Relative uniformly positive entropy of induced amenable group actions

Abstract

Let G be a countably infinite discrete amenable group. It should be noted that a G-system $(X,G)$ naturally induces a G-system $(\mathcal {M}(X),G)$, where $\mathcal {M}(X)$ denotes the space of Borel probability measures on the compact metric space X endowed with the weak*-topology. A factor map $\pi : (X,G)\to (Y,G)$ between two G-systems induces a factor map $\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$. It turns out that $\widetilde {\pi }$ is open if and only if $\pi $ is open. When Y is fully supported, it is shown that $\pi $ has relative uniformly positive entropy if and only if $\widetilde {\pi }$ has relative uniformly positive entropy.

1 Introduction

In the process of studying the classification of topological dynamical systems, entropy as a conjugacy invariant plays an important role which divides them into two classes. For $\mathbb {Z}$ -systems, the notion of uniformly positive entropy (u.p.e. for short) was introduced by Blanchard in [6] as an analogue in topological dynamics for the notion of a K-process in ergodic theory. He then naturally defined the notion of entropy pairs and used it to show that a u.p.e. system is disjoint from all minimal zero entropy systems [7]. Further research concerning u.p.e. systems and entropy pairs can be found in [8, 9, 13, 16, 17, 27].

Recently, there has been a lot of significant progress in studying relative entropy via local relative entropy theory for $\mathbb {Z}$ -systems. For a factor map between two $\mathbb {Z}$ -systems, Glasner and Weiss [14] introduced the relative uniformly positive entropy (rel-u.p.e.) and the notion of relative topological Pinsker factor based on the idea of u.p.e. extensions. Later, Park and Siemaszko [30] interpreted another relative topological Pinsker factor, defined by Lemańczyk and Siemaszko [27], using relative measure-theoretical entropy and discussed the relative product. In [19], Huang, Ye and Zhang introduced the notions of relative entropy tuples in both topological and measure-theoretical settings. They showed that the finite product of rel-u.p.e. extensions has rel-u.p.e. if and only if the factors are fully supported (for definitions see §2.3). They also proved some classical results about the rel-u.p.e. extension. We will refer readers to [10, 11, 18, 26] for more results related to local relative entropy theory.

Bauer and Sigmund [3] initiated a systematic study of the connections between dynamical properties of a $\mathbb {Z}$ -system and its induced system (whose phase space consists of all Borel probability measures on the original space, for details see §2). A well-known result due to Glasner and Weiss [15] in 1995 reveals that if a system has zero topological entropy, then so does its induced system. Later, this connection was further developed by Kerr and Li in [23]. They obtained that a system is null if and only if its induced system is null. More research concerning relations of these systems was developed in [1, 2, 33, 37]. Recently, Bernardes et al [4] proved that a $\mathbb {Z}$ -system has u.p.e. if and only if its induced system does.

After Ornstein and Weiss’s pioneering work for amenable group actions in 1987 [29], there have been many developments in the process of studying the amenable group action systems. We will refer the reader to the related papers [20, 28, 31, 35, 36, 38]. In this paper, we always assume that G is a countably infinite discrete amenable group. By a G-system $(X,G)$ , we mean a compact metric space X together with G acting on X by homeomorphisms, that is, there exists a continuous map $\Gamma :G\times X\to X$ , satisfying:

• • $\Gamma (e_{G},x)=x$ for every $x\in X$ ;

• • $\Gamma (g,\Gamma (h,x))=\Gamma (gh,x)$ for each $g,h\in G$ and $x\in X$ .

We write $\Gamma (g,x)$ as $gx$ for every $g\in G$ and $x\in X$ .

Motivated by those works which were previously mentioned for $\mathbb {Z}$ -systems and the local entropy theory developed for countable discrete amenable group action systems due to Huang, Ye and Zhang [20], and Kerr and Li [24], the present paper aims to investigate the properties of the relative uniformly positive entropy (rel-u.p.e.) for an induced factor map of a factor map between two G-systems (see §2 for definitions).

More precisely, let $(X,G)$ be a G-system, $\mathcal {B}_X$ be the set of Borel subsets of X and $\mathcal {M}(X)$ be the space of Borel probability measures on the compact metric space X endowed with the weak*-topology. Then the G-system $(X,G)$ induces a system $(\mathcal {M}(X),G)$ (see §2 for details). For any $x\in X$ , let $\delta _{x}$ denote the Dirac measure on x and

$$ \begin{align*}\mathcal{M}_n(X)=\bigg\{\frac{1}{n}\sum_{i=1}^n\delta_{x_i}: x_1, x_2,\ldots,x_n\in X \bigg\}\end{align*} $$

for each $n\in \mathbb {N}$ . Then $\mathcal {M}_n(X)$ is closed and invariant under G (that is, $g\mathcal {M}_n(X)=\mathcal {M}_n(X)$ for every $g\in G$ ). Hence, we can consider the subsystems $(\mathcal {M}_n(X),G)$ of $(\mathcal {M}(X),G)$ for each $n\in \mathbb {N}$ . For a factor map $\pi :(X,G)\to (Y,G)$ between two G-systems, when $\mathrm {supp}(Y)=Y$ (for definitions see §2.3), we have the following result.

Theorem 1.1. Let $\pi : (X,G)\to (Y,G)$ be a factor map between two G-systems, $\widetilde {\pi }: (\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$ be the factor map induced by $\pi $ and $\widetilde {\pi }_n: (\mathcal {M}_n(X),G) \ \to (\mathcal {M}_n(Y),G)$ be the restriction of $\widetilde {\pi }$ on $\mathcal {M}_n(X)$ . When $\mathrm {supp}(Y)=Y$ , the following are equivalent:

1. (1) $\pi $ has relative uniformly positive entropy;

2. (2) $\widetilde {\pi }_n$ has relative uniformly positive entropy for some $n\in \mathbb {N}$ ;

3. (3) $\widetilde {\pi }_n$ has relative uniformly positive entropy for every $n\in \mathbb {N}$ ;

4. (4) $\widetilde {\pi }$ has relative uniformly positive entropy.

Notice that when Y is a singleton, we obtain that $(X,G)$ has u.p.e. if and only if the induced system $(\mathcal {M}(X),G)$ has u.p.e. (when $G=\mathbb {Z}$ , see [4, Theorem 4]).

We say a map $\pi : X\to Y$ between two topological spaces is open if the images of open sets are open. Then we have the following result.

Theorem 1.2. Let $\pi :X\to Y$ be a surjective continuous map between two compact metrizable spaces, and $\widetilde {\pi }:\mathcal {M}(X)\to \mathcal {M}(Y)$ be the induced map of $\pi $ . Then $\pi $ is open if and only if $\widetilde {\pi }$ is open.

This paper is organized as follows. In §2, we will list some basic notions and results needed in our argument. In §§3 and 4, we will give a proof of Theorem 1.1. Finally, we prove Theorem 1.2 in §5.

2 Preliminaries

In this section, we recall some basic notation and results which will be used repeatedly in our paper. Denote by $\mathbb {N}$ and $\mathbb {R}$ the set of natural numbers and real numbers, respectively. For $n\in \mathbb {N}$ , we write $[n]$ for $\{1,2, \ldots ,n\}$ .

2.1 Amenable group

We say a countably infinite discrete group G is amenable if there always exists an invariant Borel probability measure when it acts on any compact metric space. In the case where G is a countably infinite discrete group, amenability is equivalent to the existence of a F $\phi $ lner sequence: a sequence of non-empty finite subsets $\{F_n\}_{n=1}^{\infty }$ of G such that

$$ \begin{align*}\lim_{n\to\infty}\frac{\vert F_n\Delta gF_n\vert}{\vert F_n\vert}=0\end{align*} $$

for all $g\in G$ . One should refer to Ornstein and Weiss’ paper [29] for more details about an amenable group. In this paper, we always assume that G is a countably infinite discrete amenable group and denote by $\mathcal {F}(G)$ the collection of non-empty finite subsets of G. The following result is well known (see [25, Theorem 4.48]).

Theorem 2.1. Let $\phi $ be a real-valued function on $\mathcal {F}(G)$ satisfying:

1. (1) $\phi (Fs)=\phi (F)$ for all $F\in \mathcal {F}(G)$ and $s\in G$ ; and

2. (2) $\phi (F)\leq ({1}/{k})\sum _{E\in \mathcal {E}}\phi (E)$ for every $k\in \mathbb {N}$ , $F\in \mathcal {F}(G)$ and finite collection $\mathcal {E}\subseteq \mathcal {F}(G)$ with $\bigcup _{E\in \mathcal {E}}E\subseteq F$ and $\sum _{E\in \mathcal {E}}1_E\geq k1_F$ .

Then ${\phi (F)}/{\vert F\vert }$ converges to a limit as F becomes more and more invariant and this limit is equal to $\inf _{F}{\phi (F)}/{\vert F\vert }$ , where F ranges over all non-empty finite subsets of G.

2.2 Induced systems

Assume that X is a compact metric space. Let $\mathcal {B}_X$ be the collection of Borel subsets of X, $C(X)$ be the space of continuous maps from X to $\mathbb R$ endowed with the supremum norm $\|\cdot \|_{\infty }$ and $\mathcal {M}(X)$ be the set of Borel probability measures on X endowed with the weak $^\ast $ -topology, which is the smallest topology making the map

$$ \begin{align*}D_g:\mathcal{M}(X)\to\mathbb{R},\quad \mu\mapsto\int_Xg\,d\mu\end{align*} $$

continuous for every $g\in C(X)$ , and the topology basis of weak $^\ast $ -topology consists of the following sets:

(2.1) $$ \begin{align} \mathbb{V}(\mu;f_1,\ldots,f_k;\epsilon):= \bigg\{\nu\in\mathcal{M}(X): \bigg\vert\int_Xf_i\,d\mu-\int_Xf_i\,d\nu\bigg\vert<\epsilon\ \text{for all } i\in[k]\bigg\}, \end{align} $$

where $\mu \in \mathcal {M}(X)$ , $k\geq 1$ , $\epsilon>0$ and $f_i:X\to \mathbb {R}$ are continuous functions for $i\in [k]$ . The Prohorov metric on $\mathcal {M}(X)$ ,

$$ \begin{align*} d_P(\mu,\nu):=\mathrm{inf}\{\delta>0:\mu(A)\leq\nu(A^{\delta})+\delta\ \text{and}\ \nu(A)\leq \mu(A^{\delta})+\delta\ \text{for all}\ A\in\mathcal{B}_X\}, \end{align*} $$

where $A^{\delta }=\{x\in X: d(x,A)<\delta \}$ , is compatible with the weak*-topology. We will refer the readers to the books [5, 12, 22] for the knowledge of space $\mathcal {M}(X)$ . Moreover,

$$ \begin{align*}d_P(\mu,\nu)= \mathrm{inf}\{\delta>0:\mu(A)\leq\nu(A^{\delta})+\delta\ \text{for all}\ A\in\mathcal{B}_X\}\end{align*} $$

(see [5, p. 72]). Proposition 2.2 describes a basis for the weak*-topology on $\mathcal {M}(X)$ due to Bernardes et al (see [4, Lemma 1]).

Proposition 2.2. The set of the form

$$ \begin{align*}\mathbb{W}(U_1,U_2,\ldots,U_k:\eta_1,\eta_2,\ldots,\eta_k):=\{\mu\in\mathcal{M}(X): \ \mu(U_i)>\eta_i \text{ for } i\in[k]\},\end{align*} $$

where $k\geq 1$ , $U_1,U_2,\ldots ,U_k$ are non-empty disjoint open sets in X and $\eta _1,\eta _2,\ldots ,\eta _k$ are positive real numbers with $\eta _1+\eta _2+\cdots +\eta _k<1$ , form a basis for the weak*-topology on $\mathcal {M}(X)$ .

