#  An isometric extensor of metrics

Yoshito Ishiki  Department of Mathematical Sciences Tokyo Metropolitan
University Minami-osawa Hachioji Tokyo 192-0397 Japan  ishiki-
<EMAIL_ADDRESS>
###### Abstract.

In this paper, for a metrizable space $Z$, we consider the space of metrics
that generate the same topology on $Z$, and that space of metrics is equipped
with the supremum metrics. For a metrizable space $X$ and a closed subset $A$
of it, we construct a map $E$ from the space of metrics on $A$ into the space
of metrics on $X$ such that $E$ is an extension of metrics and preserves the
supremum metrics between metrics.

###### Key words and phrases:

Spaces of metrics, Extension of metrics

###### 2020 Mathematics Subject Classification:

Primary 54E35; Secondary 54C20, 54D35

###### Contents

  1. 1 Intoroduction
    1. 1.1 Backgrounds
    2. 1.2 Main result
  2. 2 Preliminaries
    1. 2.1 Basic notations
    2. 2.2 Components of spaces of metrics
    3. 2.3 Spaces of maps with finite supports
    4. 2.4 Constructions of metric spaces
      1. 2.4.1 $\ell^{1}$-product spaces
      2. 2.4.2 $1$-Wasserstein spaces
      3. 2.4.3 Spaces of measurable functions
    5. 2.5 Osmotic constructions
  3. 3 The Whitney–Dugundji decomposition
  4. 4 Proof of the main result
  5. 5 Questions

##  1\. Intoroduction

###  1.1. Backgrounds

For a metrizable space $Z$, we denote by $\mathrm{Met}(Z)$ the set of all
metrics on $Z$ that generate the same topology of $Z$. We write
$\mathrm{BMet}(Z)$ as the set of all bounded metrics in $\mathrm{Met}(Z)$. We
define the supremum metric $\operatorname{\mathcal{D}}_{Z}$ on
$\mathrm{Met}(Z)$ by

$\operatorname{\mathcal{D}}_{Z}(d,e)=\sup_{x,y\in Z}|d(x,y)-e(x,y)|.$

Take a closed subset $A$ of $Z$. It was Felix Hausdorff [17] who first
considered the extension problem of metrics on $A$ to $Z$, and proved that for
each $d\in\mathrm{Met}(Z)$, there exists $D\in\mathrm{Met}(Z)$ such that
$D|_{A^{2}}=d$. Independently, in later years, Bing [7] also proved the same
theorem (see also [3], [1], and [41]). Based on Hausdorff’s result, several
mathematicians considered development of it. For variants of Hausdorff’s
extension theorem, see, for example, [19], [12], [11]. For extensions of
ultrametrics (non-Archimedean metrics), see [42], [39], [20], [21], [23], and
[22].

Hausdorff’s extension theorem only states that we can find a extended metric
$D\in\mathrm{Met}(X)$ of each $d\in\mathrm{Met}(A)$. Thus, the next point of
iterrest is whether these extensions of metrics can be taken simultaneously.
In other words, can we construct an extensor
$\mathrm{Met}(A)\to\mathrm{Met}(X)$ that is continuous with respect to some
topology on spaces of metrics? As long as the author knows, in 1981, Nguyen
Van Khue and Nguyen To Nhu [27] first constructed simultaneous extensions
$\Phi_{1},\Phi_{2}\colon\mathrm{Met}(A)\to\mathrm{Met}(X)$, which satisfy that
$\Phi_{1}$ is $20$-Lipschitz with respect to the supremum metrics on spaces of
metrics, and $\Phi_{2}$ is continuous with respect to the topologies of point-
wise convergence and preserving orders.

An extensor $\Phi\colon\mathrm{BMet}(A)\to\mathrm{BMet}(Z)$ on spaces of
bounded metics is isometric (with respect to the supremum metrics) if we have
$\operatorname{\mathcal{D}}_{A}(d,e)=\operatorname{\mathcal{D}}_{Z}(\Phi(d),\Phi(e))$
for all $d,e\in\mathrm{Met}(A)$. Such extensors on bounded metrics have been
constructed by Bessaga [5], Banakh [4], Pikhurko [35], and Zarichnyi [44]. In
this paper, we shall construct an isometric extensor
$E\colon\mathrm{Met}(A)\to\mathrm{Met}(X)$ of metrics. Remark that, in
contrast to the previous results, our extensor $E$ can be applied to not only
bounded metric but also unbounded metrics.

