# Compact Median Algebras are $\mu$-Boundaries in a unique way

Uri Bader  and  Aviv Taller

###### Abstract.

We consider group actions on compact median algebras. We show that, given a
generating probability measure $\mu$ on the acting group and under suitable
conditions on the median algebra, it could be realized in a unique way as a
$\mu$-boundary in the sense of Furstenberg. Along the way, we prove some
structural results.

##  1\. Introduction

In this work we study group actions on compact topological median algebras.
Our main goal is to understand stationary measures for such actions. This is a
continuation of [BT21], where we assumed the acting group is amenable and
studied the associated invariant measures. However, here we remove the
amenability assumption, thus invariant measures do not exist in general, while
stationary measures do exist. Our main result is Theorem B which shows that,
under suitable assumptions, there exists a unique stationary measure making
the given median space a boundary in the sense of Furstenberg.

We first state Proposition A below, which is a general structural result - it
describes a maximal cubical factor of a median algebra. Recall that a _cube_
is a median algebra isomorphic to $\\{0,1\\}^{n}$ for some cardinal $n$,
called the _dimension_ of the cube. A finite dimensional cube is called a
_finite cube_.

Given a topological group $G$, a topological median algebra endowed with an
action of $G$ by automorphisms such that the action map $G\times M\to M$ is
continuous is called a topological $G$-median algebra. A $G$-equivariant
continuous median map of topological $G$-median algebras is called a
_morphism_. As usual in the theory of dynamical systems, we study _minimal
actions_ , that is we assume that our median algebra does not contain any non-
empty, proper closed invariant sub-median algebra. As in [BT21], we will
always assume the underlying median algebra is sclocc, that is second
countable, locally open-convex and compact, see §2 for precise definitions.
Moreover, we impose one more condition on our algebras - we assume that all
intervals are countable (which includes finite, by our convention).

######  Proposition A.

Let $G$ be a topological group and let $M$ be a $G$-median algebra. Assume
that $M$ is sclocc and with countable intervals. Then there exist topological
$G$-median algebras $M^{\prime}$ and $C$ such that $M\simeq M^{\prime}\times
C$, $C$ is a finite cube and every surjective morphism $M\to C^{\prime}$,
where $C^{\prime}$ is a $G$-cube, is the composition of the projection map
$M\to C$ and a unique morphism $C\to C^{\prime}$.

In case $M\simeq M^{\prime}$, that is $C$ is trivial in Proposition A, we say
that $M$ has no cubical factor. Recall that, given a probability measure $\mu$
on $G$, a probability measure $\nu$ on $M$ is said to be $\mu$-stationary if
$\mu*\nu=\nu$, where $\mu*\nu$ is the push forward of the measure
$\mu\times\nu$ under the action map $G\times M\to M$.

######  Theorem B.

Let $G$ be a countable discrete group and let $M$ be a minimal topological
$G$-median algebra. Assume that $M$ is sclocc with countable intervals and
with no cubical factor. Let $\mu$ be a generating probability measure on $G$.
Then there exists a unique $\mu$-stationary probability measure $\nu$ on $M$.
Moreover, the $G$-space $(M,\nu)$ is a $\mu$-boundary in the sense of
Furstenberg.

Given $\mu$, a generating probability measure on $G$, we recall Furstenberg’s
correspondence between $G$-maps $B(G,\mu)\to\text{Prob}(M)$ and
$\mu$-stationary measures on $M$, where $B(G,\mu)$ denotes the corresponding
Furstenberg-Poisson boundary of $G$, see [BS06, Theorem 2.16]. The uniqueness
of the stationary measure $\nu$ in Theorem B is thus equivalent to having a
unique $G$-map $\phi:B(G,\mu)\to\text{Prob}(M)$ and the fact $(M,\nu)$ is a
$\mu$-boundary is equivalent to fact that the image of $\phi$ consists of
$\delta$-measures in $\text{Prob}(M)$.

We will recall in §5 the definition of a boundary pair for $G$, as discussed
in [BF14]. By [BF14, Theorem 2.7], the future and past Furstenberg-Poisson
boundaries associated with a generating probability measure form a boundary
pair. Theorem B will follow from the following.

######  Theorem C.

Let $G$ be a countable discrete group and let $M$ be a minimal topological
$G$-median algebra. Assume that $M$ is sclocc with countable intervals and
with no cubical factor. Let $(B_{-},B_{+})$ be a boundary pair for $G$. Then
there are unique (up to a.e equivalence) a.e defined measurable $G$-maps
$\phi_{-}:B_{-}\to M$ and $\phi_{+}:B_{+}\to M$. Moreover, these maps are the
unique (up to a.e equivalence) a.e defined measurable $G$-maps
$B_{-}\to\text{Prob}(M)$, $B_{+}\to\text{Prob}(M)$.

We denote by $\text{Cub}(M)$ the set of all cubes in $M$ and endow it with the
Chabouty topology. Any measurable map $B_{\pm}\rightarrow\text{Cub}(M)$, give
rise to a map $B_{\pm}\rightarrow\text{Prob}(M)$, by sending every cube to its
unique uniform probability measure. Thus, theorem C implies in particular the
uniqueness of the maps $B_{\pm}\rightarrow\text{Cub}(M)$. The following is the
generalization of the above theorem, where we consider the possibility of a
non-trivial cubical factor.

######  Theorem D.

Let $G$ be a countable discrete group and let $M$ be a minimal topological
$G$-median algebra. Assume that $M$ is sclocc with countable intervals. Let
$(B_{-},B_{+})$ be a boundary pair for $G$. Then, there exist unique (up to
a.e equivalence) a.e defined measurable $G$-maps
$\phi_{-}:B_{-}\to\text{Cub}(M)$ and $\phi_{+}:B_{+}\to\text{Cub}(M)$. Under
the decomposition $M\simeq M^{\prime}\times C$ given in Proposition A,
$\phi_{-}$ and $\phi_{+}$ correspond to the products of the unique maps
$\phi^{\prime}_{-}:B_{-}\to M^{\prime}$ and $\phi^{\prime}_{+}:B_{+}\to
M^{\prime}$ given in Theorem C and the constant maps
$B_{\pm}\to\\{C\\}\in\text{Cub}(C)$.

######  Example 1.0.1.

Let $G:=F_{2}\times\\{\pm 1\\}$ where $F_{2}:=\langle a,b\rangle$ is the free
group generated by two elements, $\\{\pm 1\\}\cong{\mathbb{Z}}/2$. Let
$M:=(T_{4}\cup\partial T_{4})\times\\{\pm 1\\}$, where the 4-regular tree
$T_{4}$ is identified with the Cayley graph of $F_{2}$ and $\partial T_{4}$
with its boundary. We take the usual compact median structure on
$T_{4}\cup\partial T_{4}$ and the corresponding product structure on $M$. We
consider the natural action of $G$ on $M$. This action is clearly minimal.
With respect to the measure $\mu(\\{(x,\pm 1)\\})=1/8$ for $x\in\\{a^{\pm
1},b^{\pm 1}\\}$, the Furstenberg Poisson boundary $B(G,\mu)$ of $G$ is
isomorphic to $\partial T_{4}$ endowed with the standard stationary measure.
Then the map

$\partial T_{4}\to\text{Cub}(M),\quad\theta\mapsto\theta\times\\{\pm 1\\}$

is the unique (up to a.e equivalence) a.e defined measurable $G$-map $\partial
T_{4}\to\text{Cub}(M)$.

It is worth noting that in case of a non-trivial cubical factor, it is no
longer true that there are unique maps $B_{\pm}\to\text{Prob}(M)$. The
following example illustrates this fact.

######  Example 1.0.2.

Consider the subgroup $G\leq\\{\pm 1\\}^{3}$ consisting of all the elements
$g=(x,y,z)$ such that $xyz=1$. Note that its action on the cube $\\{\pm
1\\}^{3}$ is minimal. For every $t\in[0,1]$, let $\mu_{t}$ be the probability
measure on $\\{\pm 1\\}^{3}$ which takes the value $t/8$ on elements of $G$
and the value $(1-t)/8$ on elements not in $G$. This is a continuum family of
$G$-invariant probability measures.

##  2\. Median Algebras

This section will focus on essential results and definitions in the context of
median algebras that are required for this paper. For an extensive survey on
median algebras we refer the reader to [Rol98, Fio20, Bow].

A median algebra is a set $M$ with a ternary operator $m:M^{3}\rightarrow M$,
satisfying the following three axioms. For every $x,y,z,u,v\in M$

  1. (1)

$m(x,y,z)=m(x,z,y)=m(y,x,z)$,

  2. (2)

$m(x,x,y)=x$, and

  3. (3)

$m(m(x,y,z),u,v)=m(x,m(y,u,v),m(z,u,v))$

A median morphism between two median algebras $M$ and $N$ is a map
$\phi:M\rightarrow N$ such that $\phi\circ m=m\circ\phi^{3}$. For two points
$x$ and $y$ in $M$, the interval $[x,y]$ is defined as the set of all points
$z\in M$, such that $m(x,y,z)=z$. A subset $C\subset M$ is said to be convex,
if for every $x,y\in C$, $[x,y]\subset C$. The following is a median algebra
version of Helly’s theorem:

######  Lemma 2.0.1 (Helly’s Theorem, [Rol98, Theorem 2.2]).

If $C_{1},...,C_{n}$ are convex, and for every $i\neq j,\ \ C_{i}\cap
C_{j}\neq\varnothing$, then also
$\overset{n}{\underset{i=1}{\cap}}C_{i}\neq\varnothing$.

The convex hull of a subset $S\subset M$, denoted by $\text{Conv}(S)$, is the
smallest convex set containing S. For two subsets $A,B\subset M$, the join of
A and B, denoted by $[A,B]$, is the union of all intervals $[a,b]$, for $a\in
A$ and $b\in B$. For two convex sets, the join gives a neat description for
the convex hull of their union. Namely:

######  Lemma 2.0.2 ([Rol98, Proposition 2.3]).

If $C,C^{\prime}\subset M$ are two convex sets, then, $\text{Conv}(C\cup
C^{\prime})=[C,C^{\prime}]$

A half-space $\mathfrak{h}\subset M$, is a convex subset, such that the set
$\mathfrak{h}^{*}:=M\backslash\mathfrak{h}$ is also convex. An unordered pair
of two complementary half-spaces
$\mathfrak{w}=\\{\mathfrak{h},\mathfrak{h}^{*}\\}$ is called a wall. Given two
sets $A$ and $B$, we say that a half-space $\mathfrak{h}$ is separating A from
B, if $A\subset\mathfrak{h}$ and $\mathfrak{h}\cap B=\varnothing$. We denote
by $\Delta(A,B)$ the collection of all half-spaces that separates $A$ from
$B$. We say the a wall $\mathfrak{w}$ separates $A$ and $B$ if
$\mathfrak{w}\cap\Delta(A,B)$ is non-empty. One of the main features of median
algebras is that for any two convex sets $A$ and $B$,
$\Delta(A,B)\neq\varnothing$. See theorem 2.8 in [Rol98].

Let $C\subset M$ be a subset and $x\in M$. An element $y\in C$ is said to be
the gate of $x$ in $C$, if $y\in[x,z]$ for every $z\in C$. Note that the gate,
if exist, is unique. We say that $C$ is gate-convex, if for every $x\in M$
there exists a gate $y$ in $C$. Given a gate-convex set $C$, we denote by
$\pi_{C}$ the map that assign for every element $x\in M$ its gate in $C$. The
map $\pi_{C}$ is called gate-projection. The following are key features of
gate-convex sets:

######  Lemma 2.0.3 ([Fio20, Proposition 2.1]).

If a map $\phi:M\rightarrow M$ is a gate-projection, then for all $x,y,z\in
M$, we have $\phi(m(x,y,z))=m(\phi(x),\phi(y),\phi(z))=m(\phi(x),\phi(y),z)$.

######  Lemma 2.0.4 ([Fio20, Lemma 2.2]).

  1. (1)

If $C_{1}\subset M$ is convex and $C_{2}\subset M$ is gate-convex, the
projection $\pi_{C_{2}}(C_{1})$ is convex. If moreover, $C_{1}\cap
C_{2}\neq\varnothing$, we have $\pi_{C_{2}}(C_{1})=C_{1}\cap C_{2}$.

  2. (2)

If $C_{1},C_{2}\subset M$ are gate-convex, the sets $\pi_{C_{1}}(C_{2})$ and
$\pi_{C_{2}}(C_{1})$ are gate-convex with gate-projections
$\pi_{C_{1}}\circ\pi_{C_{2}}$ and $\pi_{C_{2}}\circ\pi_{C_{1}}$, respectively.

  3. (3)

If $C_{1},C_{2}\subset M$ are gate-convex and $C_{1}\cap
C_{2}\neq\varnothing$, then $C_{1}\cap C_{2}$ is gate-convex with gate-
projection $\pi_{C_{2}}\circ\pi_{C_{1}}=\pi_{C_{1}}\circ\pi_{C_{2}}$. In
particular, if $C_{2}\subset C_{1}$, then
$\pi_{C_{2}}=\pi_{C_{1}}\circ\pi_{C_{2}}$.

It follows by the axioms of the median operator that for every $x,y,z\in M$,
$m(x,y,z)\in[x,y]$. Moreover, by lemma 2.0.3 and axiom (3), intervals are
gate-convex, and gate-convex sets are convex. A criterion for a convex set to
be gate-convex, will be given in the context of compact median algebras.

Let $\phi:M\rightarrow M$ be an automorphism of M, and $C$ a gate-convex set.
Observe that for every $x\in M$, and every $y\in C$, we have

$\phi(\pi_{C}(x))=\phi(m(x,y,\pi_{C}(x)))=m(\phi(x),\phi(y),\phi(\pi_{C}(x)))\in[\phi(x),\phi(y)].$

That is, $\phi(C)$ is gate-convex, and for every $x\in M$,
$\pi_{\phi(C)}(x)=\phi(\pi_{C}(\phi^{-1}(x)))$.