A G-system $(X,G)$ induces a system $(\mathcal {M}(X),G)$ , where $g: \mathcal {M}(X)\to \mathcal {M}(X)$ is defined by $(g\mu )(A):=\mu (g^{-1}A)$ for every $g\in G$ , $\mu \in \mathcal {M}(X)$ and $A\in \mathcal {B}_X$ . We call $(\mathcal {M}(X),G)$ the induced system of $(X,G)$ .

Let $(X,G)$ and $(Y,G)$ be two G-systems. A continuous map $\pi :(X,G)\to (Y,G)$ is called a factor map between $(X,G)$ and $(Y,G)$ if it is onto and $\pi \circ g=g\circ \pi $ for every $g\in G$ . Here, $\pi $ can induce a factor map $\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$ by

$$ \begin{align*}(\widetilde{\pi}\mu)(B)=\mu(\pi^{-1}B)\end{align*} $$

for every $\mu \in \mathcal {M}(X)$ and $B\in \mathcal {B}_Y$ . For every $n\in \mathbb {N}$ , we denote

$$ \begin{align*}\widetilde{\pi}_n:=\tilde{\pi}\vert_{\mathcal{M}_n(X)}: \mathcal{M}_n(X)\to\mathcal{M}_n(Y)\end{align*} $$

by the restriction of $\tilde {\pi }$ on $\mathcal {M}_n(X)$ . Note that $\widetilde {\pi }_n$ is also a factor map for each $n\in \mathbb {N}$ .

2.3 Support

Let $(X,G)$ be a G-system, $(\mathcal {M}(X),G)$ be the induced G-system of $(X,G)$ . We denote by $\mathcal {M}(X,G)$ the set of all G-invariant measures. For $\mu \in \mathcal {M}(X)$ , we denote by $\mathrm {supp}(\mu )$ the support of $\mu $ , that is, the smallest closed subset $W\subseteq X$ such that $\mu (W)=1$ . We denote by $\mathrm {supp}(X,G)$ the support of $(X, G)$ , that is,

$$ \begin{align*}\mathrm{supp}(X, G) =\bigcup_{\mu\in\mathcal{M}(X,G)}\mathrm{supp}(\mu).\end{align*} $$

Here, $(X,G)$ is called fully supported if there is an invariant measure $\mu \in \mathcal {M}(X,G)$ with full support (that is, $\mathrm {supp}(\mu )=X$ ), equivalently, $\mathrm {supp}(X, G) = X$ .

2.4 Relative uniformly positive topological entropy

For a given G-system $(X,G)$ , a cover of X is a family of Borel subsets of X, whose union is X. Denote the set of finite covers by $\mathcal {C}_X$ . For $n\in \mathbb N$ and $\mathcal {U}_1,\mathcal {U}_2,\ldots ,\mathcal {U}_n\in \mathcal {C}_X$ , we denote

$$ \begin{align*}\bigvee_{i=1}^{n}\mathcal{U}_i=\{A_1\cap A_1\cap\cdots\cap A_n: A_i\in\mathcal{U}_i,\ i\in[n]\}.\end{align*} $$

Let $\pi :(X,G)\to (Y,G)$ be a factor map between two G-systems and $\mathcal {U}\in \mathcal {C}_X$ . For any non-empty subset E of X, let $N(\mathcal {U},E)$ be the minimum among the cardinalities of the subsets of $\mathcal {U}$ which cover E, and define

$$ \begin{align*}N(\mathcal{U}|\pi)=\sup_{y\in Y}N(\mathcal{U},\pi^{-1}(y)).\end{align*} $$

The topological conditional entropy of $\mathcal {U}$ with respect to $\pi $ is defined by

$$ \begin{align*}h_{\mathrm{top}}(\mathcal{U},G|\pi)=\lim_{n\to\infty}\frac{1}{\vert F_n\vert}\log N(\mathcal{U}_{F_n}|\pi),\end{align*} $$

where $\mathcal {U}_{F_{n}}=\bigvee _{g\in F_{n}}g^{-1}\mathcal {U}$ and $\{F_n\}_{n=1}^{\infty }$ is a F $\phi $ lner sequence of G. It is well known that $h_{\mathrm {top}}(\mathcal {U},G\vert \pi )$ is well defined and is independent of the choice of the F $\phi $ lner sequences of G.

Let $\pi :(X,G)\to (Y,G)$ be a factor map between G-systems. Here, $\mathcal {U} = \{ U_{1},\ldots ,U_{n}\}\in \mathcal {C}_X$ is said to be non-dense-on- $\pi $ -fibre if there is $y\in Y$ such that $\pi ^{-1}(y)$ is not contained in any element of $\overline {\mathcal {U}}$ which consists of the closures of elements of $\mathcal {U}$ in X. Clearly, if an open cover $\mathcal {U}=\{U_{1},U_{2}\}$ is non-dense-on- $\pi $ -fibre, then $\pi (U_{1})\cap \pi (U_{2})\neq \emptyset $ . We say $(X,G)$ or $\pi $ has relative uniformly positive entropy (rel-u.p.e. for short) if for any non-dense-on- $\pi $ -fibre open cover $\mathcal {U}$ of X with two elements, we have $h_{\mathrm{top}}(\mathcal {U},G|\pi )>0$ .

For $n\in \mathbb {N}$ and G-systems $(Z_i,G)$ , $i\in [n]$ , we set

$$ \begin{align*}\prod_{i\in[n]}Z_i=\{(z_1,z_2,\ldots,z_n): z_i\in Z_i;\ i\in[n]\}\end{align*} $$

and

$$ \begin{align*}g(z_1,z_2,\ldots,z_n)=(gz_1,gz_2,\ldots,gz_n)\end{align*} $$

for every $g\in G$ and $z_i\in Z_i$ for $i\in [n]$ . Clearly, $(\prod _{i\in [n]}Z_i,G)$ is also a G-system. When $Z_i=Z$ for all $i\in [n]$ , we write $\prod _{i\in [n]}Z_i$ as $Z^{(n)}$ . Let $\pi _i:(X_i,G)\to (Y_i,G)$ be factor maps between G-systems for $i\in [n]$ . Then $\{\pi _i\}_{i\in [n]}$ induce a factor map

$$ \begin{align*}\prod_{i\in[n]}\pi_i: \bigg(\prod_{i\in[n]}X_i,G\bigg) \to \bigg(\prod_{i\in[n]}Y_i,G\bigg)\end{align*} $$

by

$$ \begin{align*}\prod_{i\in[n]}\pi_i(x_1,x_2,\ldots,x_n)=(\pi_1 x_1,\pi_2 x_2,\ldots,\pi_n x_n)\end{align*} $$

for every $(x_1,x_2,\ldots ,x_n)\in \prod _{i\in [n]}X_i$ . When $\pi _i=\pi $ for all $i\in [n]$ , we write $\prod _{i\in [n]}\pi _i$ as $\pi ^{(n)}$ . In [19], Huang, Ye and Zhang showed that the finite product of rel-u.p.e. factor maps between $\mathbb {Z}$ -systems has rel-u.p.e. It also holds for G-systems.

Theorem 2.3. Let $\pi _i: (X_i,G)\to (Y_i,G)$ be a factor map between two G-systems and $\mathrm {supp}(Y_i)=Y_i$ for $i=1,2$ . Then $\pi _1$ and $\pi _2$ have rel-u.p.e. if and only if $\pi _1\times \pi _2: (X_1\times X_2,G)\to (Y_1\times Y_2,G)$ has rel-u.p.e.

We will give a proof of Theorem 2.3 in Appendix A (see Theorem A.5).

3 $\pi $ has rel-u.p.e. if and only if $\widetilde {\pi }_n$ has rel-u.p.e

Let X be a compact metric space and $\rho _X$ be a compatible metric for X. We denote $B_{\rho _X}(x,\delta )=\{y\in X: \rho _X(x,y)<\delta \}$ for $x\in X$ and $\delta>0$ , and denote

$$ \begin{align*}\Delta(X)=\{(x,x): x\in X\}.\end{align*} $$

For $(x_1,x_2)\in X\times X\backslash \Delta (X)$ and $\mathcal {U}=\{U_1,U_2\}\in \mathcal {C}_X$ , we say $\mathcal {U}$ is an admissible cover of X with respect to $(x_1,x_2)$ if for any $i\in [2]$ , one has $\{x_1,x_2\}\nsubseteq \overline {U_i}$ . Let $\pi :(X,G)\to (Y,G)$ be a factor map between two G-systems. Here, $(x_1,x_2)\in X\times X\backslash \Delta (X)$ is called an entropy pair relevant to $\pi $ if for any admissible cover $\mathcal {U}$ with respect to $(x_1,x_2)$ , we have $h_{\mathrm{top}}(\mathcal {U},G|\pi )>0$ . Denote by $E(X,G|\pi )$ the set of all entropy pairs relevant to $\pi $ . Let

$$ \begin{align*}R_{\pi}=\{(x_1,x_2)\in X\times X :\pi(x_1)=\pi(x_2)\}.\end{align*} $$

It is easy to see that $E(X,G|\pi )\subseteq R_{\pi }\setminus \Delta (X)$ , and $\pi $ has rel-u.p.e. if and only if $E(X,G|\pi )=R_{\pi }\setminus \Delta (X)$ .

The concept of dynamical independence is introduced in [24, Definition 2.1]. Now we consider its relative version. Let $\pi : (X,G)\to (Y,G)$ be a factor map between two G-systems. For any $n\in \mathbb {N}$ and a tuple $\mathcal {V}=(V_1,V_2,\ldots ,V_n)$ of subsets of X, we say $J\subseteq G$ is an independence set of $\mathcal {V}$ with respect to $\pi $ if for every non-empty finite subset $I\subset J$ , there exists $y\in Y$ such that

$$ \begin{align*}\pi^{-1}(y)\cap \bigcap_{g\in I}g^{-1}V_{\sigma(g)}\neq\emptyset\end{align*} $$

holds for every $\sigma \in [n]^{I}$ . We denote by $\mathcal {P}_{\mathcal {V}}^{\pi }$ the set of all independence sets of $\mathcal {V}$ with respect to $\pi $ .

Remark 3.1. For every $n\in \mathbb {N}$ and a tuple $\mathcal {V}=(V_1,V_2,\ldots ,V_n)$ of subsets of X, if we set

$$ \begin{align*}\mathcal{I}_{\mathcal{V}}: \mathcal{F}(G)\to \mathbb{R}; \quad \mathcal{I}_{\mathcal{V}}(F):=\max_{I\subseteq F,I\in \mathcal{P}_{\mathcal{V}}^{\pi}}\vert I\vert,\end{align*} $$

then by Theorem 2.1, $\mathcal {I}_{\mathcal {V}}(F)/\vert F\vert $ converges as F becomes increasingly more invariant and this limit is equal to $\inf _{F}({\mathcal {I}_{\mathcal {V}}(F)}/{\vert F\vert })$ , where F ranges over $\mathcal {F}(G)$ . When this limit is positive, we say $\mathcal {V}$ is independent with respect to $\pi $ .

The next lemma follows [24, Lemma 3.4] (see also [17, Theorem 7.4]).

Lemma 3.2. Let $\pi : (X,G)\to (Y,G)$ be a factor map between two G-systems, and $V_1, V_2$ be two disjoint subsets of X. If we set $\mathcal {U}=\{X\setminus V_1, X\setminus V_2\}$ , then $h_{\mathrm {top}}(\mathcal {U},G|\pi )>0$ if and only if $\{V_1,V_2\}$ is independent with respect to $\pi $ .

Let $\pi : (X,G)\to (Y,G)$ be a factor map between two G-systems. For any $(x_1,x_2)\in X\times X\backslash \Delta (X)$ , disjoint open subsets $V_1, V_2$ of X with $x_i\in V_i$ for $i\in [2]$ , $\mathcal {V}=\{X\setminus V_1, X\setminus V_2\}$ is an admissible cover of X with respect to $(x_1,x_2)$ . Then by Lemma 3.2, we immediately have the following corollary.