###  1.2. Main result

For a metrizable space $X$, we denote by $\mathrm{Comp}(X)$ the set of all
complete metrics $d\in\mathrm{Met}(X)$. Of course, we have
$\mathrm{Comp}(X)\neq\emptyset$ if and only if $X$ is completely metrizable.
We also denote by $\mathrm{CPM}(X)$ the set of all pseudometrics $d\colon
X\times X\to[0,\infty)$ that are continuous as maps between $X\times X$ and
$[0,\infty)$. We consider that $\mathrm{CPM}(X)$ is equipped with the supremum
metric $\operatorname{\mathcal{D}}_{X}$, for which we use the same symbol as
the metric on $\mathrm{Met}(X)$. We have
$\mathrm{Met}(X)\subseteq\mathrm{CPM}(X)$ and moreover, the set
$\mathrm{Met}(X)$ is comeager in $\mathrm{CPM}(X)$ (see [24]).

We prove the following generalization of Hausdorff’s metric extension theorem.

######  Theorem 1.1.

Let $X$ be a metrizable space, and $A$ be a closed subset of $X$. Then there
exists a map $E\colon\mathrm{Met}(A)\to\mathrm{Met}(X)$ such that

  1. (1)

for every $d\in\mathrm{Met}(A)$ we have $E(d)|_{A^{2}}=d$;

  2. (2)

the map $E$ is an isometric embedding, i.e., for every pair
$d,e\in\mathrm{Met}(A)$, we have

$\operatorname{\mathcal{D}}_{A}(d,e)=\operatorname{\mathcal{D}}_{X}(E(d),E(e));$

  3. (3)

we have $E(\mathrm{BMet}(A))\subseteq\mathrm{BMet}(X)$;

  4. (4)

we have $E(\mathrm{Comp}(A))\subseteq\mathrm{Comp}(X)$;

Furthermore, we also obtain an isometric extensor
$F\colon\mathrm{CPM}(A)\to\mathrm{CPM}(X)$ of pseudometrics such that
$F|_{\mathrm{Met}(A)}=E$.

This theorem is an Archimedean analogue of [22, Theorem 4.7]. Our proof of the
main result is based on the idea of extending spaces and extending
homeomorphisms, which was used in [44], and can go back to [17], [18], and
[29]. Namely, for a metric space $Z$ and for a closed subset $A$ of $Z$, we
will find an extension metrizable space $L$ of $A$ with n topological
embedding $I\colon A\to L$ such that every metric $d\in\mathrm{Met}(A)$ can be
naturally extended to a metric on $L$ through $I$. In this situation, we will
construct an topological embedding $J\colon Z\to L$ such that $J|_{A}=I$.

The organization of this paper is as follows: In Section 2, we review several
basic concepts on metrics spaces. We also explain the constructions of the
$\ell^{1}$-products, the $1$-Wasserstein spaces, and spaces of measurable
maps. In the end of Section 2, we summarize these constructions as an osmotic
construction, which is a method to obtain extension spaces of a metrizable
space where metrics are naturally extended. The author hopes that this notion
would be helpful to consider a generalization of our main result. In Section
3, we review the classical discussion called the Whitney–Dugundji
decomposition, which plays a key role of the proof of the main result. The
whole of Section 4 is devoted to the proof of Theorem 1.1. Section 5 exhibits
several questions on extensors of metrics.

######  Acknowledgements.

This work was supported by JSPS KAKENHI Grant Number JP24KJ0182.

The author would like to thank Tatsuya Goto for the advices on Borel sets.

##  2\. Preliminaries

###  2.1. Basic notations

First, we review the basic notions and notations on metric spaces. For a
metric space $(Z,w)$, we denote by $U(x,r;w)$ the open ball centered at $x\in
Z$ with radius $r\in(0,\infty)$. For a subset $A\subseteq Z$, and for $x\in Z$
we define $w(x,A)=\inf_{a\in A}w(x,a)$. Note that the function $x\mapsto
w(x,A)$ is $1$-Lipschitz. For a subset $S$ of $Z$, we represent
$\operatorname{diam}_{w}(S)$ the diameter of $S$ with respect to $w$.