A topological median algebra is a median algebra M, endowed with a Hausdorff
topology, with respect to which the median operator is continuous. It is
called locally open-convex if, in addition, each of its points has a basis of
open and convex neighborhoods. The reader should note the differences between
our definition of locally open-convex to that of locally convex in [Fio20].

Let $M$ be a compact median algebra. For any two points $x,y\in M$, the
interval $[x,y]$ is closed, as it is the continuous image of the compact set
$\\{x\\}\times\\{y\\}\times M$. It follows by lemmata 2.6 and 2.7 in [Fio20],
that any convex set in $M$ is gate-convex if and only if it is closed, and
that gate-projections are continuous. In particular, the convex hull of two
gate-convex sets $C$ and $C^{\prime}$ is compact, and therefore a gate-convex
as well, as it is, by lemma 2.0.2, the continuous image of the compact set
$C\times C^{\prime}\times M$.

######  Lemma 2.0.5 ([BT21, Lemma 2.4]).

For disjoint non-empty subsets $A,B\subset M$, if $A$ is gate-convex then
there exists $a\in A$ such that $\Delta(A,B)=\Delta(a,B)$.

Note that it implies in particular that also $\Delta(B,A)=\Delta(B,a)$.

Our main object is second countable, locally open-convex, compact, or in short
sclocc, median algebra M. We fix such a space for the rest of this section.

Denote by $\text{Prob}(M)$ the space of probability measures on $M$. In [BT21]
we introduce the self-median operator
$\Phi:\text{Prob}(M)\rightarrow\text{Prob}(M)$, given by $\mu\mapsto
m_{*}(\mu^{3})$. A probability measure is said to be balanced, if it is
invariant under $\Phi$.

Given a cube $C=\\{0,1\\}^{k}\simeq({\mathbb{Z}}/2{\mathbb{Z}})^{k}$, one can
easily verify that its corresponding normalized Haar measure $\lambda$, is
balanced. Thus, each subcube $C\subset M$, induces a balanced measure on M.
Such a measure in $M$ is called cubical. We have the following
characterisation of balanced measures:

######  Theorem 2.0.6 ([BT21, Theorem A]).

Every balanced measure on a sclocc median algebra is cubical.

A half-space $\mathfrak{h}$ in $M$ is said to be admissible if $\mathfrak{h}$
is open and $\mathfrak{h}^{*}$ has non empty interior. The following is an
enhanced separation property:

######  Lemma 2.0.7 ([BT21, Proposition 3.6]).

Let $C$ and $C^{\prime}$ be two disjoint closed convex subsets in M. Then,
there exist an admissible half-space $\mathfrak{h}\in\Delta(C,C^{\prime})$.

######  Corollary 2.0.8.

There exists a countable collection $\mathcal{H}$ of closed half-spaces in M,
such that for every $x\neq y\in M$, there exist two disjoint, closed half-
spaces $\mathfrak{h}\in\Delta(x,y)\cap\mathcal{H}$ and
$\mathfrak{h}^{\prime}\in\Delta(y,x)\cap\mathcal{H}$, and such that
$x\in\mathfrak{h}^{\circ}$ and $y\in\mathfrak{h}^{\prime\circ}$.

###### Proof.

As $M$ is second countable and locally open-convex, we can find a countable
basis for the topology, $\mathcal{C}$, consisting of open convex sets. Thus,
every two different points can be separated by two elements of $\mathcal{C}$
with disjoint closures.

Let $U$ and $V$ be two elements of $\mathcal{C}$ with disjoint closures. By
lemma 2.0.7 we can choose a closed half-space
$\mathfrak{h}_{(U,V)}\in\Delta(\overline{U},\overline{V})$. Using lemma 2.0.7
again, we find closed half-space
$\mathfrak{h}_{(V,U)}\in\Delta(\overline{V},\mathfrak{h}_{(U,V)})$.

Denote by $\mathcal{H}$ the countable collection of all half-spaces of the
form $\mathfrak{h}_{(U,V)}$ and $\mathfrak{h}_{(V,U)}$. The reader can now
verify that $\mathcal{H}$ possesses the required properties. ∎

Given a collection of half-spaces $\mathcal{H}$, and a non-empty subset $A$ of
$M$, we denote by $\mathcal{H}(A)$ the subcollection of $\mathcal{H}$
consisting of all the half-spaces $\mathfrak{h}\in\mathcal{H}$ that cut $A$
into two, i.e. such that both $\mathfrak{h}\cap A$ and $\mathfrak{h}^{*}\cap
A$ are non empty.

Fix some collection of half spaces $\mathcal{H}$, and its corresponding
collection of walls $\mathcal{W}_{\mathcal{H}}$. That is,

$\mathcal{W}_{\mathcal{H}}:=\big{\\{}\\{\mathfrak{h},\mathfrak{h}^{*}\\}\ |\
\mathfrak{h}\in\mathcal{H}\big{\\}}.$

We have a natural surjection
$\mathcal{H}\rightarrow\mathcal{W}_{\mathcal{H}}$, given by
$\mathfrak{h}\mapsto\\{\mathfrak{h},\mathfrak{h}^{*}\\}$. Fix a section
$\sigma:\mathcal{W}_{\mathcal{H}}\rightarrow\mathcal{H}$. Given a set $A$ we
denote by $\chi_{A}$ its characteristic function. Consider the following map

$\iota_{\mathcal{W}_{\mathcal{H}}}:M\rightarrow
2^{\mathcal{W}_{\mathcal{H}}},\
x\mapsto\big{(}\chi_{\sigma(\mathbf{w})}(x)\big{)}_{\mathbf{w}\in\mathcal{W}_{\mathcal{H}}}.$

We say that $\mathcal{W}_{\mathcal{H}}$ is separating if
$\iota_{\mathcal{W}_{\mathcal{H}}}$ is injective, that is, if the walls in
$\mathcal{W}_{\mathcal{H}}$ are separate points in M. It is called transverse
if $\iota_{\mathcal{W}_{\mathcal{H}}}$ is a surjection. The collection
$\mathcal{H}$ is said to have one of the above properties if
$\mathcal{W}_{\mathcal{H}}$ possess it. We note that these properties are
independent of the choice of the section $\sigma$.

We say that two collections of walls $\mathcal{W}_{1}$ and $\mathcal{W}_{2}$
are transverse, if for every $\mathfrak{w}_{1}\in\mathcal{W}_{1}$ and
$\mathfrak{w}_{2}\in\mathcal{W}_{2}$, the set of walls
$\\{\mathfrak{w}_{1},\mathfrak{w}_{2}\\}$ is transverse. Two collections of
half-spaces $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are said to be transverse,
if their collections of walls, $\mathcal{W}_{\mathcal{H}_{1}}$ and
$\mathcal{W}_{\mathcal{H}_{2}}$, are transverse.

We finish with a handy observation:

######  Lemma 2.0.9.

Let $\phi:M\rightarrow N$ be a continuous median bijection between two sclocc
median algebras. Then $\phi$ is an isomorphism.

###### Proof.

Any continuous bijection between compact Hausdorff spaces is a homeomorphism.
So it is left to show that the inverse $\phi^{-1}$ is a median map.

Indeed, let $x,y,z\in M$. Then,

$\displaystyle\phi^{-1}(m(x,y,z))$
$\displaystyle=\phi^{-1}(m(\phi(\phi^{-1}(x)),\phi(\phi^{-1}(y)),\phi(\phi^{-1}(z))))$
$\displaystyle=\phi^{-1}\circ\phi\big{(}m(\phi^{-1}(x),\phi^{-1}(y),\phi^{-1}(z))\big{)}=m(\phi^{-1}(x),\phi^{-1}(y),\phi^{-1}(z)).$

∎

##  3\. Cubing Decomposition And Proof Of Proposition A

Given a median algebra M, denote by $\mathcal{H}_{M}$ and $\mathcal{W}_{M}$
the collection of all half-spaces and walls of M, respectively. We fix a
sclocc median algebra M and a group $G$ as in Proposition A, and a section
$\sigma_{M}:\mathcal{W}_{M}\rightarrow\mathcal{H}_{M}$.

######  Lemma 3.0.1.

Let $\mathcal{W}\subset\mathcal{W}_{M}$ be a $G$-invariant subcollection of
walls that separates points in M. Then, there exist two sclocc median algebras
$M^{\prime}$ and $C$, where C is a finite cube and an isomorphism
$\phi:M\rightarrow M^{\prime}\times C$, if and only if there exists a
partition $\mathcal{W}=\mathcal{W}_{1}\amalg\mathcal{W}_{2}$ such that the
following conditions hold:

  1. (1)

The set $\mathcal{W}_{1}$ is transverse, in which all the walls consists of
clopen half-spaces,

  2. (2)

The collections $\mathcal{W}_{1}$ and $\mathcal{W}_{2}$ are transverse.

Furthermore, we can choose $M^{\prime}$ and $C$ to be $G$-spaces, and $\phi$
to be $G$-equivariant, if and only if there exists a partition as above, with
$\mathcal{W}_{1}$ and $\mathcal{W}_{2}$ $G$-invariant.

###### Proof.

Suppose first that we have a partition as above. Then, item (1) implies that
the map $\iota:=\iota_{\mathcal{W}_{1}}$ is in fact a continuous projection
onto the cube $C:=2^{\mathcal{W}_{1}}$. Fix $x\in C$, and denote by $M_{x}$
the fiber $\iota^{-1}(x)$. As $\iota$ is a continuous median map, $M_{x}$ is
closed and convex, thus, a gate-convex set. We claim that for every $x,y\in
C$, $\pi_{M_{x}}|_{M_{y}}$ is an isomorphism.

Let $a,b\in M_{y}$. By lemma 2.0.8, we can find two disjoint closed half-
spaces $a\in\mathfrak{h}_{a}^{0}\subset\mathfrak{h}_{a}\in\Delta(a,b)$ and
$b\in\mathfrak{h}_{b}^{0}\subset\mathfrak{h}_{b}\in\Delta(b,a)$. For a set
$A\subset M$, denote by $\overline{A}$ the topological closure of $A$.
Applying lemma 2.0.5 to the pairs $\\{a,\overline{\mathfrak{h}_{a}^{*}}\cap
M_{y}\\}$ and $\\{b,\overline{\mathfrak{h}_{b}^{*}}\cap M_{y}\\}$, implies the
existence of two points $a^{\prime}\in\overline{\mathfrak{h}_{a}^{*}}\cap
M_{y}$ and $b^{\prime}\in\overline{\mathfrak{h}_{b}^{*}}\cap M_{y}$, such that
$\Delta(a,a^{\prime})=\Delta(a,\overline{\mathfrak{h}_{a}^{*}})$ and
$\Delta(b,b^{\prime})=\Delta(b,\overline{\mathfrak{h}_{b}^{*}})$. These
equalities are true in M, although we applied the lemma in $M_{y}$, thanks to
lemma 2.3 in [Fio20].

Denote by $\mathcal{H}$ the union
$\sigma_{M}(\mathcal{W})\cup\sigma_{M}(\mathcal{W})^{*}$. As $\mathcal{W}$ is
separating, we can find two half-spaces
$\mathfrak{f}_{a}\in\Delta(a,a^{\prime})\cap\mathcal{H}$ and
$\mathfrak{f}_{b}\in\Delta(b,b^{\prime})\cap\mathcal{H}$. Now, it follows by
Helly’s theorem, compactness, and the assumption that $\mathcal{W}_{1}$ and
$\mathcal{W}_{2}$ are transverse, that $\overline{\mathfrak{f}_{a}}\cap
M_{x}\neq\varnothing$ and that $\overline{\mathfrak{f}_{b}}\cap
M_{x}\neq\varnothing$. Note that $\overline{\mathfrak{f}_{a}}$ and
$\overline{\mathfrak{f}_{b}}$ are disjoint, and therefore, it follows by lemma
2.0.4(1) that $\pi_{M_{x}}(a)\neq\pi_{M_{x}}(b)$. Thus, $\pi_{M_{x}}|_{M_{y}}$
is injective. By lemma 2.0.4(2),
$\pi_{M_{x}}\circ\pi_{M_{y}}|_{\pi_{M_{x}}(M_{y})}=id_{\pi_{M_{x}}(M_{y})}$.
In particular, $\pi_{M_{y}}(\pi_{M_{x}}(M_{y}))=\pi_{M_{x}}(M_{y})$, and thus,
by injectivity, $\pi_{M_{y}}(\pi_{M_{x}}(M_{y}))=M_{y}$, that is,
$\pi_{M_{y}}|_{M_{x}}$ is surjective. The claim now follows by the symmetry of
the arguments.

Fix $x_{0}\in C$, denote $M_{0}:=M_{x_{0}}$ and by $\pi_{0}$ its gate-
projection. As $M$ is the disjoint union of the $M_{x}$, it is clear that the
continuous map $\iota\times\pi_{0}:M\rightarrow C\times M_{0}$ is a bijection.
It follows by lemma 2.0.9 that this is an isomorphism, as needed. Note that
$C$ must be finite, as the intervals in M are countable.