Corollary 3.3. Let $\pi : (X,G)\to (Y,G)$ be a factor map between two G-systems and $(x_1,x_2)\in X\times X\backslash \Delta (X)$ . Then $(x_1,x_2)\in E(X,G|\pi )$ if and only if for any disjoint open subsets $V_1, V_2$ of X with $x_i\in V_i$ for $i=1,2$ , $\{V_1,V_2\}$ is independent with respect to $\pi $ .

We note that for any two non-empty finite sets H, W, if $H\subseteq W$ and $S\subset \{1,2\}^{W}$ , one has

(3.1) $$ \begin{align} \vert S\vert_H\vert\geq\frac{\vert S\vert}{2^{\vert W\vert-\vert H\vert}}, \end{align} $$

where $S\vert _H$ is the restriction of S on H, that is,

$$ \begin{align*} S\vert_H=\{\sigma\in\{1,2\}^H:\text{there exists}\ \sigma'\in S\ \text{such that}\ \sigma(h)=\sigma'(h)\ \text{for all}\ h\in H\}. \end{align*} $$

The following consequence of Karpovsky and Milman’s generalization of the Sauer–Perles–Shelah lemma [21, 32, 34] is well known, and one can also refer to [24, Lemma 3.5].

Lemma 3.4. Given $k\geq 2$ and $\unicode{x3bb}>1$ , there exists a constant $c>0$ such that for all $n\in \mathbb {N}$ , if $S\subseteq [k]^{[n]}$ satisfies $\vert S\vert \geq ((k-1)\unicode{x3bb} )^n$ , then there is an $I\subseteq [n]$ with $\vert I\vert \geq cn$ and $S\vert _I=[k]^I$ .

Theorem 1.1 follows from Theorems 3.5, 4.2 and 4.3.

Theorem 3.5. Let $n\in \mathbb {N}$ , $\pi : (X,G)\to (Y,G)$ be a factor map between two G-systems, $\widetilde {\pi }: (\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$ be the factor map induced by $\pi $ and $\widetilde {\pi }_n: \ (\mathcal {M}_n(X),G) \ \to (\mathcal {M}_n(Y),G)$ be the restriction of $\widetilde {\pi }$ on $\mathcal {M}_n(X)$ . When $\mathrm {supp}(Y) \ =Y$ , the following are equivalent:

1. (1) $\pi $ has rel-u.p.e.;

2. (2) $\widetilde {\pi }_n$ has rel-u.p.e. for some $n\in \mathbb {N}$ ;

3. (3) $\widetilde {\pi }_n$ has rel-u.p.e. for every $n\in \mathbb {N}$ .

Proof. $(3)\Rightarrow (2)$ is trivial. We will prove $(1)\Rightarrow (3)$ and $(2)\Rightarrow (1)$ .

$(1)\Rightarrow (3)$ . Assume that $\pi $ has rel-u.p.e. For every fixed $1\le n< \infty $ , to obtain that $\widetilde {\pi }_n$ has rel-u.p.e., it is sufficient to prove that $E(\mathcal {M}_n(X),G|\widetilde {\pi }_n)\supseteq R_{\widetilde {\pi }_n}(\mathcal {M}_{n}(X),G) \ \backslash \Delta (\mathcal {M}_n(X))$ .

Let $(\mu _1,\mu _2)\in R_{\widetilde {\pi }_n}(\mathcal {M}_n(X), \ G)\backslash \Delta (\mathcal {M}_n(X))$ , and $\widetilde {V}_1$ and $\widetilde {V}_2$ be two disjoint open subsets of $\mathcal {M}_n(X)$ with $\mu _i\in \widetilde {V}_i$ for $i\in [2]$ . By Corollary 3.3, we shall show that $\{\widetilde {V}_1,\widetilde {V}_2\}$ is independent with respect to $\widetilde {\pi }_n$ .

For $i\in [2]$ and $j\in [n]$ , there exist points $x_j^i\in X$ such that $\mu _i=({1}/{n})\sum _{j=1}^n\delta _{x_j^i}$ . We note that the map $\Phi : X^{(n)}\to \mathcal {M}(X)$ , defined by

$$ \begin{align*}\Phi(z_1,z_2,\ldots,z_n)=\frac{1}{n}\sum_{i=1}^{n}\delta_{z_i}\end{align*} $$

is continuous. Thus, for every $i\in [2]$ and $j\in [n]$ , there exists open neighbourhoods $V_j^i$ of $x_j^i$ such that

$$ \begin{align*}\mu_i\in\bigg\{\frac{1}{n}\sum_{j=1}^{n}\delta_{z_j}: z_j\in V_j^i, j\in[n]\bigg\}\subseteq \widetilde{V}_i.\end{align*} $$

Since $\widetilde {V}_1{\kern-1pt}\cap{\kern-1pt} \widetilde {V}_2 {\kern-1pt}={\kern-1pt}\emptyset $ , if we set $W_i{\kern-1pt}={\kern-1pt}V^i_1{\kern-1pt}\times{\kern-1pt} V_2^i{\kern-1pt}\times \cdots \times{\kern-1pt} V_n^i$ for $i{\kern-1pt}={\kern-1pt}1,2$ , one has $W_1{\kern-1pt}\cap{\kern-1pt} W_2{\kern-1pt}={\kern-1pt}\emptyset $ . Without loss of generality, we can assume that $\pi (x_j^1)=\pi (x_j^2)$ for all $j\in [n]$ since $\widetilde {\pi }_n(\mu _1)=\widetilde {\pi }_n(\mu _2)$ . Let $\omega _i=(x_1^i,x_2^i,\ldots ,x_n^i)\in W_i$ for $i=1,2$ . Then

$$ \begin{align*}(\omega_1,\omega_2)\in R_{\pi^{(n)}}\setminus\Delta(X^{(n)})=E(X^{(n)},G\vert\pi^{(n)})\end{align*} $$

as $\pi ^{(n)}$ has rel-u.p.e. by Theorem 2.3. Thus, $\{W_1,W_2\}$ is independent with respect to $\pi ^{(n)}$ . We note that $\mathcal {P}^{\pi ^{(n)}}_{\{W_1,W_2\}}\subseteq \mathcal {P}^{\widetilde {\pi }_n}_{\{\widetilde {V_1},\widetilde {V_2}\}}$ . This implies $\{\widetilde {V_1},\widetilde {V_2}\}$ is independent with respect to $\widetilde {\pi }_n$ .

$(2)\Rightarrow (1)$ . We assume that $\widetilde {\pi }_n$ has rel-u.p.e. for some positive integer $1\le n< \infty $ . In the following, we prove that $R_{\pi }\setminus \Delta (X)\subseteq E(X,G\vert \pi )$ . Let $(x_1,x_2)\in R_{\pi }\setminus \Delta (X)$ , $V_1$ and $V_2$ be two disjoint open subsets of X with $x_i\in V_i$ , $i=1,2$ . By Corollary 3.3, we only need to show that $\{V_1,V_2\}$ is independent with respect to $\pi $ .

We set

$$ \begin{align*}\widetilde{V_i}=\bigg\{\mu\in\mathcal{M}_n(X): \mu(V_i)>1-\frac{1}{2n}\bigg\}\end{align*} $$

for $i=1,2$ . Clearly, $\widetilde {V_1}$ and $\widetilde {V_2}$ are disjoint open subsets of $\mathcal {M}_n(X)$ with $\delta _{x_i}\in \widetilde {V_i}$ for $i=1,2$ . Since $\widetilde {\pi }_n$ has rel-u.p.e., and $(\delta _{x_1},\delta _{x_2})\in R_{\widetilde {\pi }_n}\setminus \Delta (\mathcal {M}_n(X))=E(\mathcal {M}_n(X),G\vert \widetilde {\pi }_n)$ , $\{\widetilde {V_1},\widetilde {V_2}\}$ is independent with respect to $\widetilde {\pi }_n$ . Then there exists a constant $c>0$ , such that for every fixed $F\in \mathcal {F}(G)$ , there exist $I\subseteq F$ with $\vert I\vert>c\vert F\vert $ and $\nu =({1}/{n})\sum _{i=1}^{n}\delta _{y_i}\in \mathcal {M}_n(Y)$ for some $y_i\in Y$ such that

$$ \begin{align*}A_{\sigma}:=\widetilde{\pi}_n^{-1}(\nu)\cap \bigcap_{g\in I}g^{-1}\widetilde{V}_{\sigma(g)}\neq\emptyset\end{align*} $$

for every $\sigma \in \{1,2\}^{I}$ .

For every $\sigma \in \{1,2\}^{I}$ and $\mu _{\sigma }=({1}/{n})\sum _{i=1}^{n}\delta _{z_i^{\sigma }}\in A_{\sigma }$ , we can assume $\pi (z_i^{\sigma })=y_i$ for $i\in [n]$ . Moreover, for every $g\in I$ , one has $g\mu _{\sigma }=({1}/{n})\sum _{i=1}^n\delta _{gz_i^{\sigma }}\in \widetilde {V}_{\sigma (g)}$ . That is,

$$ \begin{align*}\frac{1}{n}\sum_{i=1}^n\delta_{gz_i^{\sigma}}(V_{\sigma(g)})>1-\frac{1}{2n},\end{align*} $$

which implies $gz_{i}^{\sigma }\in V_{\sigma (g)}$ for every $i\in [n]$ . In particular,

$$ \begin{align*}z_{1}^{\sigma}\in \pi^{-1}(y_1)\cap\bigcap_{g\in I}g^{-1}V_{\sigma(g)}\end{align*} $$

for every $\sigma \in \{1,2\}^I$ . Thus, $\{V_1,V_2\}$ is independent with respect to $\pi $ . This ends our proof.

4 $\pi $ is rel-u.p.e. if and only if $\widetilde {\pi }$ is rel-u.p.e

In this section, we will prove $\pi $ is rel-u.p.e. if and only if $\widetilde {\pi }$ is rel-u.p.e. We need the following lemma.

Lemma 4.1. Let $\pi : X\to Y$ be a continuous surjective map between two compact metric spaces, $\widetilde {\pi }: \mathcal {M}(X)\to \mathcal {M}(Y)$ be the map induced by $\pi $ and $\widetilde {\pi }_n:\mathcal {M}_n(X)\to \mathcal {M}_n(Y)$ be the restriction of $\widetilde {\pi }$ on $\mathcal {M}_n(X)$ . Then $\bigcup _{n\in \mathbb {N}}R_{\widetilde {\pi }_n}$ is dense in $R_{\widetilde {\pi }}$ .

Proof. Fix compatible metrics $\rho _X$ for X and $\rho _Y$ for Y. Let $(\mu _1,\mu _2)\in R_{\widetilde {\pi }}$ . Without loss of generality, we can assume $\mu _1\neq \mu _2$ . For any two disjoint open subsets $\widetilde {V}_1$ , $\widetilde {V}_2$ of $\mathcal {M}(X)$ with $\mu _i\in \widetilde {V}_i$ for $i\in [2]$ , by (2.1), there exist a constant $r>0$ small enough, integers $L_1$ and $L_2$ , $f_1,\ldots ,f_{L_1}\in C(X)$ and $g_1,\ldots ,g_{L_2}\in C(X)$ such that

$$ \begin{align*}\mu_1\in\widetilde{W}_1:=\bigg\{\mu\in\mathcal{M}(X):\bigg\vert\int_X f_i\,d\mu-\int_X f_i\,d\mu_1\bigg\vert<r,i\in [L_1]\bigg\}\subseteq\widetilde{V_1}\end{align*} $$

and

$$ \begin{align*}\mu_2\in\widetilde{W}_2:=\bigg\{\mu\in\mathcal{M}(X):\bigg\vert\int_X g_j\,d\mu-\int_X g_j\,d\mu_2\bigg\vert<r,j\in[L_2]\bigg\}\subseteq\widetilde{V_2}.\end{align*} $$

It is sufficient to prove that $(\widetilde {W}_1\times \widetilde {W}_2)\cap R_{\widetilde {\pi }_N}\neq \emptyset $ for some $N\in \mathbb {N}$ .