In this paper, we often denote by $\omega_{0}$ the set of all non-negative
integers when we regard the set of non-negative integers as a (discrete)
space. Of course, we have $\omega_{0}=\mathbb{Z}_{\geq 0}$ as a set. When
emphasizing the set of integers as an index set, we rather use
$\mathbb{Z}_{\geq 0}$ than $\omega_{0}$.

###  2.2. Components of spaces of metrics

For a metrizable space $Z$, and for $d,e\in\mathrm{Met}(Z)$, we write $d\sim
e$ if $\operatorname{\mathcal{D}}_{Z}(d,e)<\infty$. Then “$\sim$” becomes an
equivalence relationship on $\mathrm{Met}(Z)$. We represent
$\mathbf{M}_{<\infty}(Z)=\mathrm{Met}(Z)/\\!\\!\sim$. For a member
$d\in\mathrm{Met}(Z)$, we write $\mathfrak{C}(d)$ as the equivalence class of
$d$. Remark that each $\mathfrak{C}\in\mathbf{M}_{<\infty}(Z)$ is a
(path-)connected component of $\mathrm{Met}(Z)$, and it is also a clopen
subset of $\mathrm{Met}(Z)$.

In the proof of main theorem, we fix a equivalence class
$\mathfrak{C}=\mathfrak{C}(m)\in\mathbf{M}_{<\infty}(A)$. Then we construct an
isometric map $E\colon\mathfrak{C}\to\mathrm{Met}(X)$, and gluing then
together, we obtain $E\colon\mathrm{Met}(A)\to\mathrm{Met}(X)$.

###  2.3. Spaces of maps with finite supports

For a set $T$, we denote by $\mathbf{L}_{fin}^{\infty}(T)$ the set of all maps
$f$ from $T$ into $\mathbb{R}$ such that $\\{\,s\in T\mid f(s)\neq 0\,\\}$ is
finite. Let $\mathbf{D}_{L^{\infty}}$ stand for the supremum metric on
$\mathbf{L}_{fin}^{\infty}(T)$. Namely, we have
$\mathbf{D}_{L^{\infty}}(f,g)=\max\\{\,|f(s)-g(s)|\mid s\in T\,\\}$.

###  2.4. Constructions of metric spaces

In this section, we shall introduce three constructions of metric spaces, the
$\ell^{1}$-product with a fixed metric space, the $1$-Wasserstein space, and
the space of measurable functions.

####  2.4.1. $\ell^{1}$-product spaces

Fix a metric space $(S,u)$. For a metrizable space $Z$, we define
$\$\\!\left(Z,S\right)=Z\times S$. When $S$ is clear by the context, we simply
write $\$\\!\left(Z,S\right)=\$\\!\left(Z\right)$. For a metric
$d\in\mathrm{Met}(Z)$, we define $\$\\!\left[d\right]=d\times_{\ell^{1}}u$,
i.e., we have $\$\\!\left[d\right]((x,s),(y,t))=d(x,y)+u(s,t)$. Fix a point
$o\in S$, we define $\mathrm{I}^{\$}_{Z}\colon Z\to\$\\!\left(Z\right)$ by
$\mathrm{I}^{\$}_{Z}(x)=(x,o)$.

######  Proposition 2.1.

Let $(S,u)$ be a fixed metric space, $o\in S$ be a fixed point, and $Z$ be a
metrizable space. We also fix $\mathfrak{C}\in\mathbf{M}_{<\infty}(Z)$. Then
the following statements are true:

  1. (1)

For every $d\in\mathfrak{C}$, and for every pair $x,y\in Z$, we have

$\$\\!\left[d\right]\left(\mathrm{I}^{\$}_{Z}(x),\mathrm{I}^{\$}_{Z}(y)\right)=d(x,y).$

  2. (2)

The map $\$\colon\mathrm{Met}(Z)\to\mathrm{Met}(\$(Z,S))$ by
$d\mapsto\$\\!\left[d\right]$ is an isometric embedding, i.e., for every pair
$d,e\in\mathrm{Met}(Z)$, we have

$\operatorname{\mathcal{D}}_{Z}(d,e)=\operatorname{\mathcal{D}}_{\$(Z)}(\$\\!\left[d\right],\$\\!\left[e\right]).$

  3. (3)

For every pair $d\in\mathfrak{C}$, the metric $\$\\!\left[d\right]$ and
$\$\\!\left[e\right]$ generate the same topology of $\$\\!\left(Z\right)$.