Suppose now that $M=M^{\prime}\times C$, for $M^{\prime}$ and $C$ as in the
lemma. Let $p_{M^{\prime}}$ and $p_{C}$ be the natural projection to each
factor. We have the following two subcollections of $\mathcal{W}_{M}$:

$\displaystyle\mathcal{W}_{M}^{1}:=\big{\\{}\\{p_{C}^{-1}(\mathfrak{w})\ |\
\mathfrak{w}\in\mathcal{W}_{C}\big{\\}}$
$\displaystyle\mathcal{W}_{M}^{2}:=\big{\\{}\\{p_{M^{\prime}}^{-1}(\mathfrak{w})\
|\ \mathfrak{w}\in\mathcal{W}_{M^{\prime}}\big{\\}}$

Clearly, these collections are transverse, and since $\mathcal{W}_{C}$ is
transverse and its walls are consists of clopen half-spaces, so is
$\mathcal{W}_{M}^{1}$ and so are its walls. Take
$\mathfrak{h}\in\mathcal{H}_{M}$. Observe that if
$(x_{1},y_{1}),(x_{2},y_{2})\in\mathfrak{h}$, then,
$[x_{1},x_{2}]\times[y_{1},y_{2}]\subset\mathfrak{h}$. Hence, $\mathfrak{h}$
is of the form $C_{1}\times C_{2}$ for some two convex sets $C_{1}\subset
M^{\prime}$ and $C_{2}\subset C$. Since this is true also for
$\mathfrak{h}^{*}$, we must have $\mathfrak{h}\in\mathcal{W}_{M}^{1}$ or
$\mathfrak{h}\in\mathcal{W}_{M}^{2}$. That is, $\mathcal{W}_{M}^{1}$ and
$\mathcal{W}_{M}^{2}$ constitute a partition of $\mathcal{W}_{M}$. Define
$\mathcal{W}_{1}:=\mathcal{W}\cap\mathcal{W}^{1}_{M}$ and
$\mathcal{W}_{2}:=\mathcal{W}\cap\mathcal{W}^{2}_{M}$. It is clear that this
is the required partition. We note that it is possible that $\mathcal{W}_{1}$
is in fact empty, which just implies that $C$ is a point.

The last assertion in the lemma is immediate. ∎

###### Proof of proposition A.

If M is already a cube, there is nothing to prove. Otherwise, define

$\displaystyle\mathcal{W}_{1}:=\big{\\{}\mathfrak{w}\ |\
\mathfrak{w}\in\mathcal{W}_{M}\text{ consist of clopen half-spaces and
}\mathcal{W}_{M}\backslash\\{\mathfrak{w}\\}\ and\ \\{\mathfrak{w}\\}\text{
are transverse }\big{\\}}$
$\displaystyle\mathcal{W}_{2}:=\mathcal{W}_{M}\backslash\mathcal{W}_{1}$

Any isomorphism of $M$ must preserve $\mathcal{W}_{1}$, and therefore, is also
preserving $\mathcal{W}_{2}$. By construction, these sets meets the conditions
in lemma 3.0.1, and thus, correspond to a decomposition $M=M^{\prime}\times
C$, where $M^{\prime}$ is a sclocc median algebra, $C$ is a finite cube, and
on both there exist $G$-actions, such that the action on $M$ is the diagonal.

Let $\varphi:M\rightarrow C^{\prime}$ be a surjective morphism, with
$C^{\prime}$ a cube. Denote by $\mathcal{W}$ the collection of walls of the
form $\\{\varphi^{-1}(\mathfrak{f}),\varphi^{-1}(\mathfrak{f}^{*})\\}$ for
$\mathfrak{f}\in\mathcal{H}_{C^{\prime}}$. It is clear that
$\mathcal{W}\subset\mathcal{W}_{1}$, and that $\varphi$ is the composition of
$p_{C}=\iota_{\mathcal{W}_{1}}$ with the projection from $C$ to the factor
generated by $\mathcal{W}$. ∎

##  4\. Special Constructions And Technical Lemmata

The purpose of this section is to establish all the constructions and
technical results that are necessary for the proof of the main theorems, e.g.
checking the measurability of some sets and maps between them, etc. The
trusting reader can skip this section for the benefit of the reading flow.

###  4.1. Subsets of $\mathcal{C}$M

Let M be a sclocc median algebra. As it is a metrizable space, we may consider
its space $\mathcal{C}M$ of closed subsets, endowed with the Hausdorff
distance that corresponds to a metric on M. We fix such a metric and recall
that $\mathcal{C}M$ is a compact metric space (see for example lemma 5.31 in
[BH99]).

Let $F\subset M$ be a finite subset. It follows by lemma 2.0.2, and by an
inductive argument, that the convex hull of F is the join of two compact
convex sets, and therefore, it is itself a compact and convex set. In
particular, as discussed in the previous section, $conv(F)$ is a gate-convex
set, and is an element of $\mathcal{C}M$.

######  Definition 4.1.1.

Denote by $\mathcal{J}_{n}$ the subspace of $\mathcal{C}M$ consisting of all
the convex sets that generated by n elements. In addition, denote by
$\mathcal{J}_{n}^{\subseteq}$ the subspace of
$\mathcal{J}_{n}\times\mathcal{J}_{n}$ consisting of pairs (C,J) such that
$J\subset C$.

######  Lemma 4.1.2.

$\mathcal{J}_{n}$ and $\mathcal{J}_{n}^{\subseteq}$ are closed subspaces of
$\mathcal{C}M$ and $\mathcal{C}M\times\mathcal{C}M$, respectively.

###### Proof.

Notice that it is enough to show that $\mathcal{J}_{n}$ is a closed subspace.
We argue by induction on n. The case $n=1$ is clear, as $\mathcal{J}_{1}$ is
none other than M itself. Assume the claim is correct for n, and let
$C_{i}\in\mathcal{J}_{n+1}$ be a sequence that converges to some element
$C\in\mathcal{C}M$. For every i, denote by $x_{j}^{i}\in M$, j=1,..,n+1, the
collection of generators of $C_{i}$, that is,
$C_{i}=Conv(\\{x_{j}^{i}\\}_{j=1}^{n+1})$. For every i, denote by
$C_{i}^{\prime}$ the set $Conv(\\{x_{j}^{i}\\}_{j=1}^{n})$, and notice that by
the definition of the convex hall,
$C_{i}=Conv(C_{i}^{\prime}\cup\\{x_{n+1}^{i}\\})$. By the induction
hypothesis, after taking a subsequence, we may assume that there exist
$x_{n+1}\in M$ and $C^{\prime}\in\mathcal{J}_{n}$ such that the sequences
$\\{x_{n+1}^{i}\\}_{i}$ and $\\{C_{i}^{\prime}\\}_{i}$ converge to $x_{n+1}$
and $C^{\prime}$, respectively. We claim that
$C=Conv(C^{\prime}\cup\\{x_{n+1}\\})\in\mathcal{J}_{n+1}$.

Lemma 2.0.2 implies that $C_{i}=[C_{i}^{\prime},x_{n+1}^{i}]$ and
$Conv(C^{\prime}\cup\\{x_{n+1}\\})=[C^{\prime},x_{n+1}]$. Notice that, as for
every i, $C_{i}^{\prime}\subset C_{i}$, then $C^{\prime}\subset C$. Therefore,
we get one inclusion for free: $[C^{\prime},x_{n+1}]\subset C$. For the other
inclusion, fix an element $y\in C$. Let $y_{i}\in C_{i}$ be a sequence that
converges to $y$. By the observation above, for every i we can find $z_{i}\in
C_{i}^{\prime}$ such that $y_{i}\in[z_{i},x_{n+1}^{i}]$. After passing to a
subsequence, if needed, we may assume that $z_{i}$ converges to an element
$z\in C^{\prime}$. But now, by continuity of the median

$y=\underset{n}{\lim}\ y_{i}=\underset{n}{\lim}\
m(z_{i},y_{i},x_{n+1}^{i})=m(z,y,x_{n+1})\in[C^{\prime},x_{n+1}]$

∎

Denote by $\text{Cub}(M)$ the collection of all closed subcubes in M.

######  Lemma 4.1.3.

The space $\text{Cub}(M)$ is a closed subspace of $\mathcal{C}M$.

Before proving this lemma, let us introduce a new notion, and a related
property. Let $A\subset M$. We denote by $\text{Ends}(A)$ the subset of all
elements $x\in A$, for which there is an antipodal element in A. That is, an
element $x^{*}\in A$, such that $m(x,x^{*},z)=z$, for all $z\in A$. If
$A\in\mathcal{J}_{2}$, this amounts to saying that $[x,x^{*}]=A$. We claim the
following:

######  Lemma 4.1.4.

Let $A\in\mathcal{C}M$ be a median subalgebra. If
$\text{Ends}(A)\neq\varnothing$, then $\text{Ends}(A)\in\text{Cub}(M)$

###### Proof.

First, let us prove that $\text{Ends}(A)$ is a closed median subalgebra. Let
$x_{1},x_{2},x_{3}\in\text{Ends}(A)$, and let
$x_{1}^{*},x_{2}^{*},x_{3}^{*}\in\text{Ends}(A)$ be their antipodal elements
in $A$. We claim that $m(x_{1}^{*},x_{2}^{*},x_{3}^{*})$ is the antipodal in
$A$ of $m(x_{1},x_{2},x_{3})$. Namely, that for every $z\in A$

$(*)\
m\big{(}z,m(x_{1},x_{2},x_{3}),m(x_{1}^{*},x_{2}^{*},x_{3}^{*})\big{)}=z$

Fix $z\in A$. For the sake of contradiction, suppose that we can find

$\mathfrak{h}\in\Delta\big{(}z,m\big{(}z,m(x_{1},x_{2},x_{3}),m(x_{1}^{*},x_{2}^{*},x_{3}^{*})\big{)}\big{)}$

Since $z\in\mathfrak{h}$, it follows by convexity of $\mathfrak{h}$ and
$\mathfrak{h}^{*}$, that both $m(x_{1},x_{2},x_{3})$ and
$m(x_{1}^{*},x_{2}^{*},x_{3}^{*})$ are in $\mathfrak{h}^{*}$. By convexity,
again, there must be $3\geq i\geq 1$ such that both $x_{i}$ and $x_{i}^{*}$
are in $\mathfrak{h}^{*}$. But now,

$z\in A=[x_{i},x_{i}^{*}]\cap A\subset\mathfrak{h}^{*}$

which is a contradiction. Hence, $(*)$ holds.

Suppose now that there is a sequence $x_{n}\in\text{Ends}(A)$ that converges
to some element $x\in A$. Let $x^{*}_{n}\in\text{Ends}(A)$ be the
corresponding sequence of antipodal elements. After replacing $x_{n}$ with
some sub-sequence, if necessary, we may assume that $x_{n}^{*}$ converges to
an element $x^{*}\in A$. Fix $z\in A$. By continuity of the median operator,

$\displaystyle z$
$\displaystyle=\underset{n\rightarrow\infty}{\lim}z=\underset{n\rightarrow\infty}{\lim}m(x_{n},x_{n}^{*},z)$
$\displaystyle=m(x,x^{*},z)$

Thus, $A=[x,x^{*}]\cap A$, which implies that $x\in\text{Ends}(A)$, as needed.

We now show that $\text{Ends}(A)$ isomorphic to a cube. By the above, we may
assume without loss of generality that $\text{Ends}(A)=A=M$. Denote by
$\mathcal{W}^{\circ}$ the collection of all clopen walls of $M$. That is, the
collection of all pairs $\\{\mathfrak{h},\mathfrak{h}^{*}\\}$ for a clopen
half-space $\mathfrak{h}\subset M$.

It follows by corollary 3.4 in [BT21], that it is sufficient (and necessary)
to show that $\mathcal{W}^{\circ}$ is separating points in $M$ and is
transverse.

For starter, let us show that for any closed half-space $\mathfrak{h}$,
$\\{\mathfrak{h},\mathfrak{h}^{*}\\}\in\mathcal{W}^{\circ}$. Notice that for
every $x\in M$, the antipodal element $x^{*}$ is unique. Indeed, if $x^{*}$
and $x^{\prime*}$ are antipodal elements of $x$, then

$x^{*}=m(x,x^{*},x^{\prime*})=x^{\prime*}$

Moreover, the above discussion implies that if $x_{n}$ converges to $x$, then
$x_{n}^{*}$ converges to $x^{*}$. Thus, the antipodal map, $*:x\mapsto x^{*}$,
is a continuous median automorphism of $M$. If $\mathfrak{h}$ is a non-empty
half-space, such that also $\mathfrak{h}^{*}$ is non-empty, then
$\mathfrak{h}=*^{-1}(\mathfrak{h}^{*})$ (otherwise it would imply that there
exists $x\in M$, such that $M=[x,x^{*}]\subset\mathfrak{h}$ or
$M=[x,x^{*}]\subset\mathfrak{h}^{*}$). In particular, if $\mathfrak{h}$ is
closed, then it is also open, as needed.

Let $\\{\mathfrak{h},\mathfrak{h}^{*}\\}$ and
$\\{\mathfrak{f},\mathfrak{f}^{*}\\}$ be two walls in $\mathcal{W}^{\circ}$.
Suppose that they are not transverse. Without loss of generality we may assume
that $\mathfrak{h}\subset\mathfrak{f}$. But then,
$\mathfrak{h}^{*}=*(\mathfrak{h})\subset*(\mathfrak{f})=\mathfrak{f}^{*}\subset\mathfrak{h}^{*}$
which implies that
$\\{\mathfrak{h},\mathfrak{h}^{*}\\}=\\{\mathfrak{f},\mathfrak{f}^{*}\\}$.
That is, $\mathcal{W}^{\circ}$ is transverse, as required. ∎

######  Corollary 4.1.5.

For $I\in\mathcal{J}_{2}$, $\text{Ends}(I)\in\text{Cub}(M)$.

In the course of proving lemma 4.1.4, we also proved the following lemma:

######  Lemma 4.1.6.

Suppose that $M$ is a sclocc median algebra which is a cube. Then, every
closed half-space is also open.

###### Proof of lemma 4.1.3.

Let $C_{n}\in\text{Cub}(M)$ be a sequence that converges to some closed set
$C\in\mathcal{C}M$. It follows by lemma 4.1.4 that we only need to show that
$C$ is a median subalgebra and that $\text{Ends}(C)=C$.

If $x_{n},y_{n}$ and $z_{n}\in C_{n}$ converges to $x,y$ and $z\in C$,
respectively, then clearly, $m(x_{n},y_{n},z_{n})\in C_{n}$ converges to
$m(x,y,z)\in C$.