Without loss of generality, we can assume $\Vert f_i\Vert \leq 1$ and $\Vert g_j\Vert \leq 1$ for $i\in [L_1]$ and $j\in [L_2]$ . Moreover, since $f_i,g_j\in C(X)$ for $i\in [L_1]$ and $j\in [L_2]$ , there exists $\varepsilon>0$ such that for any $x,z\in X$ with $\rho _X(x,z)<\varepsilon $ , one has

(4.1) $$ \begin{align} \begin{array}{l}\vert f_i(x)-f_i(z)\vert<\dfrac{r}{2}\quad \text{for every } i\in [L_1], \\[6pt] \vert g_j(x)-g_j(z)\vert<\dfrac{r}{2}\quad \text{for every } j\in[L_2]. \end{array}\end{align} $$

For every $y\in Y$ , since $\pi $ is continuous, one can find an open neighbourhood $V_y\subseteq Y$ such that

$$ \begin{align*}\pi^{-1}(y)\subseteq\pi^{-1}(V_y)\subseteq\overline{\pi^{-1}(V_y)}\subseteq(\pi^{-1}(y))^{{\varepsilon}/{2}}, \end{align*} $$

where $(\pi ^{-1}(y))^{{\varepsilon }/{2}}=\{x\in X: \rho _X(x,\{\pi ^{-1}(y)\})<{\varepsilon }/{2}\}$ . Moreover, since Y is compact, there exist $K\in \mathbb {N}$ and pairwise different points $y_1,\ldots ,y_K$ of Y such that $Y=\bigcup _{i=1}^{K}V_{y_i}$ . Then one can find $t>0$ such that $y_i\in B_{\rho _Y}(y_i,t)\subset V_{y_i}$ for any $i\in [K]$ and $\{B_{\rho _Y}(y_1,t),\ldots ,B_{\rho _Y}(y_K,t)\}$ are pairwise disjoint. We set

$$ \begin{align*}W_1= V_{y_1}\setminus\bigcup_{i=2}^{K}B_{\rho_Y}(y_i,t) \quad \text{and}\quad W_i= V_{y_i}\setminus\bigg(\bigcup_{j=1}^{i-1}V_{y_j}\cup\bigcup_{j=i+1}^{K}B_{\rho_Y}(y_j,t)\bigg) \end{align*} $$

for $i=2,\ldots ,K$ . Then $\{W_1,\ldots ,W_K\}$ is a partition of Y and $y_i\in W_i\subseteq V_{y_i}$ for $i\in [K]$ . Moreover, $\{\pi ^{-1}(W_1),\ldots ,\pi ^{-1}(W_K)\}$ is a partition of X which satisfies

$$ \begin{align*} \pi^{-1}(y_i)\subseteq\pi^{-1}(W_i)\subseteq \overline{\pi^{-1}(V_{y_i})}\subseteq(\pi^{-1}(y_i))^{{\varepsilon}/{2}}\end{align*} $$

for every $i\in [K]$ . Then for every $i\in [K]$ , there exist $P_i\in \mathbb {N}$ and pairwise different $x_1^i,x_2^i,\ldots ,x_{P_i}^i\in \pi ^{-1}(y_i)$ , such that $\{x_j^i: j\in [P_i]\}$ is a ${\varepsilon }/{2}$ -net of $\pi ^{-1}(W_i)$ . Then one can choose Borel subsets $A_j^i$ of X for $i\in [K]$ and $j\in [P_i]$ , such that:

1. (i) $\mathrm{diam}(A_j^i)<\varepsilon $ for every $i\in [K],j\in [P_i]$ ;

2. (ii) $x_j^i\in A_j^i$ for every $i\in [K], j\in [P_i]$ ;

3. (iii) $\{A_j^i: j\in [P_i]\} $ is a partition of $\pi ^{-1}(W_i)$ for every $i\in [K]$ .

For every $i\in [K]$ , $j\in [P_i]$ , we set $a_{ij}=\mu _{1}(A_j^{i})$ and $b_{ij}=\mu _{2}(A_j^{i})$ . Since $\widetilde {\pi }(\mu _1)=\widetilde {\pi }(\mu _2)$ , we have

$$ \begin{align*}\sum_{j=1}^{P_i}a_{ij}=\sum_{j=1}^{P_i}\mu_{1}(A_j^{i})=\mu_{1}(\pi^{-1}(W_i))=\mu_{2}(\pi^{-1}(W_i))=\sum_{j=1}^{P_i}\mu_{2}(A_j^{i})=\sum_{j=1}^{P_i}b_{ij}\end{align*} $$

for $i\in [K]$ . Then for any $i\in [K]$ and $j\in [P_i]$ , there exist integers $q_{ij}$ , $\tilde {q}_{ij}$ , $Q_i$ and $N\in \mathbb {N}$ large enough satisfying the following conditions:

• (i*) ${q_{ij}}/{N}\leq a_{ij}<({q_{ij}+1}/{N})$ ;

• (ii*) ${\tilde {q}_{ij}}/{N}\leq b_{ij}<{\tilde {q}_{ij}+1}/{N}$ ;

• (iii*) ${Q_i}/{N}\leq \sum _{j=1}^{P_i}a_{ij}=\sum _{j=1}^{P_i}b_{ij}<({Q_i+1})/{N}$ .

Now, we choose an $x_0\in X$ arbitrarily and set

$$ \begin{align*}\widetilde{\mu}_1=\frac{1}{N}\bigg(\sum_{i=1}^{K}\bigg(\sum_{j=1}^{P_i-1}q_{ij}\delta_{x_{j}^{i}}+\bigg(Q_i-\sum_{j=1}^{P_i-1}q_{ij}\bigg)\delta_{x_{P_i}^{i}}\bigg)\bigg)+\frac{N-\sum_{i=1}^{K}Q_{i}}{N}\delta_{x_0}\end{align*} $$

and

$$ \begin{align*}\widetilde{\mu}_2=\frac{1}{N}\bigg(\sum_{i=1}^{K}\bigg(\sum_{j=1}^{P_i-1}\tilde{q}_{ij}\delta_{x_{j}^{i}}+\bigg(Q_i-\sum_{j=1}^{P_i-1}\tilde{q}_{ij}\bigg)\delta_{x_{P_i}^{i}}\bigg)\bigg)+\frac{N-\sum_{i=1}^{K}Q_{i}}{N}\delta_{x_0}.\end{align*} $$

It is clear that $(\widetilde {\mu }_1,\widetilde {\mu }_2)\in R_{\widetilde {\pi }_N}$ . Now we shall show that $\widetilde {\mu }_i\in \widetilde {W_i}$ for $i\in [2]$ .

In fact, for any $\ell \in [L_1]$ , one has

(4.2) $$ \begin{align} \bigg\vert\int f_{\ell}\,d\mu_1-\int f_{\ell}\,d\tilde{\mu}_1\bigg\vert&= \bigg\vert\sum_{i=1}^{K}\sum_{j=1}^{P_i}\int_{A_{j}^{i}}f_l\,d\mu_1-\frac{1}{N}\sum_{i=1}^{K}\bigg(\sum_{j=1}^{P_i-1}q_{ij}f_l(x_{j}^{i})\nonumber\\ &\quad+\bigg(Q_i-\sum_{j=1}^{P_i-1}q_{ij}\bigg)f_l(x_{P_i}^{i})\bigg)-\frac{N-\sum_{i=1}^{K}Q_{i}}{N}f_l(x_0)\bigg\vert\nonumber\\ &\leq \bigg\vert\sum_{i=1}^{K}\sum_{j=1}^{P_i}\int_{A_{j}^{i}}f_l\,d\mu_1-\frac{1}{N}\sum_{i=1}^{K}\sum_{j=1}^{P_i}q_{ij}f_l(x_{j}^{i})\bigg\vert\nonumber\\ &\quad+\bigg\vert\frac{1}{N}\sum_{i=1}^{K}(Q_i-\sum_{j=1}^{P_i}q_{ij})f_l(x_{P_i}^{i})\bigg\vert+\bigg\vert\frac{N-\sum_{i=1}^{K}Q_{i}}{N}f_l(x_0)\bigg\vert. \end{align} $$

Since $\mathrm {diam}(A_j^{i})<\varepsilon $ for $i\in [K]$ and $j\in [P_i]$ , by (4.1) and $(\text {i}^*)$ , we have

(4.3) $$ \begin{align} &\bigg\vert\sum_{i=1}^{K}\sum_{j=1}^{P_i}\int_{A_{j}^{i}}f_l(x)\,d\mu_1-\frac{1}{N}\sum_{i=1}^{K}\sum_{j=1}^{P_i}q_{ij}f_l(x_{j}^{i})\bigg\vert\nonumber\\ &\quad\leq\sum_{i=1}^{K}\sum_{j=1}^{P_i}\int_{A_{j}^{i}}\vert f_l(x)-f_l(x_{j}^{i})\vert\, d\mu_1+\sum_{i=1}^{K}\sum_{j=1}^{P_i}\bigg(a_{ij}-\frac{q_{ij}}{N}\bigg)\vert f_l(x_{j}^{i})\vert\nonumber\\ &\quad\leq\frac{r}{2}+\frac{\sum_{i=1}^{K}P_i}{N}. \end{align} $$

By $(\text {i}^*)$ and $(\text {iii}^*)$ , one has

(4.4) $$ \begin{align} \sum_{i=1}^{K}\bigg\vert\frac{Q_{i}}{N}-\frac{1}{N}\sum_{j=1}^{P_i}q_{ij}\bigg\vert &\leq\sum_{i=1}^{K}\bigg\vert\frac{Q_{i}}{N}-\sum_{j=1}^{P_i}a_{ij}\bigg\vert+\sum_{i=1}^{K}\bigg\vert\sum_{j=1}^{P_i}a_{ij}-\frac{1}{N}\sum_{j=1}^{P_i}q_{ij}\bigg\vert\nonumber \\ &\leq\frac{K}{N}+\frac{\sum_{i=1}^{K}P_i}{N} \end{align} $$

and

$$ \begin{align*} \bigg\vert\frac{N-\sum_{i=1}^{K}Q_{i}}{N}\bigg\vert =\bigg\vert\sum_{i=1}^{K}\sum_{j=1}^{P_i}a_{ij}-\frac{1}{N}\sum_{i=1}^{K}Q_{i}\bigg\vert\leq\sum_{i=1}^{K}\bigg\vert\sum_{j=1}^{P_i}a_{ij}-\frac{Q_{i}}{N}\bigg\vert\leq\frac{K}{N}. \end{align*} $$

When N is large enough such that ${K}/{N}+{\sum _{i=1}^{K}P_i}/{N}\leq {r}/{6},$ by (4.2), (4.3) and (4.4), we have $\widetilde {\mu }_1\in \widetilde {W_1}$ . Similarly, we can prove that $\widetilde {\mu }_2\in \widetilde {W_2}$ . This ends our proof.

Theorem 4.2. Let $\pi :(X,G)\to (Y,G)$ be a factor map between two G-systems with $\mathrm {supp}(Y)=Y$ and $\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$ be the induced map of $\pi $ . Suppose $\pi $ has rel-u.p.e., then $\widetilde {\pi }$ also has rel-u.p.e.

