin{equation*} \operatorname{\mathbb{E}mdim}(T,G) \leq \underline{\operatorname{\mathbb{E}mdim}_{\mathrm{M}}^{\mathbb{S}}}(T,d,G). \end{equation*}