Pick $x\in C$, and $x_{n}\in C_{n}$ such that $x_{n}$ converges to $x$. Let
$x_{n}^{*}\in C_{n}$ be the corresponding sequence of antipodal elements, and
suppose it converges to $x^{*}\in C$ (restrict to a sub-sequence, if
necessary). Fix $z\in C$, and some (sub-)sequence $z_{n}\in C_{n}$ that
converges to $z$. Then,

$\displaystyle z$
$\displaystyle=\underset{n\rightarrow\infty}{\lim}z=\underset{n\rightarrow\infty}{\lim}m(x_{n},x_{n}^{*},z)$
$\displaystyle=m(x,x^{*},z)$

∎

###  4.2. Mapping Properties

For the rest of this section, fix a sclocc median algebra M, and a discrete
group $G$ that acts on M continuously by automorphisms. Note that in
particular, we are not assuming that intervals in M are countable, nor that
subcubes of M are of finite dimension (unless stated otherwise).

######  Lemma 4.2.1.

The map $\mathcal{J}_{n}^{2}\rightarrow\mathcal{J}_{n}^{\subseteq}$, defined
by $(I,J)\mapsto(I,\pi_{I}(J))$, is continuous.

###### Proof.

This is simply because the map $\pi_{I}$ is a continuous function, that
depends continuously on the generators of I. ∎

######  Lemma 4.2.2.

Let $C\subset M$ be a gate convex set. Suppose that for every $\gamma\in G$,
$\pi_{C}$ is a surjection from $\gamma C$ to C. Then the assignment
$\gamma.x:=\pi_{C}(\gamma x)$, for $\gamma\in G$ and $x\in C$, defines a
continuous action of $G$ on C by median automorphisms. Moreover, $\pi_{C}$ is
a median $G$-map between $M$ to $C$.

###### Proof.

The continuity part is clear, as the new action is defined by the composition
of continuous functions. We now show that this is an action. Let $x\in C$ and
$\gamma^{\prime},\gamma\in G$. By lemma 2.0.4.(2), as $\pi_{C}|_{\gamma C}$ is
surjective, $\pi_{C}\circ\pi_{\gamma C}=\pi_{C}$. So we have:

$\gamma.(\gamma^{\prime}.x)=\pi_{C}(\gamma\pi_{C}(\gamma^{\prime}x))=\pi_{C}(\pi_{\gamma
C}(\gamma\gamma^{\prime}x))=\pi_{C}(\gamma\gamma^{\prime}x)=(\gamma\gamma^{\prime}).x$

As required.

If $y,z\in C$ are another two elements, then by lemma 2.0.3 we have

$\displaystyle\gamma.m(x,y,z)$ $\displaystyle=\pi_{C}(\gamma
m(x,y,z))=\pi_{C}(m(\gamma x,\gamma y,\gamma z))$
$\displaystyle=m(\pi_{C}(\gamma x),\pi_{C}(\gamma y),\pi_{C}(\gamma
z))=m(\gamma.x,\gamma.y,\gamma.z)$

Finely, note that by the definition of the induced action, $\pi_{C}$ is a
$G$-map, and it is a median map by lemma 2.0.3. ∎

######  Lemma 4.2.3.

The map $C\mapsto I_{C}:=\text{Conv}(C)$, is a continuous $G$-map from
$\text{Cub}(M)$ to $\mathcal{J}_{2}$.

###### Proof.

The fact that this map is a $G$-map, is clear. We left to prove that it is
continuous.

Let $C_{n}$ be a sequence of subcubes that converges to $C\in\text{Cub}(M)$.
Fix some $x\in C$, its antipodal $x^{*}\in C$, some sequence $x_{n}\in C_{n}$
that converges to $x$, and the corresponding sequence of antipodal points
$x_{n}^{*}\in C_{n}$. We recall that as in the proof of lemma 4.1.4, it
follows that $x_{n}^{*}$ converges to $x^{*}$.

Fix $z\in I_{C}$, and denote $z_{n}:=m(x_{n},x_{n}^{*},z)\in I_{C_{n}}$. By
the continuity of the median, we have

$\underset{n}{\lim}\ z_{n}=\underset{n}{\lim}\
m(x_{n},x_{n}^{*},z)=m(x,x^{*},z)=z$

On the other hand, let $z_{n}\in I_{C_{n}}$ be a (sub-)sequence that converges
to some $z\in M$. Then, again, by contiuity of the median:

$\displaystyle z$ $\displaystyle=\underset{n}{\lim}\ z_{n}=\underset{n}{\lim}\
m(x_{n},x_{n}^{*},z_{n})$ $\displaystyle=m(x,x^{*},z)\in I_{C}$

Thus, indeed, $I_{C_{n}}$ converges to $I_{C}$. ∎

It is natural to wonder whether the map in the other direction, namely, the
map that assigns to every interval its collection of ends, is continuous as
well. Apparently, the answer is no. This is reflected by the fact that the
image of the Ends map is not always a closed subsets of $\text{Cub}(M)$.

This image, on the other hand, can be identified as the collection of maximal
subcubes. Here, we mean that $C$ is maximal, in the the sense that if
$C^{\prime}$ is another subcube containing $C$, than $C$ is a proper face of
$C^{\prime}$. Denote this collection by $\text{Cub}_{m}(M)$.

######  Lemma 4.2.4.

The set $\text{Cub}_{m}(M)$ is a $G_{\delta}$ subset of $\text{Cub}(M)$.

###### Proof.

Define $F_{\epsilon}:=\\{C\in\text{Cub}(M)\ |\
d_{H}(C,\text{Ends}(I_{C}))\geq\epsilon\\}$, where, as always, $I_{C}$ stands
for the convex hull of C. When $\epsilon>0$, then,
$F_{\epsilon}\subset\text{Cub}(M)\backslash\text{Cub}_{m}(M)$. On the other
hand, if $C\notin\text{Cub}_{m}(M)$, then
$\epsilon^{\prime}:=d_{H}(C,\text{Ends}(I_{C}))>0$, and in particular, $C\in
F_{\epsilon^{\prime}}$. Thus, we have

$\text{Cub}(M)\backslash\text{Cub}_{m}(M)=\underset{n\in{\mathbb{N}}}{\bigcup}F_{1/n}$

where the right hand side is an increasing union of subsets. Therefore, it is
enough to show that $F_{\epsilon}$ is a closed set.

Fix some sequence $C_{n}\in F_{\epsilon}$, that converges to some
$C\in\text{Cub}(M)$. After restricting to a subsequence, if necessary, we may
assume that $C_{n}^{\prime}:=\text{Ends}(I_{C_{n}})$ converges to some subcube
$C^{\prime}$. We have the following short sequence of inclusions:

$C\subset C^{\prime}\subset\text{Ends}(I_{C})=\text{Ends}(I_{C^{\prime}})$

We note that the later equality follows from the fact that $I_{C_{n}}$
converges to $I_{C}$. See lemma 4.2.3. Therefore, for any limit $x\in I_{C}$
of some $x_{n}\in\text{Ends}(I_{C_{n}})$, there exists an antipodal point
$x^{*}$, which is the limit of $x_{n}^{*}$. Thus, $x\in\text{Ends}(I_{C})$.

For every n, the fact that $d_{H}(C_{n},\text{Ends}(I_{C_{n}}))\geq\epsilon$,
implies that there exists $x_{n}\in\text{Ends}(I_{C_{n}})$ with
$d(\\{x_{n}\\},C_{n})\geq\epsilon$, that is, for every $y_{n}\in C_{n}$,
$d(x_{n},y_{n})\geq\epsilon$. After further restricting to another sub-
sequences, if necessary, we may assume that $x_{n}$ converges to some $x\in
C^{\prime}$. Take $y\in C$, and $y_{n}\in C_{n}$ that converges to $y$. Then
by continuity of the metric, we have

$\epsilon\leq\underset{n}{\lim}\
d_{H}(\\{x_{n}\\},\\{y_{n}\\})=d_{H}(\\{x\\},\\{y\\})=d(x,y)$

In particular,

$\epsilon\leq d_{H}(C,C^{\prime})\leq d_{H}(C,\text{Ends}(I_{C}))$

Hence, $C\in F_{\epsilon}$, as required.

∎

######  Corollary 4.2.5.

The assignment $I\mapsto\text{Ends}(I)$, is a Borel $G$-map isomorphism from
$\mathcal{J}_{2}$ to $\text{Cub}_{m}(M)$.

###### Proof.

Denote the above assignment by Ends. This is a bijection onto
$\text{Cub}_{m}(M)$, with an inverse denoted by Conv.

We note that a $G_{\delta}$ subset of a Polish space, is Polish. See [Kec95,
Theorem 3.11]. Therefore, by lemmata 4.2.3 and 4.2.4, Conv is a continuous
bijection between two Polish spaces.

The corollary now follows by the fact that such a map is a Borel isomorphism.
See [Kec95, Theorem 15.1]. ∎

######  Lemma 4.2.6.

The map from $\text{Cub}(M)$ to $\text{Prob}(M)^{\Phi}$, defined by the rule
$C\mapsto\lambda_{C}$, where $\lambda_{C}$ is the balanced measure on $C$, is
a homehomorphism (with respect to the $weak^{*}$ topology on
$\text{Prob}(M)^{\Phi}$).

Before proving this Lemma we need the following:

######  Lemma 4.2.7.

Let $\eta\in\text{Prob}(M)^{\Phi}$, and let
$\mathfrak{h}_{1},...,\mathfrak{h}_{m}$ be a collection of closed half-spaces.
Then, $\eta(\cap\mathfrak{h}_{i})\in\\{0\\}\cup\\{2^{-s}\ |\ m\geq s\geq 0\\}$

###### Proof.

Without loss of generality, suppose that $M=\text{supp}(\eta)$. Denote by
$\chi_{\mathfrak{h}_{i}}$ the indicator function of $\mathfrak{h}_{i}$. By
lemma 4.1.6, each $h_{i}$ is clopen. Therefore, the map
$\varphi:M\rightarrow\\{0,1\\}^{m}$, mapping each x to
$\\{\chi_{\mathfrak{h}_{i}}(x)\\}_{i}$ is a continuous median map. It follows
that $\varphi_{*}(\eta)$ is a balanced measure, and by theorem 2.0.6, it is
the uniform measure on a subcube of $\\{0,1\\}^{m}$. Therefore, on each point
in its support, its value is $2^{-s}$, for some natural number $m\geq s\geq
0$. The statement follows. ∎

###### Proof of Lemma 4.2.6.

We first claim that the assignment $\eta\mapsto\text{supp}(\eta)$ is
continuous from $\text{Prob}(M)^{\Phi}$ to $\text{Cub}(M)$. Indeed, suppose
that $\eta_{n}\xrightarrow{w^{*}}\eta$.

Fix $x\in\text{supp}(\eta)$. Since every open neighborhood of $x$ is of
positive $\eta$-measure, for $n>>0$, it is also of positive
$\eta_{n}$-measure, so we can find in it some $x_{n}\in\text{supp}(\eta_{n})$.
Therefore, $\text{supp}(\eta)$ is contained in any limit of
$\text{supp}(\eta_{n})$.

On the other hand, let $x\notin\text{supp}(\eta)$. For every
$y\in\text{supp}(\eta)$, we can find a closed half-space
$\mathfrak{h}_{y}\in\Delta(x,y)$, and such that
$x\in\mathfrak{h}_{y}^{\circ}$. See corollary 2.0.8. By compactness of
$\text{supp}(\eta)$, there are $y_{1},..,y_{m}\in\text{supp}(\eta)$ such that
$\text{supp}(\eta)\subset\bigcup\mathfrak{h}^{*}_{y_{i}}$. Thus, the set
$\bigcap\mathfrak{h}_{y_{i}}$ is an $\eta$-null compact set, so as $\eta$ is a
Radon measure, we can find a continuous function $f$ with
$\bigcap\mathfrak{h}_{y_{i}}\subset\text{supp}(f)$, and $\eta(f)<2^{-m-1}$. It
follows by lemma 4.2.7 that every balanced measure $\xi$ with
$\xi(\bigcap\mathfrak{h}_{y_{i}})>0$, then,
$\xi(\bigcap\mathfrak{h}_{y_{i}})\geq 2^{-m}$. For $n>>0$,
$\eta_{n}(\bigcap\mathfrak{h}_{y_{i}})\leq\eta_{n}(f)<2^{-m}$, thus,
$\eta_{n}(\bigcap\mathfrak{h}_{y_{i}})=0$. Since
$x\in(\bigcap\mathfrak{h}_{y_{i}})^{\circ}$, $\lim\inf
d(x,\text{supp}(\eta_{n}))>0$. In particular, any limit of
$\text{supp}(\eta_{n})$ is contained in $\text{supp}(\eta)$.

By lemma 4.1.3, $\text{Cub}(M)$ is a compact Hausdorff space. Since
$\text{Prob}(M)^{\Phi}$ is a $weak^{*}$-closed subspace of $\text{Prob}(M)$,
it is also a compact Hausdorff space. Clearly, the assignment
$\eta\mapsto\text{supp}(\eta)$ is a bijection between the above two sets (this
is due to the uniqueness of the fully supported balanced measure on any cube).
Thus, this assignment is also a homeomorphism, as required.

∎

######  Lemma 4.2.8 ([CFI16, Lemma A.1]).

Let $\mathcal{H}$ be a countable collection of closed half-spaces. Fix
$i\in\\{0,1/2,1\\}$, and let
$\mathcal{H}^{i}_{\eta}:=\\{\mathfrak{h}\in\mathcal{H}\ |\ \eta(h)=i\\}$.
Then, the map $\text{Prob}(M)\rightarrow 2^{\mathcal{H}}$, defined by
$\eta\mapsto\mathcal{H}^{i}_{\eta}$, is a measurable map with respect to the
weak-* topology on $\text{Prob}(M)^{\Phi}$.