Proof. Assume that $\pi $ has rel-u.p.e. To show $\widetilde {\pi }$ has rel-u.p.e., it suffices to prove that $R_{\widetilde {\pi }}\setminus \Delta (\mathcal {M}(X))\subseteq E(\mathcal {M}(X),G\vert \widetilde {\pi })$ . Let $(\mu _1,\mu _2)\in R_{\widetilde {\pi }}\setminus \Delta (\mathcal {M}(X))$ and $\widetilde {V_1}$ , $\widetilde {V_2}$ be two disjoint open subsets of $\mathcal {M}(X)$ with $\mu _i\in \widetilde {V}_i$ for $i\in [2]$ . By Lemma 4.1, there exist $n\in \mathbb {N}$ and $(\mu _1',\mu _2')\in R_{\widetilde {\pi }_n}\cap (\widetilde {V_1}\times \widetilde {V_2})$ . Notice that, since $\pi $ has rel-u.p.e., by Theorem 3.5, $\widetilde {\pi }_n$ has rel-u.p.e. Then $\{\widetilde {V_1}\cap \mathcal {M}_n(X),\widetilde {V_2}\cap \mathcal {M}_n(X)\}$ is independent with respect to $\widetilde {\pi }_n$ , which implies $\{\widetilde {V_1},\widetilde {V_2}\}$ is independent with respect to $\widetilde {\pi }$ . This ends our proof.

We note that for any non-empty finite subsets A, H of $\mathbb {N}$ with $A\subseteq H$ and $S\subseteq \{1,2\}^H$ , one can find $S_0\subset S$ with $\vert S_0\vert \geq {\vert S\vert }/{ 2^{\vert H\vert -\vert A\vert }}$ such that for every $\sigma _1\neq \sigma _2\in S_0$ , there exists $a\in A$ with

(4.5) $$ \begin{align} \sigma_1(a)\neq\sigma_2(a). \end{align} $$

In fact, if we let $\mathcal {W}=S\vert _A$ , then $\vert \mathcal {W}\vert \geq {\vert S\vert }/{2^{\vert H\vert -\vert A\vert }}$ . For each $w\in \mathcal {W}$ , there exists $\sigma _w\in S$ such that $\sigma _w\vert _A=w$ . Put $S_0:=\{\sigma _w: w\in \mathcal {W}\}\subseteq S$ . Then $\vert S_0\vert =\vert \mathcal {W}\vert \geq {\vert S\vert }/{2^{\vert H\vert -\vert A\vert }}$ , and for every $\sigma _1\neq \sigma _2\in S_0$ , one has $\sigma _1\vert _A\neq \sigma _2\vert _A$ .

Theorem 4.3. Let $\pi :(X,G)\to (Y,G)$ be a factor map between two G-systems and $\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$ be the induced map of $\pi $ . If $\widetilde {\pi }$ has rel-u.p.e., then so does $\pi $ .

Proof. Assume that $\widetilde {\pi }$ has rel-u.p.e. To show $\pi $ has rel-u.p.e., we shall show that $R_{\pi }\setminus \Delta (X)\subseteq E(X,G\vert \pi )$ . Let $(x_1,x_2)\in R_{\pi }\setminus \Delta (X)$ , $V_1,V_2$ be two non-empty disjoint open subsets of X with $x_i\in V_i$ for $i\in [2]$ . By Corollary 3.3, it is sufficient to show that $(V_1,V_2)$ is independent with respect to $\pi $ .

Take $\epsilon \in (0,\tfrac 12)$ with

(4.6) $$ \begin{align} 2^{1-\epsilon^2}\cdot (1-\epsilon^2)^{(1-\epsilon^2)}\cdot(\epsilon^2)^{(\epsilon^2)}>1. \end{align} $$

We set

(4.7) $$ \begin{align} &\widetilde{V_i}=\{\mu\in\mathcal{M}(X): \mu(V_i)> 1-\epsilon^{4}\} \end{align} $$

for $i\in [2]$ . Clearly, $\delta _{x_i}\in \widetilde {V}_i$ . Since $(\delta _{x_1},\delta _{x_2})\in R_{\widetilde {\pi }}$ and $\widetilde {\pi }$ has rel-u.p.e., $(\widetilde {V}_1,\widetilde {V}_2)$ is independent with respect to $\widetilde {\pi }$ . That is, there exists $c>0$ such that for every $F\in \mathcal {F}(G)$ , there exists an independence set $E\subseteq F$ of $(\widetilde {V}_1,\widetilde {V}_2)$ with respect to $\widetilde {\pi }$ with $\vert E\vert>c\vert F\vert $ .

Fix an $F\in \mathcal {F}(G)$ and an independence set $E\subseteq F$ of $(\widetilde {V}_1,\widetilde {V}_2)$ with respect to $\widetilde {\pi }$ with $\vert E\vert>c\vert F\vert $ . Then there exists $\nu \in \mathcal {M}(Y)$ , such that for every $\sigma \in \{1,2\}^E$ ,

(4.8) $$ \begin{align} \boldsymbol{\widetilde{V}}_{\sigma}:=\bigg(\bigcap_{g\in E}g^{-1}\widetilde{V}_{\sigma(g)}\bigg)\cap\widetilde{\pi}^{-1}(\nu)\neq\emptyset. \end{align} $$

For every $\sigma \in \{1,2\}^{E}$ , we take $\mu _{\sigma }\in \boldsymbol {\widetilde {V}}_{\sigma }$ . Then $\mu _{\sigma }\in g^{-1}\widetilde {V}_{\sigma (g)}$ for every $g\in E$ and $\sigma \in \{1,2\}^{E}$ , which implies $\mu _{\sigma }(g^{-1}V_{\sigma })>1-\epsilon ^4$ for every $g\in E$ and $\sigma \in \{1,2\}^E$ . Thus,

$$ \begin{align*}\int_{X}\frac{1}{\vert E\vert}\sum_{g\in E}1_{g^{-1}V_{\sigma(g)}}(x)\,d{ \mu_{\sigma}}=\frac{1}{|E|}\sum_{g\in E}\mu_{\sigma}(g^{-1}V_{\sigma(g)})> 1-\epsilon^{4}\end{align*} $$

and $\mu _{\sigma }(\widetilde {X}_{\sigma })> 1-\epsilon ^{2}$ for every $\sigma \in \{1,2\}^{E}$ , where

(4.9) $$ \begin{align} \widetilde{X}_{\sigma}=\bigg\{x\in X: \frac{1}{|E|}\sum_{g\in E}1_{g^{-1}V_{\sigma(g)}}(x)> 1-\epsilon^{2}\bigg\}. \end{align} $$

By the inner regular of measure, we can find a closed subset

(4.10) $$ \begin{align} X_{\sigma}\subseteq\widetilde{X}_{\sigma} \quad \text{with} \quad \mu_{\sigma}(X_{\sigma})> 1-\epsilon^{2} \end{align} $$

for every $\sigma \in \{1,2\}^{E}$ . Since $\pi $ is continuous, for every $\sigma \in \{1,2\}^E$ , we have

(4.11) $$ \begin{align} Y_{\sigma}:=\pi(X_{\sigma}) \end{align} $$

is a closed subset of Y and

$$ \begin{align*}\nu(Y_{\sigma})=\widetilde{\pi}\mu_{\sigma}(Y_{\sigma})\geq\mu_{\sigma}(X_{\sigma})> 1-\epsilon^{2}.\end{align*} $$

Then

$$ \begin{align*}\int_{Y}\frac{1}{2^{|E|}}\sum_{\sigma\in \{1,2\}^{E}}1_{Y_{\sigma}}(y)\,d \nu> 1-\epsilon^{2}.\end{align*} $$

Put

(4.12) $$ \begin{align} Y^{'}:=\bigg\{y\in Y: \frac{1}{2^{|E|}}\sum_{\sigma\in \{1,2\}^{E}}1_{Y_{\sigma}}(y)> 1-\epsilon\bigg\}, \end{align} $$

then $\nu (Y^{'})> 1-\epsilon >\tfrac 12$ .

Now, we fix a point $y_0\in Y^{'}$ and set

(4.13) $$ \begin{align} \mathcal{E}:=\{\sigma\in\{1,2\}^{E}: y_0\in Y_{\sigma}\}. \end{align} $$

Then $|\mathcal {E}|>(1-\epsilon )\cdot 2^{\vert E\vert }$ by (4.12). For any $\sigma \in \mathcal {E}$ , by (4.13), (4.11), (4.10) and (4.9), there is $x_{\sigma }\in X_{\sigma }$ with

$$ \begin{align*}\frac{1}{|E|}\sum_{g\in E}1_{g^{-1}V_{\sigma(g)}}(x_{\sigma})>1-\epsilon^2\end{align*} $$

such that $\pi (x_{\sigma })=y_0$ . For every $\sigma \in \mathcal {E}$ , we set

$$ \begin{align*}A(\sigma)=\{g\in E: x_{\sigma}\in g^{-1}V_{\sigma(g)}\},\end{align*} $$

then $\vert A(\sigma )\vert>(1-\epsilon ^2)\vert E\vert $ . Now we define

$$ \begin{align*}\Omega:=\{H\subseteq E:\vert H\vert=\lfloor (1-\epsilon^2)\cdot\vert E\vert\rfloor\}\end{align*} $$

and

$$ \begin{align*}\mathcal{Q} (H)=\{\sigma\in\mathcal{E}:H\subseteq\{g\in E:gx_{\sigma}\in V_{\sigma(g)}\}\}\end{align*} $$

for every $H\in \Omega $ . Then $\vert \Omega \vert =\tbinom {\vert E\vert }{\lfloor (1-\epsilon ^2)\cdot \vert E\vert \rfloor }$ and $\bigcup _{H\in \Omega }\mathcal {Q} (H)=\mathcal {E}$ . Thus, there exists $H_0\in \Omega $ such that $|\mathcal {Q} (H_0)|\geq {|\mathcal {E}|}/{|\Omega |}\geq {(1-\epsilon )2^{|E|}}/{\tbinom {\vert E\vert }{\lfloor (1-\epsilon ^2)\cdot \vert E\vert \rfloor }}.$ By (4.5), we can choose $S\subseteq \mathcal {Q}(H_0)$ such that

(4.14) $$ \begin{align} |S|\geq\frac{(1-\epsilon)2^{|E|}}{2^{|E|-\lfloor (1-\epsilon^2)\cdot\vert E\vert\rfloor} \cdot\tbinom{|E|}{\lfloor (1-\epsilon^2)\cdot\vert E\vert\rfloor} } \end{align} $$

and for any $\sigma '\neq \sigma "\in S$ , there exists $g\in H_0$ that satisfies $\sigma '(g)\neq \sigma "(g)$ . That is, $\vert S\vert _{H_0}\vert =\vert S\vert $ . Let $t=1-\epsilon ^2$ and $\unicode{x3bb} =\log _2(2^t\cdot t^t\cdot (1-t)^{(1-t)})>0$ . Then by Stirling’s formula, when $\vert E\vert $ is large enough, one has

$$ \begin{align*} &\vert S\vert_{H_0}\vert \approx 2^{\lfloor t\vert E\vert\rfloor-1}\cdot \sqrt{2\pi t(1-t)\vert E\vert}\cdot t^{t\vert E\vert}\cdot(1-t)^{(1-t)\vert E\vert}\\ &\quad\geq 2^{t\vert E\vert-2}\cdot \sqrt{2\pi t(1-t)\vert E\vert}\cdot t^{t\vert E\vert}\cdot (1-t)^{(1-t)\vert E\vert}\\ &\quad\geq (2^t\cdot t^t\cdot(1-t)^{(1-t)})^{\vert E\vert}>2^{\unicode{x3bb}\vert E\vert}. \end{align*} $$

By Lemma 3.4, there exists a subset $H_1\subseteq H_0$ with $\vert H_1\vert>d\vert H_0\vert $ such that $S\vert _{H_1}=\{1,2\}^{H_1}$ , where d is a positive constant independent with $ E$ when $\vert E\vert $ is large enough. By Remark 3.1, $(\mathcal {V}_1,\mathcal {V}_2)$ is independent with respect to $\pi $ . This ends our proof.