###### Proof.

Statement (1) follows from the definition of $\$\\!\left[d\right]$. Now we
show (2). For every pair $(x,s),(y,t)\in\$(Z)$, we have

$\displaystyle|\$\\!\left[d\right]((x,s),(y,t))-\$\\!\left[e\right]((x,s),(y,t))|$
$\displaystyle=|d(x,y)+u(s,t)-e(x,y)-u(s,t)|=|d(x,y)-e(x,y)|.$

Thus we have
$\operatorname{\mathcal{D}}_{Z}(d,e)=\operatorname{\mathcal{D}}_{\$(Z)}(\$\\!\left[d\right],\$\\!\left[e\right])$.

Statement (3) is trivial. Moreover, for every metric $d\in\mathrm{Met}(Z)$,
the metric $\$\\!\left[d\right]$ generates the product topology of $Z\times
S$. This statement is just a preparation for introducing osmotic
constructions. ∎

####  2.4.2. $1$-Wasserstein spaces

Since we will consider the Wasserstein space on a metrizable space that is not
assumed to be separable, we follow the construction of Wasserstein spaces
discussed in [14] and [32] using Radon measures.

Let $Z$ be a metrizable space. A Borel measure $\mu$ on $Z$ is said to be
_Radon_ if for every Borel subset $A$ of $Z$, and for every
$\epsilon\in(0,\infty)$ there exists a compact subset $K$ of $Z$ such that
$\mu(A\setminus K)<\epsilon$. We denote by $\mathcal{RP}\\!\left(Z\right)$ the
set of all Radon probability measures on $Z$. We define
$\mathrm{I}^{\mathrm{W}_{1}}_{Z}\colon Z\to\mathcal{RP}\\!\left(Z\right)$ by
$\mathrm{I}^{\mathrm{W}_{1}}_{Z}(x)=\boldsymbol{\delta}_{x}$, where
$\boldsymbol{\delta}_{x}$ is the Dirac measure on $x$.

For $\alpha,\beta\in\mathcal{RP}\\!\left(Z\right)$, we denote by
$\Pi(\alpha,\beta)$ the set of all $\pi\in\mathcal{RP}\\!\left(Z\times
Z\right)$ such that $\pi(A\times Z)=\alpha(A)$ and $\pi(A\times Z)=\beta(A)$
for all Borel subsets $A$ of $Z$. Now we define the $1$-Wasserstein distance
$\mathrm{W}_{1}\\!\left[d\right]$ with respect to $d$ by

$\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)=\inf_{\pi\in\Pi(\alpha,\beta)}\int_{Z\times
Z}d(s,t)\ d\pi(s,t),$

where $\alpha,\beta\in\mathcal{RP}\\!\left(Z\right)$. It should be noted that,
in general, $\mathrm{W}_{1}\\!\left[d\right]$ can take the value $\infty$.
Thus, we focus on the subset of $\mathcal{RP}\\!\left(Z\right)$ where
$\mathrm{W}_{1}\\!\left[d\right]$ takes only finite values. Fix an equivalence
class $\mathfrak{C}\in\mathbf{M}_{<\infty}(Z)$. We denote by
$\mathrm{W}_{1}^{\mathfrak{C}}\\!\left(Z\right)$ the set of
$\alpha\in\mathcal{RP}\\!\left(Z\right)$ such that
$\mathrm{W}_{1}\\!\left[d\right](\alpha,\boldsymbol{\delta}_{p})<\infty$ for
some $p\in Z$. This condition is also equivalent to the inequality
$\mathrm{W}_{1}\\!\left[d\right](\alpha,\boldsymbol{\delta}_{q})<\infty$ for
every $q\in Z$. By the definition, the set
$\mathrm{W}_{1}^{\mathfrak{C}}\\!\left(Z\right)$ is depending on the class
$\mathfrak{C}$; namely, for every pair $d,e\in\mathfrak{C}$, we have
$\mathrm{W}_{1}\\!\left[d\right](\alpha,\boldsymbol{\delta}_{p})<\infty$ if
and only if
$\mathrm{W}_{1}\\!\left[e\right](\alpha,\boldsymbol{\delta}_{p})<\infty$. When
we fix a class $\mathfrak{C}$, we simply write
$\mathrm{W}_{1}\\!\left(Z\right)=\mathrm{W}_{1}^{\mathfrak{C}}\\!\left(Z\right)$.
We call $(\mathrm{W}_{1}\\!\left(Z\right),\mathrm{W}_{1}\\!\left[d\right])$
the _$1$ -Wasserstein space on $(Z,d)$_.