Therefore, also the map
$N:\text{Prob}(M)\rightarrow{\mathbb{N}}\cup\\{\infty\\}$, defined by
$N(\eta)=|\mathcal{H}^{i}_{\eta}|$, is measurable

###### Proof.

The proof of lemma A.1 in [CFI16] works almost perfectly in our situation,
except the fact that here, for $\mathfrak{h}\in\mathcal{H}$, the assignment
$\eta\mapsto\eta(h)$ is measurable, and not necessarily continuous, with
respect to the weak-* topology.

Indeed, there is a sequence $1\geq f_{n}\in C(M)$, that converges point-wise
to $\chi_{\mathfrak{h}}$. Therefore, by dominated convergence theorem, for
every $\eta\in\text{Prob}(M)$, $\underset{n}{\lim}\
\eta(f_{n})=\eta(\mathfrak{h})$. That is, the assignment
$\eta\mapsto\eta(\mathfrak{h})$ is the point-wise limit of the continuous maps
$\eta\mapsto\eta(f_{n})$. In particular, it is measurable. ∎

######  Corollary 4.2.9 ([CFI16, Corollary A.2]).

Let $\mathcal{H}$ be a countable collection of closed half-spaces. The
following maps are measurable:

  1. (1)

$C_{1}:\text{Prob}(M)\rightarrow{\mathbb{N}}\cup\\{\infty\\}$, defined by
$C_{1}(\eta):=|\mathcal{H}_{\eta}|$

  2. (2)

$C_{3}:\text{Prob}(M)\times\text{Prob}(M)\rightarrow{\mathbb{N}}\cup\\{\infty\\}$
defined by $C_{3}(\eta,\rho):=|\mathcal{H}_{\eta}\Delta\mathcal{H}_{\rho}|$

###### Proof.

The measurability of all the above maps follows by lemma 4.2.8, the fact that
composition and product of measurable maps are measurable, and the fact that
the operation $\Delta$ on subsets of $\mathcal{H}$, can be described in terms
of continuous ring operation on $2^{\mathcal{H}}$. ∎

######  Lemma 4.2.10.

Let $C$ and $C^{\prime}$ be two gate-convex subsets of $M$. Then, the gate
projections $\pi_{C}|_{C^{\prime}}$ and $\pi_{C^{\prime}}|_{C}$ are
isomorphism if and only if $\mathcal{H}(C)=\mathcal{H}(C^{\prime})$.

###### Proof.

Suppose that $\pi_{C}|_{C^{\prime}}$ and $\pi_{C^{\prime}}|_{C}$ are
isomorphism. Let $\mathfrak{h}\in\mathcal{H}(C)$. It follows by lemma
2.0.4(1), and the fact that both $\pi_{C}(C^{\prime})\cap\mathfrak{h}$ and
$\pi_{C}(C^{\prime})\cap\mathfrak{h}^{*}$ are non-empty, that
$\mathfrak{h}\in\mathcal{H}(C^{\prime})$. The claim now follows by symmetry of
the argument.

Suppose now that $\mathcal{H}(C)=\mathcal{H}(C^{\prime})$. Assume that $x\in
C\backslash\pi_{C}(C^{\prime})$. Then,
$\varnothing\neq\Delta(x,\pi_{C}(C^{\prime}))\subset\mathcal{H}(C)\backslash\mathcal{H}(C^{\prime})$,
which is a contradiction. Thus, $\pi_{C}|_{C^{\prime}}$ is surjective. By
lemma 2.0.4(2), $\pi_{C}=\pi_{C}\circ\pi_{C^{\prime}}$. That is,
$\pi_{C}|_{C^{\prime}}\circ\pi_{C^{\prime}}|_{C}=\pi_{C}|_{C}=id_{C}$. By
symmetry, the statement follows.

∎

For a tuple $(x,y,\eta)\in M^{2}\times\text{Prob}(M)$, we denote by
$\mu_{(x,y,\eta)}\in\text{Prob}(M^{3})$ the probability measure associated to
the following functional:

$\forall s\in C(M^{3}),\ s\mapsto\int_{M}s(x,m(x,y,c),c)d\eta(c)$

######  Lemma 4.2.11.

The map $\mu_{(.)}:M^{2}\times\text{Prob}(M)\rightarrow\text{Prob}(M^{3})$ is
continuous.

###### Proof.

Suppose $(x_{n},y_{n},\eta_{n}),(x,y,\eta)\in M^{2}\times\text{Prob}(M)$ are
such that $(x_{n},y_{n},\eta_{n})\xrightarrow{n}(x,y,\eta)$. Fix $s\in
C(M^{3})$ and $\epsilon>0$. By continuity of $m$ and $s$ and the compactness
of $M^{3}$, for $n\gg 0$,
$|s(x_{n},m(x_{n},y_{n},c),c)-s(x,m(x,y,c),c)|<\epsilon/2$ for every $c\in M$.
Also, for $n\gg 0$,

$\big{|}\int_{M}s(x,m(x,y,c),c)d\eta(c)-\int_{M}s(x,m(x,y,c),c)d\eta_{n}(c)\big{|}<\epsilon/2$

Therefore, for $n\gg 0$,

$\displaystyle|\mu_{(x,y,\eta)}(s)-\mu_{(x_{n},y_{n},\eta_{n})}(s)|$
$\displaystyle\leq\big{|}\int_{M}s(x,m(x,y,c),c)d\eta(c)-\int_{M}s(x,m(x,y,c),c)d\eta_{n}(c)\big{|}$
$\displaystyle+\big{|}\int_{M}s(x,m(x,y,c),c)d\eta_{n}(c)-\int_{M}s(x_{n},m(x_{n},y_{n},c),c)d\eta_{n}(c)\big{|}$
$\displaystyle\leq\epsilon/2+\int_{M}|s(x,m(x,y,c),c)d-s(x_{n},m(x_{n},y_{n},c),c)|d\eta_{n}(c)$
$\displaystyle\leq\epsilon$

as needed. ∎

###  4.3. General results about measurability

######  Lemma 4.3.1.

Let $p:X\rightarrow Y$ be a countable-to-one Borel map between two standard
Borel spaces. Then $p$ is Borel bimeasurable, i.e. the image of every Borel
set $B\subset X$, is Borel in $Y$.

###### Proof.

Fix a Borel subset $B\subset X$, and define the following set:

$P:=\\{(x,y)\in B\times Y\ :\ p(x)=y\\}$

Note that this is a Borel subset of $X\times Y$. To see that, it is enough to
observe that $P$ is equal to the set $B\times Y\cap(p\times
id_{Y})^{-1}(\Delta(Y))$, where $p\times id_{Y}$ is the map from $X\times Y$
to $Y\times Y$ defined by $(x,y)\mapsto(p(x),y)$, and $\Delta(Y)\subset
Y\times Y$ is the diagonal.

By assumption, each section $P_{y}$ is countable. Therefore, according to
[Kec95, Lemma 18.12], $p(B)=\text{proj}_{Y}(P)$ is Borel. ∎

######  Lemma 4.3.2.

Let $p:X\rightarrow Y$ be a countable-to-one Borel map between two standard
Borel spaces. Then, there exists a Borel subset $A\subset{\mathbb{N}}\times
Y$, and Borel isomorphism $\psi:X\rightarrow A$, such that the following
diagram commutes:

${X}$${A}$${Y}$${Y}$$\scriptstyle{\psi}$$\scriptstyle{p}$$\scriptstyle{\text{proj}_{Y}}$$\scriptstyle{id_{Y}}$

###### Proof.

First, consider the following equivalence relation on X:

$E:=\\{(x,x^{\prime})\in X\times X\ :\ p(x)=p(x^{\prime})\\}=(p\times
p)^{-1}(\Delta(Y))$

Where $p\times p$ is the map from $X\times X$ to $Y\times Y$ defined by
$(x,x^{\prime})\mapsto(p(x),p(x^{\prime}))$, and $\Delta(Y)\subset Y\times Y$
is the diagonal. Therefore, it is clear that $E$ is a Borel subset of $X\times
X$, and by assumption, all of its equivalence classes are countable.

Fix any section $s:Y\rightarrow X$. It follows Lemma 4.3.1 that $s$ is Borel.
Indeed, choose any Borel subset $B\subset X$. Then, $s^{-1}(B)=p(B)$, which is
a Borel.

According to [FM77, Theorem 1], there exists a countable group
$H=\\{h_{i}\\}_{i\in{\mathbb{N}}}$ of Borel automorphisms of $X$, such that
the following holds:

$E=\\{(x,h_{i}.x)\ :\ \forall x\in X,i\in{\mathbb{N}}\\}$

Without loss of generality, assume that $h_{1}=e_{H}$. For every $x\in X$, we
denote by $c_{x}$ the elements of $Y$ for which $x\in p^{-1}(c_{x})$. Also, we
denote by $n_{x}$ the minimal positive integer $i$ such that
$h_{i}.s(c_{x})=x$. Finaly, we define $\psi:X\rightarrow{\mathbb{N}}\times Y$,
by $x\mapsto(n_{x},c_{x})$.

We claim that $\psi$ is measurable. Note that it is enough to show that sets
of the form $\psi^{-1}(\\{i\\}\times A)$ are Borel, whenever $A\subset Y$ is
Borel. Fix such a Borel set $A\subset Y$. We will prove by induction on $i$
that these sets are indeed Borel.

For $i=1$, $\psi^{-1}(\\{1\\}\times A)=s(A)$, and this is a Borel set,
according to Lemma 4.3.1. Now, note that we have the following equality:

$\psi^{-1}(\\{i\\}\times A)=h_{i}.s(A)\backslash\underset{1\leq
j<i}{\cup}\psi^{-1}(\\{j\\}\times A)$

By the induction hypothesis, $\psi^{-1}(\\{i\\}\times A)$ is Borel.

We finish by noting that $\psi$ is injective, and therefore, it is
bimeasurable by Lemma 4.3.1. Hence, $\psi$ is an isomorphism between $X$ and
$\psi(X)\subset{\mathbb{N}}\times Y$. ∎

######  Lemma 4.3.3.

Let $Y$ be a compact metric space, and consider the factor map
$\text{proj}_{Y}:{\mathbb{N}}\times Y\rightarrow Y$. Fix some
$\mu\in\text{Prob}(Y)$, and denote by $\mathfrak{G}$ the space of classes of
$\mu$-a.e. defined measurable sections for $\text{proj}_{Y}$. Consider the
following metric $d_{\mu}$ on $\mathfrak{G}$

$\ d_{\mu}(f,f^{\prime}):=\int_{Y}(1-\delta_{f(c)f^{\prime}(c)})d\mu(c)$

where $\delta_{ab}=1$ if $a=b$ and zero otherwise. Then, the metric space
$(\mathfrak{G},d_{\mu})$ is separable.

###### Proof.

Fix a countable basis for the topology $\\{U_{i}\\}_{i}$, and denote by
$\mathcal{B}$ the Boolean algebra that is generated by this basis. Consider
the collection of sections $s:Y\rightarrow{\mathbb{N}}\times Y$, of the form

$s(y)=(\sum_{i=1}^{m}j_{i}\chi_{B_{i}}(y),y)$

where $m$ and $j_{i}$ run over all positive numbers, and the $B_{i}$’s run
over all the elements of $\mathcal{B}$, and where $\chi_{B_{i}}$ denotes the
characteristic function of $B_{i}$. Observe that this is a countable
collection. We now show that it is dense in $(\mathfrak{G},d_{\mu})$.

Let $f\in\mathfrak{G}$ and $\epsilon>0$. For every $n\in{\mathbb{N}}$, denote
by $A_{n}:=f^{-1}(\\{n\\}\times Y)$. As $Y=\cup_{n}A_{n}$, there exists
$k\in{\mathbb{N}}$, such that $\mu(\cup_{n\geq k}A_{n})<\epsilon/2$. Note that
$\mu$ is a Radon probability measure, therefore, for every $k>n$, we can
choose an open set $V_{n}$ such that $\mu(V_{n}\Delta
A_{n})<\epsilon/2^{k+1}$. For every such $V_{n}$, we can choose some
$B_{n}\in\mathcal{B}$, such that $B_{n}\subset V_{n}$ and $\mu(V_{n}\backslash
B_{n})<\epsilon/2^{k+1}$. Finely, we define the following section
$s:Y\rightarrow{\mathbb{N}}\times Y$:

$s(y)=(\sum_{n=1}^{k-1}n\chi_{B_{n}}(y),y)$

Note that we have the following inclusion:

$\\{y\in\text{supp}(f)\ :\ s(y)\neq f(y)\\}\subset(\cup_{n<k}B_{n}\Delta
A_{n})\bigcup(\cup_{n\geq k}A_{n})\subset(\cup_{n<k}(V_{n}\Delta A_{n}\cup
V_{n}\backslash B_{n}))\bigcup(\cup_{n\geq k}A_{n})$

Therefore,

$\displaystyle d_{\mu}(f,s)$
$\displaystyle=\int_{Y}1-\delta_{f(y)s(y)}d\mu(y)=\mu(\\{y\in\text{supp}(f)\
:\ s(y)\neq f(y)\\})$ $\displaystyle\leq\sum_{n=1}^{k-1}\mu(V_{n}\Delta
A_{n}\cup V_{n}\backslash B_{n})+\mu(\cup_{n\geq k}A_{n})$
$\displaystyle\leq\sum_{n=1}^{k-1}\epsilon/2^{k}+\epsilon/2<\epsilon$

As needed. ∎

###  4.4. The space $\text{Map}_{G}(S,\text{Prob}(X))$

Consider a standard Borel space $S$, endowed with a probability measure
$\mu\in\text{Prob}(S)$ and a separable Banach space $E$. Define the following
space

$L^{1}(S,E):=\big{\\{}f:S\rightarrow E\ |\ f\text{ measurable and
}\int\|f(s)\|\mathrm{d}\mu\leq\infty\big{\\}}/\sim$

Where $f\sim f^{\prime}$ if and only if the map
$s\mapsto\|f(s)-f^{\prime}(s)\|$ is of $\mu$-measure zero. As discussed in
[Zim78, p. 357], the space $L^{1}(S,E)$ is a Banach space. Its dual can be
identified with the collection of all weakly measurable maps
$\lambda:S\rightarrow E^{*}$, such that the assignment
$s\mapsto\|\lambda(s)\|$ is an element of $L^{\infty}(S)$. See also [Edw65,
Theorem 8.18.2]. We denote this space by $L^{\infty}(S,E^{*})$. The
identification is given by
$\lambda\mapsto\big{(}f\mapsto\tilde{\lambda}(f)\big{)}$, where we define
$\tilde{\lambda}(f):=\langle\lambda,f\rangle:=\int\langle\lambda(s),f(s)\rangle\mathrm{d}\mu(s)$,
for $\lambda\in L^{\infty}(S,E^{*})$ and $f\in L^{1}(S,E)$. Endowed with the
essential sup norm, the space $L^{\infty}(S,E^{*})$ becomes a Banach space,
and the above identification turns into isometric isomorphism.