5 $\pi $ is open if and only if $\widetilde {\pi }$ is open

In this section, we will prove Theorem 1.2. In fact, we have the following result.

Theorem 5.1. Let $\pi : X\to Y$ be a surjective continuous map between two compact metrizable spaces, $\widetilde {\pi }: \mathcal {M}(X)\to \mathcal {M}(Y)$ be the induced map of $\pi $ and $\widetilde {\pi }_n: \mathcal {M}_n(X)\to \mathcal {M}_n(Y)$ be the restriction of $\widetilde {\pi }$ on $\mathcal {M}_n(X)$ . Then the following are equivalent:

1. (1) $\pi $ is open;

2. (2) $\widetilde {\pi }$ is open;

3. (3) $\widetilde {\pi }_n$ is open for each $n\in \mathbb {N}$ ;

4. (4) $\widetilde {\pi }_n$ is open for some $n\in \mathbb {N}$ .

Proof. $(3)\Rightarrow (4)$ is trivial. We will show $(2)\Rightarrow (1)$ , $(4)\Rightarrow (1)$ , $(1)\Rightarrow (3)$ and $(1)\Rightarrow (2)$ . Fix compatible metrics $\rho _X$ for X and $\rho _Y$ for Y.

(2) $\Rightarrow $ (1). Suppose that $\widetilde {\pi }$ is open. For every non-empty open subset U of X, we shall show that $\pi (U)$ is an open subset of Y. That is, for every $y\in \pi (U)$ , there exists $r>0$ such that $B_{\rho _Y}(y,r)\subseteq \pi (U)$ .

Now fix $y_0\in \pi (U)$ . Since U is open, there exist $x_0\in U$ and $\delta>0$ with $\pi (x_0)=y_0$ and $\overline {B_{\rho _X}(x_0,\delta )}\subseteq U$ . Then by Urysohn’s lemma, there exists a countinuous map $f: X\to [0,1]$ with $f(z)=1$ when $z\in B_{\rho _X}(x_0,{\delta}/{2})$ and $f(z)=0$ when $z\in X\backslash B_{\rho _X}(x_0,\delta )$ . We set

$$ \begin{align*}\widetilde{U}:=\bigg\{\mu\in\mathcal{M}(X): \int f\,d\mu>\frac{2}{3}\bigg\}.\end{align*} $$

Clearly, $\widetilde {U}$ is an open subset of $\mathcal {M}(X)$ and $\delta _{x_0}\in \widetilde {U}$ .

Since $\widetilde {\pi }$ is open, $\widetilde {\pi }(\widetilde {U})$ is an open subset of $\mathcal {M}(Y)$ . Note that $\delta _{y_0}=\widetilde {\pi }(\delta _{x_0})\in \widetilde {\pi }(\widetilde {U})$ . Thus, there exists $r>0$ such that

$$ \begin{align*}\{\delta_{y}: \ y\in B_{\rho_Y}(y_0,r)\}\subset \widetilde{\pi}(\widetilde{U}).\end{align*} $$

Then for every $y'\in B_{\rho _{Y}}(y_0,r)$ , there exists $\mu _{y'}\in \widetilde {U}$ such that $\widetilde {\pi }(\mu _{y'})=\delta _{y'}$ . On the one hand, since $\mu _{y'}(\{\pi ^{-1}(y')\})=\delta _{y'}(\{y'\})=1$ , we have

(5.1) $$ \begin{align} \mathrm{supp}(\mu_{y'})\subseteq\pi^{-1}(\{y'\}). \end{align} $$

On the other hand, since $\mu _{y'}\in \widetilde {U}$ , we have $\int f\,d\mu _{y'}>\tfrac 23$ . Thus,

$$ \begin{align*}\emptyset\neq \mathrm{supp}(\mu_{y'})\cap B_{\rho_X}(x_0,\delta)\subseteq \mathrm{supp}(\mu_{y'})\cap U.\end{align*} $$

By (5.1), we have $U\cap \pi ^{-1}(\{y'\})\neq \emptyset $ . That is, $y'\in \pi (U)$ . Then by the arbitrariness of $y'\in B_{\rho _{Y}}(y_0,r)$ , one has $B_{\rho _{Y}}(y_0,r)\subseteq \pi (U)$ . Thus, $\pi (U)$ is an open subset of Y and $\pi $ is open.

(4) $\Rightarrow $ (1). We assume that there exists $n\in \mathbb {N}$ such that $\widetilde {\pi }_n$ is open. Let U be an open subset of X. We shall show that for every $y\in \pi (U)$ , there exists $r>0$ such that $y\in B_{\rho _Y}(y,r)\subseteq \pi (U)$ .

Let $y\in \pi (U)$ , there exists $x\in U$ with $\pi (x)=y$ . We set

$$ \begin{align*}\widetilde{U}=\mathcal{M}_n(X)\cap\{\mu\in\mathcal{M}(X): \mu(U)>0\}.\end{align*} $$

Here, $\widetilde {U}$ is an open subset of $\mathcal {M}_n(X)$ which contains $\delta _x$ . Since $\widetilde {\pi }_n$ is open, $\widetilde {\pi }_n(\widetilde {U})$ is open which contains $\delta _y$ . Then there exists $r>0$ such that $\{\delta _{z}: \rho _{Y}(z,y)<r\}\subseteq \widetilde {\pi }_n(\widetilde {U})$ . Hence, for every $z\in B_{\rho _Y}(y,r)$ , there exist $x_1,x_2,\ldots ,x_n\in X$ such that

$$ \begin{align*}\mu=\frac{1}{n}\sum_{i=1}^n\delta_{x_i}\in\widetilde{U}\quad \text{and}\quad \widetilde{\pi}_n(\mu)=\delta_z.\end{align*} $$

Then one has $\pi (x_i)=z$ for every $i\in [n]$ . Since $\mu \in \widetilde {U}$ , there exists $i_0\in [n]$ with $x_{i_0}\in U$ . That is, $z=\pi (x_{i_0})\in \pi (U)$ . Hence, $B_{\rho _Y}(y,r)\subseteq \pi (U)$ . This implies $\pi $ is open.

(1) $\Rightarrow $ (3). Now we assume that $\pi $ is open. Let $n\in \mathbb {N}$ and $\widetilde {U}$ be an open subset of $\mathcal {M}_n(X)$ . We shall show that for every $\nu \in \widetilde {\pi }_n(\widetilde {U})\subseteq \mathcal {M}_n(Y)$ , there exists an open neighbourhood of $\nu $ in $\mathcal {M}_n(Y)$ contained in $\widetilde {\pi }_n(\widetilde {U})$ .

For any $\nu \in \widetilde {\pi }_n(\widetilde {U})\subseteq \mathcal {M}_n(Y)$ , there exist positive integers h, $k_1,k_2,\ldots ,k_h$ with $\sum _{i\in [h]}k_i=n$ and pairwise distinct $y_1,y_2,\ldots ,y_h\in Y$ such that

$$ \begin{align*}\nu=\frac{1}{n}(k_1\delta_{y_1}+k_2\delta_{y_2}+\cdots+k_h\delta_{y_h}).\end{align*} $$

Since $\nu \in \widetilde {\pi }_n(\widetilde {U})$ , there exists $\mu \in \widetilde {U}\subseteq \mathcal {M}_n(X)$ such that $\widetilde {\pi }_n(\mu )=({1}/{n})\sum _{i=1}^{h}k_i\delta _{y_i}$ . Then for every $i\in [h]$ , there exist integers $\ell _i$ , $m_{i,j}$ , and points $x_{i,j}\in X$ for $j\in [\ell _i]$ satisfying:

1. (a) $m_{i,1}+m_{i,2}+\cdots +m_{i,\ell _i}=k_i$ for every $i\in [h]$ ;

2. (b) $x_{i,1},x_{i,2},\ldots ,x_{i,\ell _i}$ are pairwise distinct and $\pi (x_{i,j})=y_i$ for every $i\in [h]$ and $j\in [\ell _i]$ ;

3. (c) $\mu =({1}/{n})\sum \nolimits _{i\in [h]}\sum \nolimits _{j\in [\ell _i]}m_{i,j}\delta _{x_{i,j}}$ .

Since $\widetilde {U}$ is an open neighbourhood of $\mu $ , there exists $r_0>0$ such that if $z_{i,j}^1,z_{i,j}^2,\ldots , z_{i,j}^{m_{i,j}}\in B_{\rho _X}(x_{i,j},r_0)$ for every $i\in [h]$ , $j\in [\ell _i]$ , then

(5.2) $$ \begin{align} \frac{1}{n}\sum\limits_{i\in[h]}\sum\limits_{j\in[\ell_i]}\bigg(\sum_{t\in[m_{i,j}]}\delta_{z_{i,j}^t}\bigg)\in\widetilde{U}. \end{align} $$

Note that $y_1,y_2,\ldots ,y_{h}$ are pairwise distinct, then there exists $\delta>0$ such that $\{B_{\rho _Y}(y_i,\delta )\}_{i\in [h]}$ are pairwise disjoint. By item (b) and the continuity of $\pi $ , there exists $r\in (0,r_0)$ such that

(5.3) $$ \begin{align} \pi(B_{\rho_X}(x_{i,j},r))\subseteq B_{\rho_X}(y_i,\delta)\ \ \ \text{and}\ \ \ B_{\rho_X}(x_{i,t},r)\cap B_{\rho_X}(x_{i,j},r)=\emptyset \end{align} $$

for every $i\in [h]$ and different $j,t\in [\ell _i]$ .

Since $\pi $ is open, $\bigcap _{j=1}^{\ell _i}\pi (B_{\rho _X}(x_{i,j},r))$ for every $i\in [h]$ is open. We set

$$ \begin{align*}\widetilde{V}:=\bigg\{\tau\in\mathcal{M}_n(Y):\tau\bigg(\bigcap_{j=1}^{\ell_i} \pi(B_{\rho_X}(x_{i,j},r))\bigg)>\frac{k_i}{n}-\frac{1}{2n},\ i\in[h]\bigg\}.\end{align*} $$

It is an open subset of $\mathcal {M}_n(Y)$ . Moreover, for every $i_0\in [h]$ ,

$$ \begin{align*} \nu\bigg(\bigcap_{j=1}^{\ell_{i_0}}\pi(B_{\rho_X}(x_{i_0,j},r))\bigg) &=\frac{1}{n}\sum_{i\in[h]}k_i\delta_{y_i}\bigg(\bigcap_{j=1}^{\ell_{i_0}}\pi(B_{\rho_X}(x_{i_0,j},r))\bigg)\nonumber\\ &=\frac{1}{n}k_{i_0}\delta_{y_{i_0}}\bigg(\bigcap_{j=1}^{\ell_{i_0}}\pi(B_{\rho_X}(x_{i_0,j},r))\bigg)\nonumber\\ &=\frac{1}{n}k_{i_0}>\frac{k_{i_0}}{n}-\frac{1}{2n}. \end{align*} $$

Thus, $\nu \in \widetilde {V}$ . Next, we shall show that $\widetilde {V}\subseteq \widetilde {\pi }_n(\widetilde {U})$ .