The $1$-Wasserstein space is sometimes called the _Lipschitz-free space_ with
emphasizing the aspect of the dual space of the space of Lipschitz functions
through the Kantorovich–Rubinstein duality. This space is also sometimes
referred to as the _Arens–Eells space_ , named after the Arens–Eells embedding
theorem [2].

From now on, we review properties of the Wasserstein spaces which we will use
in the present paper.

For a metric sapce $(X,d)$, we denote by $\mathbf{Lip}_{1}\left(X,d\right)$
the set of all real-valued $1$-Lipschitz functions. To compute values of
Wasserstein distance of specific measures, we make use of the
Kantorovich–Rubinstein duality.

######  Theorem 2.2 (The Kantorovich–Rubinstein duality).

Let $(Z,d)$ be a metric space. Then for every pair
$\alpha,\beta\in\mathrm{W}_{1}\\!\left(Z\right)$, we have

$\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)=\sup\left\\{\,\int_{Z}f(x)\
d(\alpha-\beta)(x)\ \middle|\
f\in\mathbf{Lip}_{1}\left(X,d\right)\,\right\\}.$

###### Proof.

Remark that we does not assume that $X$ is Polish in the statement of Theorem
2.2. For the proof, we refer the readers to [14, Theorem 4.1] and [26, Theorem
1], or we can prove this generalized duality using the duality for Polish
spaces together with [32, Lemma 2.5], the McShane–Whitney extension of
Lipschitz functions, and the definition of Radon measures. ∎

To show the main result, we need the following lemmas.

######  Lemma 2.3.

Let $Z$ be a metrizable space, and fix
$\mathfrak{C}\in\mathbf{M}_{<\infty}(Z)$ and $n\in\mathbb{Z}_{\geq 0}$. Take
two sequences $s_{1},\dots,s_{n}$ and $t_{1},\dots,t_{n}$ of non-negative
reals such that $\sum_{i=1}^{n}s_{i}=\sum_{i=1}^{n}t_{i}=1$, and take a finite
sequence $x_{1},\dots,x_{n}$ in $X$. Then for every point $b\in Z$, we have

$\mathrm{W}_{1}\\!\left[d\right]\left(\sum_{i=1}^{n}s_{i}\boldsymbol{\delta}_{x_{i}},\sum_{i=1}^{n}t_{i}\boldsymbol{\delta}_{x_{i}},\right)\leq\sum_{i=1}^{n}|s_{i}-t_{i}|d(x_{i},b).$

###### Proof.

Put $\alpha=\sum_{i=1}^{n}s_{i}\boldsymbol{\delta}_{x_{i}}$, and
$\beta=\sum_{i=1}^{n}t_{i}\boldsymbol{\delta}_{x_{i}}$. Then for every
$1$-Lipschitz map $f\colon(Z,d)\to\mathbb{R}$, we have

$\displaystyle\int_{Z}f(x)\
d(\alpha-\beta)=\sum_{i=1}^{n}s_{i}f(x_{i})-\sum_{i=1}^{n}t_{i}f(x_{i})$
$\displaystyle=\sum_{i=1}^{n}s_{i}f(x_{i})-f(b)+f(b)-\sum_{i=1}^{n}t_{i}f(x_{i})$
$\displaystyle=\sum_{i=1}^{n}s_{i}f(x_{i})-\sum_{i=1}^{n}s_{i}f(b)+\sum_{i=1}^{n}t_{i}f(b)-\sum_{i=1}^{n}t_{i}f(x_{i})$
$\displaystyle=\sum_{i=1}^{n}s_{i}(f(x_{i})-f(b))+\sum_{i=1}^{n}t_{i}(f(b)-f(x_{i}))$
$\displaystyle=\sum_{i=1}^{n}(s_{i}-t_{i})(f(x_{i})-f(b)).$