Suppose now that we are given a compact metric space $(X,d)$, and fix our
Banach space $E$ to be $(C(X),\|\ \|_{\text{sup}})$. Define the following
subspace of $L^{\infty}(S,C(X)^{*})$:

$\text{Map}(S,\text{Prob}(X)):=\big{\\{}\lambda\in L^{\infty}(S,C(X)^{*})\
|\text{ for a.e. s, }\lambda(s)\in\text{Prob}(X)\big{\\}}$

Note that when endowed $\text{Prob}(X)$ with the weak-* topology, then the
space $\text{Map}(S,\text{Prob}(X))$ is exactly the collection of all
measurable maps $\lambda:S\rightarrow\text{Prob}(X)$. Indeed, it is enough to
notice that for such $\lambda$, the map $s\mapsto\|\lambda(s)\|$ is just the
constant map $1$. Moreover, it equals exactly to the space $B$ as in [Zim78,
Proposition 2.2], when the Borel field $\\{A_{s}\\}$ is defined to be the
constant field $A_{s}=\text{Prob}(X)$. According to the proposition, this
space is a compact and convex subspace of $L^{\infty}(S,C(X)^{*})$.

Let $G$ be a lcsc group. Suppose that $(S,\mu)$ is an ergodic $G$-space, and
that $X$ is endowed with a continuous $G$-action. We have a natural action of
$G$ on $L^{\infty}(S,C(X)^{*})$, given by $(g.\lambda)(s):=\lambda(gs)$, for
$g\in G$, $\lambda\in L^{\infty}(S,C(X)^{*})$ and $s\in S$. According to the
proof of [Zim78, Theorem 2.1], this is the induced adjoint action to a
continuous action of $G$ on $L^{1}(S,C(X))$, with respect to the trivial
cocycle $\alpha:S\times G\rightarrow\text{Iso}(C(X))$,
$\alpha(s,g)=Id_{C(X)}$. Therefore, for every $g\in G$, the assignment
$\lambda\mapsto g.\lambda$ is a continuous automorphism of
$L^{\infty}(S,C(X)^{*})$.

Clearly, the space $\text{Map}(S,\text{Prob}(X))$ is invariant under the
action of $G$. Denote by $\text{Map}_{G}(S,\text{Prob}(X))$ the collection of
all measurable $G$-equivariant maps $\lambda:S\rightarrow\text{Prob}(X)$. By
all the above discussion, it follows that $\text{Map}_{G}(S,\text{Prob}(X))$
is a closed, thus compact, and convex subspace of
$\text{Map}(S,\text{Prob}(X))$.

The motivation to introduce the space $\text{Map}_{G}(S,\text{Prob}(X))$
derived by our use of the concept of amenable action. See [Zim78]. A key
feature of amenable action $G\curvearrowright B$, where $B$ is a Lebesgue
$G$-space, is that given an affine $G$-action on a convex $weak^{*}$ compact
set $Q\subset E^{*}$, where $E$ is a separable Banach space, there exists a
$G$-map $\phi\in\text{Map}_{G}(B,Q)$. In particular, if $G$ acts on a
metrizable compact space $X$, then the space
$\text{Map}_{G}(B,\text{Prob}(X))$ is not empty.

##  5\. Isometric ergodicity

Let $G$ be a lcsc group. A Lebesgue $G$-space $(X,\nu)$ is isometrically
ergodic, if for every separable metric space $Y$ on which $G$ acts by
isometries, and every $G$-equivariant map $\varphi:X\rightarrow Y$, $\varphi$
is essentially constant.

Given a Borel map $q:\mathcal{M}\rightarrow V$ between standard Borel spaces,
a metric on q is a Borel function
$d:\mathcal{M}\times_{V}\mathcal{M}\rightarrow[0,\infty]$ whose restriction
$d_{v}$ to each fiber $\mathcal{M}_{v}=q^{-1}(\\{v\\})$ is a separable metric.
A fiber-wise isometric $G$-action on such $\mathcal{M}$ consists of
q-compatible actions $G\curvearrowright\mathcal{M},G\curvearrowright V$, so
that the maps between the fibers
$g:\mathcal{M}_{v}\rightarrow\mathcal{M}_{gv}$ are isometries.

Suppose we have actions of $G$ on two such Borel spaces $\mathcal{M}$ and $V$,
and a Borel map $q:\mathcal{M}\rightarrow V$ in which all the fibers are
countable. Define $d:\mathcal{M}\times_{V}\mathcal{M}\rightarrow[0,\infty]$ to
be the trivial metric, that is, $d(m,n)=\delta_{mn}$. Then, the action of $G$
on $\mathcal{M}$, equipped with the metric $d$ above on q, is an example of a
fiber-wise isometric $G$-action.

A map $p:A\rightarrow B$ between Lebesgue $G$-spaces, is relatively
isometrically ergodic if for every fiber-wise isometric action on
$q:\mathcal{M}\rightarrow V$, and $q$-compatible maps
$f:A\rightarrow\mathcal{M}$, $f_{0}:B\rightarrow V$, there exists a compatible
$G$-map $f_{1}:B\rightarrow\mathcal{M}$ making the following diagram
commutative:

(1)
${A}$${\mathcal{M}}$${B}$${V}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{f_{0}}$$\scriptstyle{f_{1}}$

We say that a pair of Lebesgue $G$-spaces $(B_{-},B_{+})$ is a boundary pair
if the actions of $G$ on $B_{+}$ and $B_{-}$ are amenable, and the projection
maps $p_{\pm}:B_{-}\times B_{+}\rightarrow B_{\pm}$ are relatively
isometrically ergodic.

######  Remark 5.0.1.

Let $(B_{-},B_{+})$ be a boundary pair for $G$.

  1. (1)

For a Lebesgue $G$-spaces A and B, if the projection $A\times B\rightarrow B$
is relatively isometrically ergodic, then, A is isometrically ergoic. See
[BF14, Proposition 2.2.(iii)].

In particular, both $B_{+}$ and $B_{-}$ are isometrically ergodic.

  2. (2)

The action of $G$ on $B_{-}\times B_{+}$ is ergodic. Indeed, if $S\subset
B_{-}\times B_{+}$ is a Borel $G$-invariant subset, then its characteristic
map $\chi_{S}$ is a $G$-invariant map into the set $\\{0,1\\}$. Completing
diagram 1 with $f=\chi_{A}$, $A=B_{-}\times B_{+}$, $B=B_{+}$ and
$V=\\{pt\\}$, will generate a Borel $G$-invariant map
$f_{1}:B_{+}\rightarrow\\{0,1\\}$. It follows by the previous remark, that
$f_{1}$ must be constant.

######  Theorem 5.0.2 ([BF14, Theorem 2.7]).

Let $\mu_{+}$ be a probability measure on $G$, that is continuous with respect
to the Haar measure and is not supported on a proper closed sub-semigroup.
Define the probability measure $\mu_{-}$ to be the inverse of $\mu_{+}$. That
is, $\mu_{-}(A)=\mu_{+}(A^{-1})$. We denote by $(B_{+},\nu_{+})$ (resp. by
$(B_{-},\nu_{-})$) the Furstenberg-Poisson boundary corresponds to
$(G,\mu_{+})$ (resp. to $(G,\mu_{-})$). See [BS06, section 2] and references
therein for a comprehensive discussion on the Furstenberg-Poisson boundary.
The pair $(B_{-},B_{+})$ is a boundary pair for $G$ and for any of its
lattices.

We now turn to apply the theory of boundary pairs in the case of convex
minimal actions on sclocc median algebras. For the definition of the space
$\mathcal{J}_{n}^{\subseteq}$, see 4.1.1

######  Proposition 5.0.3.

Let $G$ be a countable discrete group, $M$ a sclocc $G$-median algebra and
suppose that $G\curvearrowright M$ is convex minimal. Let $(B_{-},B_{+})$ be a
boundary pair for $G$, and suppose that we have two Borel $G$-maps
$\varphi_{\pm}:B_{\pm}\rightarrow\mathcal{J}_{n}$. Then, for almost every
$(\theta_{-},\theta_{+})$,

$\pi_{\varphi_{+}(\theta_{+})}(\varphi_{-}(\theta_{-}))=\varphi_{+}(\theta_{+})$

###### Proof.

Consider the map $\mathfrak{f}:B_{-}\times
B_{+}\rightarrow\mathcal{J}_{n}^{\subseteq}$ defined by

$(\theta_{-},\theta_{+})\mapsto(\varphi_{+}(\theta_{+}),\pi_{\varphi_{+}(\theta_{+})}(\varphi_{-}(\theta_{-}))$

It is a measurable $G$-map. For the measurably part, see lemma 4.2.1. As
$(B_{-},B_{+})$ is a boundary pair for $G$, there is a compatible $G$-map
$\varphi^{\prime}:B_{+}\rightarrow\mathcal{J}_{n}^{\subseteq}$ making the
following diagram commutative:

${B_{-}\times
B_{+}}$${\mathcal{J}_{n}^{\subseteq}}$${B_{+}}$${\mathcal{J}_{n}}$$p_{+}$$\mathfrak{f}$$\varphi_{+}$$\varphi^{\prime}$$p$

where $p_{+}:B_{-}\times B_{+}\rightarrow B_{+}$ and
$p:\mathcal{J}_{n}^{\subseteq}\rightarrow\mathcal{J}_{n}$ are the natural
propjections. Note that it follows by the diagram, that for almost every
$(\theta_{+},\theta_{-})$

$(\dagger)\mathfrak{f}(\theta_{+},\theta_{-})=\varphi^{\prime}(\theta_{+})=(\varphi_{+}(\theta_{+}),\pi_{\varphi_{+}(\theta_{+})}(\varphi_{-}(\theta_{-})))$

For the sake of contradiction, suppose there exists a subset $A\subset
B_{-}\times B_{+}$ of positive $\nu_{-}\otimes\nu_{+}$-measure, such that for
every $(\theta_{+},\theta_{-})\in A$,

$(\ddagger)\pi_{\varphi_{+}(\theta_{+})}(\varphi_{-}(\theta_{-}))\varsubsetneq\varphi_{+}(\theta_{+})$

As $A$ is $G$-invariant, it follows by ergodicity that it is of full measure.
By Fubin’s theorem, we can find $\theta_{+}\in B_{+}$ and $\Omega\subset
B_{-}$ with full $\nu_{-}$-measure, such that $(\dagger)$ and $(\ddagger)$
holds for the pair $(\theta_{+},\theta_{-})$, for every $\theta_{-}\in\Omega$.
In particular, for every $\theta,\theta^{\prime}\in\Omega$

$I^{\prime}:=\pi_{\varphi_{+}(\theta_{+})}(\varphi_{-}(\theta))=\pi_{\varphi_{+}(\theta_{+})}(\varphi_{-}(\theta^{\prime}))\varsubsetneq\varphi_{+}(\theta_{+})=:I$

Fix $x\in I\backslash I^{\prime}$, and a closed half-space
$\mathfrak{h}\in\Delta(I^{\prime},x)$. It follows from lemma 2.0.4.(1), that
for $\theta\in\Omega$, $\varphi_{-}(\theta)\subset\mathfrak{h}$. Since we may
assume $\Omega$ is $G$-invariant, we got a contradiction to the minimality of
the action: the closure of the convex hull of the union of all
$\varphi_{-}(\theta)$ for $\theta\in\Omega$, is a $G$-invariant closed convex
set that not contains the element x. Therefore, the proposition holds.

∎

##  6\. Proof of Theorems B, C and D

In what follows, the first two subsections are devoted to the proof of theorem
C and the latter subsections are devoted to the proofs of theorems B and D
correspondingly.

###  6.1. Proof of Theorem C: Existence of boundary maps

Let M be a minimal sclocc $G$-median algebra, with countable intervals and
with no cubical factors. Let $(B_{-},\nu_{-})$ and $(B_{+},\nu_{+})$ be two
$G$-Lebesgue spaces, such that the pair $(B_{-},B_{+})$ is a boundary pair for
$G$. In this subsection we will prove the existence of a $G$-map $B_{-}\to M$.
The proof for $B_{+}$ is done in a similar way.

Observe that the self-median operator $\Phi$ acts continuously on the compact
convex spaces

$\text{Map}_{G}(B_{\pm},\text{Prob}(M)).$

These spaces are not empty, as the actions of $G$ on $B_{+}$ and $B_{-}$ are
amenable. See subsection 4.4. It follows by Tychonoff’s theorem that
$\text{Map}_{G}(B_{\pm},\text{Prob}(M))^{\Phi}=\text{Map}_{G}(B_{\pm},\text{Prob}(M)^{\Phi})\neq\varnothing$.
See [Tyc35]. For the rest of this section we fix two maps

$\varphi_{\pm}\in\text{Map}_{G}(B_{\pm},\text{Prob}(M)^{\Phi}).$

In sake of abbreviation, we might confuse between an element $\theta\in
B_{\pm}$ and its image under the appropriate map.