Now fix any $\tau \in \widetilde {V}\subseteq \mathcal {M}_n(Y)$ . We have $\tau =({1}/{n})\sum \nolimits _{s=1}^{n}\delta _{u_{s}}$ for some $u_{s}\in Y$ . For every $i\in [h]$ , we set

(5.4) $$ \begin{align} L(i):=\bigg\{s\in[n]: u_{s}\in \bigcap_{j=1}^{\ell_i} \pi(B_{\rho_X}(x_{i,j},r))\bigg\}. \end{align} $$

By $\tau \in \widetilde {V}$ , one has

(5.5) $$ \begin{align} \vert L(i)\vert=n\cdot\tau\bigg(\bigcap_{j=1}^{\ell_i} \pi(B_{\rho_X}(x_{i,j},r))\bigg)>k_i-\frac{1}{2} \end{align} $$

for every $i\in [h]$ . Since $\vert L(i)\vert \in \mathbb {N}$ , by (5.5), $\vert L(i)\vert \geq k_i$ . We note that $L(i)$ , $i\in [h]$ are pairwise disjoint since $\bigcap _{j=1}^{\ell _i}\pi (B_{\rho _X}(x_{i,j},r))$ , $i\in [h]$ are pairwise disjoint. Moreover, by $\sum _{i\in [h]}k_i=n$ , one has $\vert L(i)\vert =k_i$ for every $i\in [h]$ . Hence,

(5.6) $$ \begin{align} \bigcup_{i\in[h]}L(i)=\bigsqcup_{i\in[h]} L(i)=[n]\quad\text{and}\quad \tau=\frac{1}{n}\sum_{i\in[h]}\bigg(\sum_{s\in L(i)}\delta_{u_s}\bigg). \end{align} $$

For every $i\in [h]$ , since $\vert L(i)\vert =k_i\overset {(a)}{=}\sum _{j\in [\ell _i]}m_{i,j}$ , we can rewrite $L(i)=\{s_1,s_2,\ldots , s_{k_i}\}$ . For every $i\in [h]$ and $j\in [\ell _i]$ , we denote $R_i(j)=\sum _{t=1}^{j}m_{i,t}$ and $R_i(0)=0$ . Then $R_i(\ell _i)=k_i$ . By (5.4), for every $j\in [\ell _i]$ and integer q with $R_i(j-1)+1\leq q\leq R_i(j)$ , there exists $x_{i,q}'\in B(x_{i,j},r)$ such that $\pi (x_{i,q}')=u_{s_q}$ . Then by (5.2), one has

$$ \begin{align*}\mu':=\frac{1}{n}\sum_{i\in[h]}\sum_{j\in[\ell_i]}\sum_{q=R_i(j-1)+1}^{R(j)} \delta_{x_{i,q}'}\in\widetilde{U}\end{align*} $$

and

$$ \begin{align*} \widetilde{\pi}_n(\mu')&=\frac{1}{n}\sum_{i\in[h]}\sum_{j\in[\ell_i]}\sum_{q=R_i(j-1)+1}^{R_i(j)} \delta_{u_{s_q}}=\frac{1}{n}\sum_{i\in[h]}\sum_{q\in[k_i]}\delta_{u_{s_q}}\\ &=\frac{1}{n}\sum_{i\in[h]}\sum_{s\in L(i)}\delta_{u_s}\overset{(5.6)}{=}\tau. \end{align*} $$

This implies $\widetilde {V}\subset \widetilde {\pi }_n(\widetilde {U})$ . Hence, $\widetilde {\pi }_n(\widetilde {U})$ is an open subset of $\mathcal {M}_n(Y)$ and $\widetilde {\pi }_n$ is open.

$(1)\,\Rightarrow (2)$ . Now we assume that $\pi $ is open. Let $\widetilde {U}$ be an open subset of $\mathcal {M}(X)$ . We shall show $\widetilde {\pi }(\widetilde {U})$ is open in $\mathcal {M}(Y)$ .

For every $\nu \in \widetilde {\pi }(\widetilde {U})$ , there exists $\mu \in \widetilde {U}$ such that $\nu =\widetilde {\pi }(\mu )$ . Next we shall show that there exists $\delta>0$ small enough such that if we set

$$ \begin{align*}\widetilde{V}:=\{\tau\in\mathcal{M}(Y): d_{P}(\nu,\tau)<\delta\},\end{align*} $$

where

$$ \begin{align*} {d_P(\tau,\nu):=\mathrm{inf}\{\delta>0:\tau(A)\leq\nu(A^{\delta})+\delta \text{ and } \nu(A)\leq\tau(A^{\delta})+\delta \text{ for all } A\in\mathcal{B}_Y\}},\end{align*} $$

then $\widetilde {V}$ is an open neighbourhood of $\nu $ contained in $\widetilde {\pi }(\widetilde {U})$ .

Since $\widetilde {U}$ is open, by Proposition 2.2, there exist $k\in \mathbb {N}$ and an open set of the form $\mathbb {W}(U_1,U_2,\ldots ,U_k; \ \eta _1,\eta _2,\ldots ,\eta _k)$ of $\mathcal {M}(X)$ , where $U_1,U_2,\ldots ,U_k$ are disjoint non-empty open subsets of X and $\eta _1,\eta _2, \ldots ,\eta _k$ are positive real numbers with $\eta _1+\eta _2+\cdots +\eta _k<1$ , such that

$$ \begin{align*} \mu\in\mathbb{W}(U_1,U_2,\ldots,U_k;\eta_1, \eta_2,\ldots,\eta_k)\subset\overline{\mathbb{W}(U_1,U_2, \ldots,U_k;\eta_1,\eta_2,\ldots,\eta_k)}\subset\widetilde{U}. \end{align*} $$

For any $t_1,t_2\in \{0,1\}^{[k]}$ , we denote $t_1>t_2$ if $t_1\neq t_2$ and $t_1(i)\geq t_2(i)$ for every $i\in [k]$ . For every $\sigma \in \{0,1\}^{[k]}$ , we set

$$ \begin{align*}V_{\sigma}:=\bigcap_{\substack{i\in[k]\\ \sigma(i)=1}}\pi(U_i), \quad V^{\prime}_{\sigma}:=V_{\sigma}\backslash\bigcup\limits_{\substack{\alpha\in\{0,1\}^{[k]}\\\alpha>\sigma}}V_{\alpha}\end{align*} $$

and

(5.7) $$ \begin{align} \mathcal{E}:=\{t\in\{0,1\}^{[k]}: \nu(V_t')>0\}. \end{align} $$

Recall that for any subset A of Y and $a>0$ , we denote $A^{a}=\{y\in Y :\rho _Y(y,A)<a\}$ , where $\rho _Y$ is the compatible metric on Y. For every $i\in [k]$ , since $\pi $ is open and $U_i$ is open in X, then $\pi (U_i)$ is an open subset of Y. Then by inner regularity, there exist $\varepsilon>0$ small enough, $\delta \in (0,\varepsilon )$ and compact subsets $C_i$ of Y for $i\in [k]$ such that:

• (c1) $\mu (U_i)>\eta _i+6^{k}\varepsilon $ for every $i\in [k]$ ;

• (c2) $\nu (V_{\sigma }')>5k\varepsilon $ , for every $\sigma \in \mathcal {E}$ ;

• (c3) $C_i\subseteq C_i^{\delta }\subseteq C_i^{2\delta }\subseteq \pi (U_i)$ for every $i\in [k]$ ;

• (c4) $\nu (C_i)>\nu (\pi (U_i))-\varepsilon $ for $i\in [k]$ .

Now we set

$$ \begin{align*}\widetilde{V}:=\{\tau\in\mathcal{M}(Y): d_{P}(\nu,\tau)<\delta\}.\end{align*} $$

Clearly, $\widetilde {V}$ is an open subset of $\mathcal {M}(Y)$ containing $\nu $ . Now it is sufficient to prove that $\widetilde {V}\subset \widetilde {\pi }(\widetilde {U})$ .

For every $\sigma \in \{0,1\}^{[k]}$ , we set

$$ \begin{align*}C_{\sigma}(\delta):=\bigcap\limits_{\substack{i\in[k]\\\sigma(i)=1}}C_i^{\delta}\quad \text{and} \quad C_{\sigma}':= C_{\sigma}(\delta)\backslash\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\\alpha>\sigma}}C_{\alpha}(\delta).\end{align*} $$

Then for every $\sigma \in \{0,1\}^{[k]}$ , by items (c3) and (c4), we have

(5.8) $$ \begin{align} \nu(V_{\sigma})&=\nu\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}\pi(U_i)\bigg)\nonumber\\ &\overset{(c3)}{=}\nu\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}((\pi(U_i)\backslash C_i)\cup C_i)\bigg) \overset{(c4)}{\leq}\nu\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}C_i\bigg)+k\varepsilon. \end{align} $$

We note that for any $t_1\neq t_2\in \{0,1\}^{[k]}$ , one has $C_{t_1}'\cap C_{t_2}'=\emptyset $ . In fact, if we define $t_1\vee t_2\in \{0,1\}^{[k]}$ by

$$ \begin{align*}(t_1\vee t_2)(i)=\max\{t_1(i),t_2(i)\}\end{align*} $$

for every $i\in [k]$ , then it is clear that $t_1\vee t_2>t_1$ or $t_1\vee t_2>t_2$ . Without loss of generality, we can assume $t_1\vee t_2>t_1$ , then $C_{t_1}'\subseteq C_{t_1}(\delta )\backslash {C_{t_1\vee t_2}(\delta )}$ . However,

$$ \begin{align*} C_{t_1}'\cap C_{t_2}'\subseteq C_{t_1}(\delta)\cap C_{t_2}(\delta)= C_{t_1\vee t_2}(\delta). \end{align*} $$

Hence, $C_{t_1}'\cap C_{t_2}'=\emptyset $ .

Now for any fixed $\tau \in \widetilde {V}$ , we shall show $\tau \in \widetilde {\pi }(\widetilde {U})$ . By $d_P(\nu ,\tau )<\delta $ , one has $\tau (A^\delta )\geq \nu (A)-\delta $ for every $A\in \mathcal {B}_Y$ . Then for every $\sigma \in \mathcal {E}$ ,

(5.9) $$ \begin{align} \tau(C_{\sigma}(\delta))&=\tau\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}C_{i}^{\delta} \bigg)\geq \tau\bigg(\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}C_i\bigg)^{\delta}\bigg)\nonumber\\ &\geq\nu\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}C_i\bigg)-\delta\overset{(5.8)}{\geq} \nu(V_{\sigma})-k\varepsilon-\delta. \end{align} $$

Moreover, for every $\sigma \in \{0,1\}^{[k]}$ ,

(5.10) $$ \begin{align} &\bigg(\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\ \alpha>\sigma}} C_{\alpha}(\delta)\bigg)^{\delta} =\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\ \alpha>\sigma}}(C_{\alpha}(\delta))^{\delta} =\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\ \alpha>\sigma}} \bigg(\bigcap_{\substack{i\in[k]\\\alpha(i)=1}}C_{i}^{\delta}\bigg)^{\delta}\nonumber\\&\quad\subseteq \bigg(\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\ \alpha>\sigma}}\bigg(\bigcap_{\substack{i\in[k]\\\alpha(i)=1}}C_i^{2\delta}\bigg)\bigg) \overset{(c3)}{\subseteq}\bigg(\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\ \alpha>\sigma}}\bigg(\bigcap_{\substack{i\in[k]\\\alpha(i)=1}}\pi(U_i)\bigg)\bigg) \nonumber\\&\quad=\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\\alpha>\sigma}}V_{\alpha}. \end{align} $$

Since $d_P(\nu ,\tau )<\delta $ , one has

(5.11) $$ \begin{align} \tau(A)\leq \nu(A^\delta)+\delta\quad \text{for every } A\in\mathcal{B}_Y. \end{align} $$

Note that for every $\alpha ,\sigma \in \{0,1\}^{[k]}$ with $\alpha>\sigma $ , one has $C_{\alpha }(\delta )\subseteq C_{\sigma }(\delta )$ and $V_{\alpha }\subseteq V_{\sigma }$ . Then for every $\sigma \in \mathcal {E}$ ,

(5.12)

By $\overline {\bigcup _{n\in N}\mathcal {M}_n(Y)}=\mathcal {M}(Y)$ , there exist $\tau _n=({1}/{n})\sum \nolimits _{j=1}^{n}\delta _{y_{n,j}}\in \mathcal {M}_n(Y)$ for $n\in \mathbb {N}$ and some $y_{n,j}\in Y$ , $j\in [n]$ , such that $\tau _n\to \tau $ as $n\to \infty $ . Moreover, since $\widetilde {V}$ is open in $\mathcal {M}(Y)$ , we can find $N_0\in \mathbb {N}$ such that $\tau _n\in \widetilde {V}$ for $n\geq N_0$ .