Fix a countable collection of closed half-spaces, as in corollary 2.0.8,
taking any additional half-spaces so that the collection will be
$G$-invariant, and denote it by $\mathcal{H}$. For every $\theta_{\pm}\in
B_{\pm}$ and every $i\in\\{0,1/2,1\\}$, we define the following subset of
$\mathcal{H}$:

$\mathcal{H}_{\theta}^{i}:=\\{\mathfrak{h}\ |\
\theta_{\pm}(\mathfrak{h})=i\\}$

As almost every $\theta_{\pm}$ is balanced, it follows by theorem 2.0.6 that
its support constitutes a subcube of $M$. Therefore, the sets
$\mathcal{H}_{\theta_{\pm}}^{0},\mathcal{H}_{\theta_{\pm}}^{1/2}$ and
$\mathcal{H}_{\theta_{\pm}}^{1}$ constitute a partition of $\mathcal{H}$.

Among all these three sets, only those of the form
$\mathcal{H}_{\theta_{\pm}}^{1/2}$ will be in use. So let us denote them with
the abbreviated notation $\mathcal{H}_{\theta\pm}$. Consider the following
map:

$\displaystyle(\dagger)\ $ $\displaystyle\psi:B_{-}\times
B_{+}\rightarrow({\mathbb{N}}\cup\infty)$
$\displaystyle(\theta_{-},\theta_{+})\mapsto|\mathcal{H}_{\theta_{-}}\Delta\mathcal{H}_{\theta_{+}}|$

According to corollary 4.2.9, this map is Borel and clearly it is
$G$-invariant. As $G$ acts ergodicly on $B_{-}\times B_{+}$, the essential
image of $\psi$ is some constant $m\in{\mathbb{N}}\cup\\{\infty\\}$. We claim
that the essential image is in fact zero. This will promise us that the image
of $\varphi_{+}$ inside $\text{Prob}(M)$ consists of delta measures.

We would like to draw the reader’s attention, that the use of the map
$(\dagger)$ is greatly inspired by the proof of theorem 7.1 in [Fer18],
although the analysis is substantially different.

For a generic $\theta_{\pm}\in B_{\pm}$, we set
$C_{\pm}:=C_{\theta_{\pm}}:=\text{supp}\big{(}\varphi_{\pm}(\theta_{\pm})\big{)}$,
$I_{\pm}:=I_{\theta_{\pm}}:=\overline{\text{Conv}(C_{\pm})}=\text{Conv}(C_{\pm})$,
and we denote by $\pi_{\pm}$ or $\pi_{\theta_{\pm}}$ the gate-projections of
$I_{\pm}$. As the convex hull of a cube is an interval, the assignments
$\theta_{\pm}\mapsto I_{\pm}$ are maps from $B_{\pm}$ to $\mathcal{J}_{2}$. We
consider also the assignments $I\mapsto\text{supp}(I)$ from $\mathcal{J}_{2}$
to $\text{Cub}(M)$, and $C$ mapped to its unique fully supported balanced
measure $\lambda_{C}$, from $\text{Cub}(M)$ to $\text{Prob}(M)^{\Phi}$. It
follows by lemmata 4.2.3, 4.2.6 and corollary 4.2.5, that all these maps and
their various compositions, are measurable. It is easy to check that they are
also $G$-maps.

Let us show that $m=0$. For the sake of contradiction suppose that for almost
every $(\theta_{-},\theta_{+})$,

$\mathcal{H}_{\theta_{+}}\Delta\mathcal{H}_{\theta_{-}}\neq\varnothing$

By proposition 5.0.3, for almost every $(\theta_{-},\theta_{+})$,
$I_{+}=\pi_{+}(I_{-})$ and $I_{-}=\pi_{-}(I_{+})$. The former assertion
implies that $\mathcal{H}_{\theta_{+}}\subset\mathcal{H}_{\theta_{-}}$, while
the latter means that
$\mathcal{H}_{\theta_{-}}\subset\mathcal{H}_{\theta_{+}}$. This is a
contradiction, thus, we may assume from now on that $m=0$.

Let $\Omega\subset B_{-}$ be a Borel set of $\nu_{-}$-measure 1, and
$\theta_{+}\in B_{+}$ such that for every $\theta_{-}\in\Omega$

$\mathcal{H}_{\theta_{+}}=\mathcal{H}_{\theta_{-}}$

In particular, for every $\theta,\theta^{\prime}\in\Omega$

$\mathcal{H}^{\prime}:=\mathcal{H}_{\theta}=\mathcal{H}_{\theta^{\prime}}$

Moreover, we hvae

$\mathcal{H}(I_{\theta})=\mathcal{H}_{\theta}=\mathcal{H}_{\theta^{\prime}}=\mathcal{H}(I_{\theta^{\prime}})$

Therefore, it follows by lemma 4.2.10 that $I_{\theta}$ and
$I_{\theta^{\prime}}$ are isomorphic, and the isomorphisms are given by
$\pi_{\theta}$ and $\pi_{\theta^{\prime}}$.

Let us now show that all these intervals are in fact points. Fix
$\theta_{0}\in\Omega$. By lemma 4.2.2 and the above discussion, we may
consider the induced action of $G$ on $I_{0}:=I_{\theta_{0}}$. Observe that
this action is minimal. Indeed, as $\pi_{0}:=\pi_{I_{0}}$ is a continuous
median $G$-map, the pre-image of any closed $G$-invariant median subalgebra of
$I_{0}$, is a median subalgebra with the same properties of M.

In particular, since $C_{0}:=C_{\theta_{0}}$ is a $G$-invariant median
subalgera of $I_{0}$, we get that $I_{0}=C_{0}$. As $\theta_{0}$ is generic,
it follows that for almost every $\theta\in B_{-}$, $I_{\theta}=C_{\theta}$.

But now, $\pi_{0}$ is a surjective morphism onto a cube. Since $M$ has no
cubical factors, $C_{0}$ must be a point. Similarly, $C_{\theta}$ is a point,
for every $\theta\in\Omega$. That is, $\theta\mapsto C_{\theta}$ can be viewed
as a Borel $G$-map between $B_{-}$ to $M$. For $\theta\in B_{-}$, we denote
its corresponding point in $M$ by $x_{\theta}$.

###  6.2. Proof of Theorem C: Uniqueness of boundary maps

We now show that there are no other $G$-maps from $B_{-}$ into
$\text{Prob}(M)$. Denote by $\phi_{\pm}:B_{\pm}\rightarrow M$ our boundary
maps, and suppose we have another $G$-map
$\psi:B_{-}\rightarrow\text{Prob}(M)$ .

Define $Y=M\times\text{Prob}(M)$ and set $T:=\\{(x,y,z)\in M^{3}\ :\
y\in[x,z]\\}$. We denote by $p_{i}:T\rightarrow M$ the projection onto the
$i^{\prime}th$ coordinate of elements in $T$. Define the following space:

$X:=\\{\eta\in\text{Prob}(T)\ |\ \exists a_{\eta}\in M\text{ s.t.
}(p_{1})_{*}(\eta)=\delta_{a_{\eta}},\text{ and }p_{3}\text{ is a measure
iso.}\\}$

We have a natural projection $q:X\rightarrow Y$, namely
$\eta\mapsto(a_{\eta},(p_{3})_{*}(\eta))$. This map will fit to the right side
of diagram 1. For some purposes, it will be convenient to give an equivalent
description of $X$. For every $(a_{y},\eta_{y})=y\in Y$, define the following
space:

$X_{y}:=\\{f\in\text{Map}((M,\eta_{y}),M)\ |\eta_{y}-\text{a.e. }c\in
M,(a_{y},f(c),c)\in T\\}$

Here, the space $\text{Map}((M,\eta_{y}),M)$ denotes the space of classes of
measurble functions defined up to $\eta_{y}$-almost everywhere. We claim that
$X_{y}$ is isomorphic to the fiber of $y$ in $X$ under the projection $q$.
Indeed, fix $f\in X_{y}$, and define the following positive functional on
$C(T)$:

$\forall s\in C(T):s\mapsto\int_{M}s(a_{y},f(c),c)d\eta_{y}(c)$

Denote by $\eta_{y,f}$ the corresponding measure, and let us check that
$\eta_{y,f}\in q^{-1}(y)$. It is rather easy to verify that
$(p_{1})_{*}(\eta_{y,f})=\delta_{a_{y}}$ and that
$(p_{3})_{*}(\eta_{y,f})=\eta_{y}$. Lastly we note that the $\eta_{y}$-a.e.
defined map $\overline{f}:M\rightarrow T$, $c\mapsto(a_{y},f(c),c)$, is the
measurable inverse of $p_{3}$ as a map with the domain $(T,\eta_{y,f})$. Given
a measure $\eta\in q^{-1}(y)$, denote by $p_{3}^{-1}$ the measurable inverse
of $p_{3}:(T,\eta)\rightarrow(M,\eta_{y})$, then indeed,
$\eta=\eta_{y,p_{2}\circ p_{3}^{-1}}$.

We now move to define a metric on the projection $q$, in the sense of section
5. Given two elements $a,b\in M$, denote by $\delta_{ab}$ the function that
equals 1 if $a=b$, and 0 otherwise. For $(a_{y},\eta_{y})=y\in Y$ and two
functions $f,f^{\prime}\in X_{y}$, define $d_{y}(f,f^{\prime})$ to be the
following value:

$(*)\
d_{\eta_{y}}(f,f^{\prime}):=d_{y}(f,f^{\prime}):=\int_{M}1-\delta_{f(c)f^{\prime}(c)}d\eta_{y}(c)$

It is rather simple to check that this is indeed a metric. We claim that the
space $(X_{y},d_{y})$, is separable metric space. In addition, as described in
section 5, we need also to check that the map we get
$d:X\times_{Y}X\rightarrow[0,\infty]$ is Borel. See Lemmata 6.2.1 and 6.2.2
for the details.

The diagonal action of $G$ on $M^{3}$, induces a natural action on $T$. As
this action commutes with both $p_{1}$ and $p_{3}$, it gives rise to an
induced action on $X$. Finally, consider the natural action of $G$ on $Y$. Let
us define an action on $X$, which turns $q$ into a $\Gamma$-map: Given
$\eta\in X$ and $g\in G$, the measurable inverse of $p_{3}$ with respect to
$g_{*}\eta$ is $g_{*}(g\cdot p_{3}^{-1})$, where $p_{3}^{-1}$ is the
measurable inverse with respect to $\eta$. Therefore, the action of $g\in G$
on the fiber $X_{y}$ for $y\in Y$ translates to

$f\in X_{y}\mapsto g_{*}(g\cdot f)\in X_{gy}$

The above turns the action of $G$ on $q$ into a fiber-wise isometric action.
Indeed, if $y\in Y$, and $f,f^{\prime}\in X_{y}$:

$\displaystyle d_{gy}(g_{*}(g\cdot f),g_{*}(g\cdot f^{\prime}))$
$\displaystyle=\int_{M}1-\delta_{g_{*}(g\cdot f)(c)g_{*}(g\cdot
f^{\prime})(c)}dg_{*}\eta_{y}(c)$
$\displaystyle=\int_{M}1-\delta_{g_{*}(f)(c)g_{*}(f^{\prime})(c)}dg_{*}\eta_{y}(c)$
$\displaystyle=\int_{M}1-\delta_{g_{*}(f)(gc)g_{*}(f^{\prime})(gc)}d\eta_{y}(c)$
$\displaystyle=\int_{M}1-\delta_{f(c)f^{\prime}(c)}d\eta_{y}(c)$
$\displaystyle=d_{y}(f,f^{\prime})$

As needed.

Let us now define the horizontal arrows in diagram 1. The lower arrow, namely
$\varphi:B_{-}\rightarrow Y$, is defined by
$\theta_{-}\mapsto(\phi_{-}(\theta_{-}),\psi(\theta_{-}))$. We define the
upper horizontal arrow in terms of the fibers of $q$. Given a generic pair
$(\theta_{-},\theta_{+})\in B_{-}\times B_{-}$, we define a measurable map
$f_{(\theta_{-},\theta_{+})}:M\rightarrow M$ by
$f_{(\theta_{-},\theta_{+})}(c)=m(\phi_{-}(\theta_{-}),\phi_{+}(\theta_{+}),c)$.
Finally, the map $\varphi^{\prime}:B_{-}\times B_{+}\rightarrow X$ is defined
as follows:

$\varphi^{\prime}(\theta_{-},\theta_{+})=f_{(\theta_{-},\theta_{+})}\in
X_{(\phi_{-}(\theta_{-}),\varphi(\theta_{-}))}$

We claim that $\varphi^{\prime}$ is measurable. Consider the map from
$B_{-}\times B_{+}\rightarrow M^{2}\times\text{Prob}(M)$, defined by

$(\theta_{-},\theta_{+})\mapsto(\phi_{-}(\theta_{-}),\phi_{+}(\theta_{+}),\varphi(\theta_{-}))$

Clearly it is measurable, and note that $\varphi^{\prime}$ is just the
composition of this map with the map $\mu_{(.)}$ in 4.2.11, and therefore, it
is also measurable. It is easy to verify that it’s also a $G$-map. Indeed,
given $(\theta_{-},\theta_{+})\in B_{-}\times B_{+}$, $g\in G$ and $c\in M$

$\displaystyle f_{(g\theta_{-},g\theta_{+})}(c)$
$\displaystyle=m(g\phi_{-}(\theta_{-}),g\phi_{+}(\theta_{+}),gg^{-1}c)=gm(\phi_{-}(\theta_{-}),\phi_{+}(\theta_{+}),g^{-1}c)$
$\displaystyle=\big{(}g_{*}(g\cdot f_{(g\theta_{-},g\theta_{+})})\big{)}(c)$

and note that the last line is precisely the action of $g$ on the elements of
$X_{\varphi(\theta_{-})}$.