Let $n\geq N_0$ . For every $\sigma \in \{0,1\}^{[k]}$ , we set

$$ \begin{align*}S_\sigma^{n}:=\{h\in[n]: y_{n,h}\in C_{\sigma}'\}.\end{align*} $$

Since $\tau _n\in \widetilde {V}$ , by (5.12) and recall that for any $t_1\neq t_2\in \{0,1\}^{[k]}$ , one has $C_{t_1}'\cap C_{t_2}'=\emptyset $ , then:

1. (i) $S_{t_1}^n\cap S_{t_2}^n=\emptyset $ for any $t_1\neq t_2\in \{0,1\}^{[k]}$ ;

2. (ii) $\vert S_{\sigma }^n\vert \geq n(\nu (V_{\sigma }')-3k\varepsilon )$ for every $\sigma \in \mathcal {E}$ , where $\mathcal {E}$ is defined as (5.7).

Now, for every $\sigma \in \{0,1\}^{[k]}$ and $i\in [k]$ , we set

(5.13) $$ \begin{align} U_{i,\sigma}:=U_i\cap\pi^{-1}(V_{\sigma}')\quad \text{and} \quad a_{i,\sigma}:=\mu(U_{i,\sigma}). \end{align} $$

Fix any $\sigma \in \{0,1\}^{[k]}$ . We can rewrite $\{i\in [k]: \sigma (i)=1\}$ as $\{i_1<i_2<\cdots <i_{q}\}$ for some $q\in \mathbb {N}$ . For $i_1$ , we choose arbitrarily a subset $S_{\sigma ,i_1}^{n}$ of $S_{\sigma }^n$ with $\vert S_{\sigma ,i_1}^n\vert =\lfloor {a_{i_1,\sigma }}/{\sum \nolimits _{\ell \in [k],\sigma (\ell )=1}a_{\ell ,\sigma }}\vert S_{\sigma }^n\vert \rfloor $ , where we note: $\tfrac 00=0$ . For $i_2$ , we choose arbitrarily a subset $S_{\sigma ,i_2}^{n}$ of $S_{\sigma }^n\setminus S_{\sigma ,i_1}^n$ with $\vert S_{\sigma ,i_2}^n\vert =\lfloor {a_{i_2,\sigma }}/{\sum \nolimits _{\ell \in [k],\sigma (\ell )=1}a_{\ell ,\sigma }}\vert S_{\sigma }^n\vert \rfloor $ . We continue inductively obtaining

$$ \begin{align*}S_{\sigma,i_j}^n\subseteq S_{\sigma}^n\setminus(S_{\sigma,i_1}^n\cup S_{\sigma,i_2}^n \cup\cdots\cup S_{\sigma,i_{j-1}}^n)\end{align*} $$

for $j=3,4,\ldots ,q-1$ , with $\vert S_{\sigma ,i_j}^n\vert =\lfloor {a_{i_j,\sigma }}/{\sum \nolimits _{\ell \in [k],\sigma (\ell )=1}a_{\ell ,\sigma }}\vert S_{\sigma }^n\vert \rfloor $ . We set $S_{\sigma ,i_q}^n=S_{\sigma }^n\setminus (\bigcup _{j=1}^{q-1}S_{\sigma ,i_j}^n)$ . Additionally, we note that

$$ \begin{align*}y_{n,h}\in C_{\sigma}'\subseteq\bigcap\limits_{\substack{i\in[k]\\\sigma(i)=1}}\pi(U_i)=\bigcap_{\ell=1}^{q} \pi(U_{i_\ell})\end{align*} $$

for every $h\in S_{\sigma }^n$ . Then we have the following properties for $S_{\sigma ,i}^n$ , $i\in [k]$ .

• (i*) $\vert S_{\sigma ,i}^n\vert \geq \lfloor {a_{i,\sigma }}/{\sum \nolimits _{\ell \in [k],\sigma (\ell )=1}a_{\ell ,\sigma }}\vert S_{\sigma }^n\vert \rfloor $ for every $i\in [k]$ with $\sigma (i)=1$ .

• (ii*) For every $i\in [k]$ with $\sigma (i)=1$ , if $h\in S_{\sigma ,i}^n$ , then there exists $x_{n,h}^{\sigma }\in U_i$ satisfying $\pi (x_{n,h}^{\sigma })=y_{n,h}$ .

• (iii*) $S_{\sigma ,i'}^n\cap S_{\sigma ,i"}^n=\emptyset $ for every $i'\neq i"\in \{i\in [k]:\sigma (i)=1\}$ and $\bigcup \nolimits _{i\in [k],\sigma (i)=1} S_{\sigma ,i}^n=S_{\sigma }^n$ .

Since $\pi $ is surjective, for every $h'\in S_0^n:=[n]\backslash (\bigcup \nolimits _{\sigma \in \{0,1\}^{[k]}}S_{\sigma }^n)$ , there exists $x_{n,h'}\in X$ such that $\pi (x_{n,h'})=y_{n,h'}$ . Now we set

(5.14) $$ \begin{align} \mu_n:&=\frac{1}{n}\bigg(\sum_{\sigma\in \{0,1\}^{[k]}}\sum_{h\in S_{\sigma}^n}\delta_{x_{n,h}^{\sigma}}+\sum_{h'\in S_0^n}\delta_{x_{n,h'}}\bigg)\nonumber\\ &\overset{\text{(iii*)}}{=}\frac{1}{n}\bigg(\sum_{\sigma\in \{0,1\}^{[k]}}\sum_{\substack{i\in[k]\\ \sigma(i)=1}}\sum_{h\in S_{\sigma,i}^n} \delta_{x_{n,h}^{\sigma}}+\sum_{h'\in S_0^n}\delta_{x_{n,h'}}\bigg). \end{align} $$

Clearly, $\widetilde {\pi }(\mu _n)=\tau _n$ . We claim that $\mu _{n}(U_{i_0})>\eta _{i_0}$ for every $i_0\in [k]$ when n is sufficiently large. Once it is true, we have

$$ \begin{align*} \mu_n\in\mathbb{W}(U_1,U_2,\ldots,U_k;\eta_1,\eta_2,\ldots,\eta_k). \end{align*} $$

Then we can find a sequence $n_1<n_2<\cdots $ such that $\lim \nolimits _{i\to \infty }\mu _{n_i}=\mu '$ for some $\mu '\in \mathcal {M}(X)$ . Thus,

$$ \begin{align*} \mu'\in\overline{\mathbb{W}(U_1,U_2,\ldots,U_k;\eta_1,\eta_2,\ldots,\eta_k)}\subset\widetilde{U}\end{align*} $$

and $\widetilde {\pi }(\mu ')=\lim \nolimits _{i\to \infty }\widetilde {\pi }(\mu _{n_i})=\lim \nolimits _{i\to \infty }\tau _{n_i}=\tau $ . By the arbitrariness of $\tau $ , one has $\widetilde {V}\subseteq \widetilde {\pi }(\widetilde {U})$ . This will end our proof.

Now, we shall show the claim: $\mu _{n}(U_{i_0})>\eta _{i_0}$ for every $i_0\in [k]$ when n is sufficiently large. To show that, for any fixed $i_0\in \{1,2,\ldots ,k\}$ , we need the following facts.

Fact 1: $\sum \nolimits _{\sigma \in \mathcal {E},\sigma (i_0)=1}\mu (U_{i_0}\cap \pi ^{-1}(V_{\sigma }'))=\mu (U_{i_0}\cap \bigcup \nolimits _{\sigma \in \mathcal {E},\sigma (i_0)=1}\pi ^{-1}(V_{\sigma }'))$ . In fact, for any $t_1\neq t_2\in \{0,1\}^{[k]}$ , if $y\in V_{t_1}'\cap V_{t_2}'\subseteq V_{t_1}\cap V_{t_2}$ , then $y\in V_{t_1\vee t_2}$ . Since $t_1\vee t_2>t_1\ \text {or}\ t_2$ , one has $y\notin V_{t_1}'$ or $y\notin V_{t_2}'$ , which is a contradiction of $y\in V_{t_1}'\cap V_{t_2}'$ . Hence, $V_{t_1}'\cap V_{t_2}'=\emptyset $ . Then Fact 1 follows.

Fact 2: $\nu ((\bigcup \nolimits _{\substack {\sigma \in \mathcal {E}\\\sigma (i_0)=1}}V_{\sigma }')\Delta (\bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}}V_{\sigma }))=0$ , where $A\Delta B$ denotes $(A\setminus B)\cup (B\setminus A)$ for every $A,B\in \mathcal {B}_Y$ . In fact, it is clear that

(5.15) $$ \begin{align} \bigcup\limits_{\substack{\sigma\in\mathcal{E}\\\sigma(i_0)=1}}V_{\sigma}'\subseteq \bigcup\limits_{\substack{\sigma\in\{0,1\}^{[k]}\\\sigma(i_0)=1}}V_{\sigma}. \end{align} $$

Since $\sigma \in \{0,1\}^{[k]}\backslash \mathcal {E}$ implies $\nu (V_{\sigma }')=0$ , one has

(5.16) $$ \begin{align} \nu\bigg(\bigcup\limits_{\substack{\sigma\in\mathcal{E}\\\sigma(i_0)=1}}V_{\sigma}'\bigg) =\nu\bigg(\bigcup\limits_{\substack{\sigma\in\{0,1\}^{[k]}\\\sigma(i_0)=1}}V_{\sigma}'\bigg). \end{align} $$

Clearly, $\bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}}V_{\sigma }\supseteq \bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}} V_{\sigma }'$ . Moreover, for any given $x\in \bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}}V_{\sigma }$ , if we define $\sigma '$ as

$$ \begin{align*}\sigma'(i)=\max\{t(i): t\in\{0,1\}^{[k]} \text{ with } x\in V_{t}\}\end{align*} $$

for every $i\in [k]$ , then $\sigma '(i_0)=1$ and $x\in V_{\sigma '}^{'}\subseteq \bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}} V_{\sigma }'$ . Hence,

(5.17) $$ \begin{align} \bigcup\limits_{\substack{\sigma\in\{0,1\}^{[k]}\\\sigma(i_0)=1}}V_{\sigma}=\bigcup\limits_{\substack{\sigma\in\{0,1\}^{[k]}\\\sigma(i_0)=1}} V_{\sigma}'. \end{align} $$

By (5.15), (5.16) and (5.17), Fact 2 holds.

Fact 3: For every $\sigma \in \mathcal {E}$ , $\sum \nolimits _{\ell \in [k],\sigma (\ell )=1}a_{\ell ,\sigma }\leq \nu (V_{\sigma }')$ . Note that $U_1,U_2,\ldots ,U_k$ are disjoint. Then by (5.13), we have

Thus, Fact 3 holds.

Now by Facts 1–3, we have

(5.18)

We define $t\in \{0,1\}^{[k]}$ as $t(i_0)=1$ and $t(i)=0$ for each $i\in [k]\backslash \{i_0\}$ . Then $\pi (U_{i_0})=V_{t}\subseteq \bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}}V_{\sigma }$ . Thus, $U_{i_0}\subseteq \pi ^{-1}(\bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}}V_{\sigma })$ and by (5.18), we have

(5.19) $$ \begin{align} \sum_{\sigma\in\mathcal{E},\sigma(i_0)=1}a_{i_0,\sigma}=\mu(U_{i_0}). \end{align} $$

Then for any $n>N_0$ , we have

Then by letting $n\to \infty $ , for every $i_0\in [k]$ since $\mu (U_{i_0})>\eta _{i_0}+6^{k}\varepsilon $ by (c1), we have $\mu _n(U_{i_0})>\eta _{i_0}$ . This ends the proof of the claim.