The relatively isometrically ergodic projection $B_{-}\times B_{+}\rightarrow
B_{-}$ gives us a map $\varphi^{\prime\prime}:B_{-}\rightarrow X$ that
completes the diagram:

${B_{-}\times
B_{+}}$${X}$${B_{-}}$${Y}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{q}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi^{\prime\prime}}$

We claim that for almost every $(\theta_{-},\theta_{+})\in B_{-}\times B_{+}$,
and $\psi(\theta_{-})$-a.e. $c\in M$
$(\ddagger)\varphi^{\prime}(\theta_{-},\theta_{+})(c)=\varphi^{\prime\prime}(\theta_{-})(c)=\phi_{-}(\theta_{-})$.
Suppose this is not the case and note that that the collection of points for
which $(\ddagger)$ doesn’t hold, defines a measurable set in $B_{-}\times
B_{+}$. For details, see 6.2.3.

This collection is invariant under the action of $G$, and therefore, by
ergodicity, for a.e. $(\theta_{-},\theta_{+})\in B_{-}\times B_{+}$, there
exists $c\in\text{supp}(\psi(\theta_{-}))$, with
$\varphi^{\prime}(\theta_{-},\theta_{+})(c)\neq\phi_{-}(\theta_{-})$. Fix
$\Omega\subset B_{+}$ of measure 1, and $\theta_{-}\in B_{-}$, such that for
every $\theta_{+}\in\Omega$,
$\varphi^{\prime}(\theta_{-},\theta_{+})=\varphi^{\prime\prime}(\theta_{-})$.
By the above, there exists $c\in\text{supp}(\psi(\theta_{-}))$, such that for
every $\theta_{+}\in\Omega$,
$x=\varphi^{\prime\prime}(\theta_{-})(c)=\varphi^{\prime}(\theta_{-},\theta_{+})(c)\neq\phi_{-}(\theta_{-})$.
This is in turn implies a contradiction, as $M$ is a minimal $G$\- median
algebra. Indeed, take a closed half-space
$\mathfrak{h}\in\Delta(x,\phi_{-}(\theta_{-}))$. By the construction of
$\varphi^{\prime}$ and convexity, it follows that
$\phi_{+}(\theta_{+})\in\mathfrak{h}$, for every $\theta_{+}\in\Omega$. But
now, the closure of the convex hull of all those $\phi_{+}(\theta_{+})$ is
contained in $\mathfrak{h}\varsubsetneq M$, and thus, it constitutes a proper
$G$-invariant median subalgebra. Contradiction!

We finish by showing that for a.e. $\theta_{-}\in B_{-}$,
$\psi(\theta_{-})=\delta_{\phi_{-}(\theta_{-})}$. Suppose this is not the
case. Consider the $G$-map from $B_{-}$ to $(\mathcal{C}M)^{2}$, defined by
$\theta\mapsto(\text{supp}(\psi(\theta)),\phi_{-}(\theta))$. By assumption,
the preimage of the diagonal $\Delta\mathcal{C}M\subset(\mathcal{C}M)^{2}$ is
a $G$ invariant measurable set that is not of full measure. Therefore, by
ergodicity, again, for a.e. $\theta_{-}$,
$\psi(\theta_{-})\neq\delta_{\phi_{-}(\theta_{-})}$.

Fix $\Omega\subset B_{+}$ and $\theta_{-}\in B_{-}$ as above,
$c\in\text{supp}(\psi(\theta_{-}))$ different than $\phi_{-}(\theta_{-})$, and
a closed half-space $\mathfrak{h}\in\Delta(\phi_{-}(\theta_{-}),c)$. It
follows the discussion above that now, $\phi_{+}(\theta_{+})\in\mathfrak{h}$
for every $\theta_{+}\in\Omega$. This implies a contradiction, as required.

######  Lemma 6.2.1.

The space $(X_{y},d_{y})$ is a separable metric space.

###### Proof.

Consider the space $Z:=\\{(b,c)\in M^{2}\ :\ b\in[a_{y},c]\\}$ and the natural
projection $p:Z\rightarrow M$, $(b,c)\mapsto c$. Note that $Z$ is a compact
metric space, as it is the image of the continuous map $k:M\times M\rightarrow
M\times M$, that defined by $(z,c)\mapsto(m(a_{y},z,c),c)$. Hence, we are in
the situation of Lemma 4.3.2. Let $\psi:Z\rightarrow{\mathbb{N}}\times M$ be
the bimeasurable, injective map such that the following diagram commute:

${Z}$${{\mathbb{N}}\times
M}$${M}$${M}$$\scriptstyle{\psi}$$\scriptstyle{p}$$\scriptstyle{\text{proj}_{M}}$$\scriptstyle{id_{M}}$

Denote by $\mathfrak{G}$ the space of classes of $\eta_{y}$-a.e. defined
measurable sections for $\text{proj}_{M}$. We have a natural injective map
$\mathfrak{f}:X_{y}\rightarrow\mathfrak{G}$, defined by $f\mapsto\psi\circ f$.
Observe that as $\psi$ injective, for every $f,f^{\prime}\in X_{y}$ we have

$\\{c\in\text{supp}(f)\cap\text{supp}(f^{\prime})\ :\ f(c)\neq
f^{\prime}(c)\\}=\\{c\in\text{supp}(f)\cap\text{supp}(f^{\prime})\ :\
\psi(f(c))\neq\psi(f^{\prime}(c))\\}$

That is, $\mathfrak{f}$ is an isometry from $(X_{y},d_{y})$ into
$(\mathfrak{G},d_{\eta_{y}})$. By lemma 4.3.3, the latter is separable metric
space, hence, so is the former. ∎

######  Lemma 6.2.2.

The map $d:X\times_{Y}X\rightarrow[0,1]$ is Borel.

###### Proof.

Here the Borel structure on $X\times_{Y}X$ is the one induced by the ambiant
space $X\times X$, where $X\subset\text{Prob}(T)$ is eqquiped with the weak*
topology.

So first we want to describe the distance $d_{y}$ in terms of elements in $X$.
Let $x_{1},x_{2}\in X$, such that $q(x_{1})=q(x_{2})=y\in Y$, and let
$l^{i}:M\rightarrow T$ be the corresponding inverses for $p_{3}$. Then,

$\displaystyle d_{y}(x_{1},x_{2})$ $\displaystyle=d_{y}(p_{2}\circ
l^{1},p_{2}\circ l^{2})=\eta_{y}(\\{c\in M\ :\ p_{2}(l^{1}(c))\neq
p_{2}(l^{2}(c))\\})$ $\displaystyle=\eta_{y}(\\{c\in M\ :\ l^{1}(c)\neq
l^{2}(c)\\})=\eta_{y}(p_{3}(\text{supp}(x_{1})\Delta\text{supp}(x_{2})))$
$\displaystyle=x_{1}(\text{supp}(x_{1})\backslash\text{supp}(x_{2}))+x_{2}(\text{supp}(x_{2})\backslash\text{supp}(x_{1}))$
$\displaystyle=2-x_{1}(\text{supp}(x_{2}))-x_{2}(\text{supp}(x_{1}))$

So let us show that the map $(\mu,\eta)\mapsto\mu(\text{supp}(\eta))$ is a
measurable map from $\text{Prob}(T)^{2}$ to [0,1].

First, let $\mathcal{C}T$ denote the space of closed subsets of $T$, equipped
with the Chabauty topology. We claim that the map
$v:\text{Prob}(T)\mapsto\mathcal{C}T$, defined by
$\mu\mapsto\text{supp}(\mu)$, is measurable. Given an open set $U\subset T$,
the map $t_{U}:\text{Prob}(T)\rightarrow[0,1]$ that evaluates every measure
$\mu$ at $U$, is a measurable map (in fact, it is a lower semi-continuous
map). Therefore, the following sets are evidentially measurable:

$\displaystyle v^{-1}\big{(}\\{F\in\mathcal{C}T\ :\ F\cap
U\neq\varnothing\\}\big{)}=t_{U}^{-1}((0,1])$ $\displaystyle
v^{-1}\big{(}\\{F\in\mathcal{C}T\ :\ F\subset
U\\}\big{)}=\bigcup_{n}t_{U_{n}}^{-1}(\\{1\\})$

Where $U_{n}=T\backslash\overline{V_{1/n}(U^{c})}$, and where
$V_{\epsilon}(F)$ is the epsilon neighborhood of $F$ in $T$. This implies the
measurability of $v$.

Finely, let us show that the map
$u:\text{Prob}(T)\times\mathcal{C}T\rightarrow[0,1]$, given by
$(\mu,F)\mapsto\mu(F)$ is measurable, and more specifically, upper semi-
continuous. Suppose that
$(\mu_{n},F_{n}),(\mu,F)\in\text{Prob}(T)\times\mathcal{C}T$ are such that
$(\mu_{n},F_{n})\xrightarrow{n}(\mu,F)$. For every $m\in{\mathbb{N}}$, we know
that
$\underset{n\rightarrow\infty}{\lim}\sup\mu_{n}(\overline{V_{1/m}(F)})\leq\mu(\overline{V_{1/m}(F)})$.
In addition, for $n\gg 0$ $F_{n}\subset
V_{1/m}(F)\subset\overline{V_{1/m}(F)}$. Therefore, for every
$m\in{\mathbb{N}}$,
$\underset{n\rightarrow\infty}{\lim}\sup\mu_{n}(F_{n})\leq\mu(\overline{V_{1/m}(F)})$.
We end by noting that
$\underset{m\rightarrow\infty}{\lim}\mu(\overline{V_{1/m}(F)})=\mu(F)$.

∎

######  Lemma 6.2.3.

In the above notations, the condition $(\ddagger)$ defines a measurable set in
$B_{-}\times B_{+}$.

###### Proof.

The relation described in $(\ddagger)$, defines the collection of pairs
$(\theta_{-},\theta_{+})\in B_{-}\times B_{+}$ such that
$\text{supp}\varphi^{\prime}(\theta_{-},\theta_{+})=\\{\phi_{-}(\theta_{-})\\}^{2}\times\text{supp}\psi(\theta_{-})$.
Consider the map $\text{supp}\circ\varphi^{\prime}:B_{-}\times
B_{+}\rightarrow\mathcal{C}T\subset\mathcal{C}M^{3}$, which is measurable, and
note that the collection above is precisely the preimage of the set $\Delta
M\times\mathcal{C}M\subset\mathcal{C}(M^{3})$. ∎

###  6.3. Proof of Theorem B

Let $G$, $M$ and $\mu$ be as in the theorem and set $\mu_{+}=\mu$ and
$\mu_{-}=\iota_{*}(\mu)$, for $\iota(g)=g^{-1}$. Let $(B_{+},\nu_{+})$ be the
Furstenberg-Poisson boundary corresponds to $(G,\mu_{+})$, and
$(B_{-},\nu_{-})$ the Furstenberg-Poisson boundary corresponds to
$(G,\mu_{-})$. According to Theorem 5.0.2, the pair $(B_{-},B_{+})$ is a
boundary pair. Thus, by Theorem C, there exists a unique $G$-map from $B_{+}$
to $\text{Prob}(M)$. The theorem now follows by Furstenberg’s correspondence,
see [BS06, Theorem 2.16].

###  6.4. Proof of Theorem D

Let $M$, $G$, $(B_{-},\nu_{-})$ and $(B_{+},\nu_{+})$ be as in Theorem D, and
$C$ and $M^{\prime}$ as promised by Proposition A. As in the previous section,
we will show the statement for $B_{-}$. Therefore, the space $B_{+}$ is
mentioned here merely for the application of theorem C.

Denote by $p_{M^{\prime}}$ the natural projection from $M$ onto $M^{\prime}$.
This is a $G$-equivariant continuous median map. Thus, the action of $G$ on
$M^{\prime}$ must be minimal as well.

As $M^{\prime}$ has no cubical factor, it follows by theorem C that we have a
unique $G$-map $\theta\mapsto x_{\theta}$, from $B_{-}$ to $M^{\prime}$, which
is also the unique map from $B_{-}$ to $\text{Cub}(M^{\prime})$. This map
gives rise to a map $\varphi:B_{-}\rightarrow\text{Cub}(M)$, defined by
$\theta\mapsto\\{x_{\theta}\\}\times C$.

Suppose we have another $G$-map
$\varphi^{\prime}:B_{-}\rightarrow\text{Cub}(M)$. For a generic $\theta\in
B_{-}$, denote by $I_{\theta}^{\prime}$ the corresponding convex hull of
$\varphi^{\prime}(\theta)$. Observe that $p_{M^{\prime}}$ maps intervals to
intervals. Therefore, we have the following $G$-map:

$B_{-}\rightarrow\text{Cub}(M^{\prime}),\
\theta\mapsto\text{Ends}(p_{M^{\prime}}(I_{\theta}^{\prime}))$

By the uniqueness of $x_{\theta}\in M^{\prime}$, it follows that
$p_{M^{\prime}}(I_{\theta}^{\prime})=\\{x_{\theta}\\}$ for a.e. $\theta\in
B_{-}$. That is, for a.e. $\theta\in B_{-}$,
$\varphi^{\prime}(\theta)=\\{x_{\theta}\\}\times C_{\theta}^{\prime\prime}$,
for some subcube $C_{\theta}^{\prime\prime}\subset C$.

That is, we have a $G$-equivariant assignment from $B_{-}$ to the finite set
$\text{Cub}(C)$, namely, $\theta\mapsto C_{\theta}^{\prime\prime}$. By metric
ergodicity, this map is essentially constant. Denote by $C^{\prime}\subset C$
its essential image. As in the case of $M^{\prime}$, the action of $G$ on $C$
must be minimal as well, and since $C^{\prime}$ is $G$-invariant, we get an
equality $C^{\prime}=C$. In other words, $\varphi^{\prime}$ is nothing but the
map $\varphi$.

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