Here, for each $i\in\mathbb{Z}_{\geq 0}$, we take a number $\mathbf{sgn}_{i}$
in $\\{1,-1\\}$ such that $\mathbf{sgn}_{i}\cdot|s_{i}-t_{i}|=(s_{i}-t_{i})$.
Then we can continue to compute.

$\displaystyle\sum_{i=1}^{n}(s_{i}-t_{i})(f(x_{i})-f(b))=\sum_{i=1}^{n}|s_{i}-t_{i}|\cdot\mathbf{sgn}_{i}\cdot(f(x_{i})-f(b))$
$\displaystyle\leq\sum_{i=1}^{n}|s_{i}-t_{i}|\cdot|f(x_{i})-f(b)|\leq\sum_{i=1}^{n}|s_{i}-t_{i}|\cdot
d(x_{i},b).$

Thus, for every $f\in\mathbf{Lip}_{1}\left(Z,d\right)$, we have

$\int_{Z}f(x)\ d(\alpha-\beta)\leq\sum_{i=1}^{n}|s_{i}-t_{i}|\cdot
d(x_{i},b).$

Therefore, the Kantorovich–Rubinstein duality (Theorem 2.2) implies the lemma.
∎

######  Lemma 2.4.

Let $Z$ be a metrizable space, and fix
$\mathfrak{C}\in\mathbf{M}_{<\infty}(Z)$. Take a finite sequence
$c_{1},\dots,c_{n}$ of non-negative reals such that $\sum_{i=1}^{n}c_{i}=1$,
and take a finite sequence $x_{1},\dots,x_{n}$ in $X$ Then for every $p\in X$,
we have

$\mathrm{W}_{1}\\!\left[d\right]\left(\sum_{i=1}^{n}c_{i}\boldsymbol{\delta}_{x_{i}},\boldsymbol{\delta}_{p}\right)=\sum_{i=1}^{n}c_{i}d(x_{i},p).$

In particular, we also have

$\mathrm{W}_{1}\\!\left[d\right]\left(\sum_{i=1}^{n}c_{i}\boldsymbol{\delta}_{x_{i}},\boldsymbol{\delta}_{p}\right)\leq
d(x_{i},p).$

###### Proof.

Put $\alpha=\sum_{i=1}^{n}c_{i}\boldsymbol{\delta}_{x_{i}}$, and
$\beta=\boldsymbol{\delta}_{p}$. Define $\mu\in\Pi(\alpha,\beta)$ by
$\mu=\sum_{i=1}^{n}c_{i}\boldsymbol{\delta}_{(x_{i},p)}$. Under this
situation, we have

$\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)\leq\int_{Z\times Z}d(x,y)\
\mu(x,y)=\sum_{i=1}^{n}c_{i}d(x_{i},p).$

To obtain the opposite inequality, consider
$f\in\mathbf{Lip}_{1}\left(X,d\right)$ define by $f(x)=d(x,p)$. Then we have

$\displaystyle\int_{Z}f\
d(\alpha-\beta)=\left(\sum_{i=1}^{n}c_{i}f(x_{i})\right)-f(p)=\sum_{i=1}^{n}c_{i}d(x_{i},p).$

Thus the Kantorovich–Rubinstein duality (Theorem 2.2) implies that
$\sum_{i=1}^{n}c_{i}d(x_{i},p)\leq\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)$.
This finishes the proof. ∎

######  Corollary 2.5.

Let $Z$ be a metrizable space, and fix a class
$\mathfrak{C}\in\mathbf{M}_{<\infty}(Z)$. Then for every $d\in\mathfrak{C}$,
and for every pair $x,y\in Z$, we have

$\mathrm{W}_{1}\\!\left[d\right]\left(\mathrm{I}^{\mathrm{W}_{1}}_{Z}(x),\mathrm{I}^{\mathrm{W}_{1}}_{Z}(y)\right)=d(x,y).$

###### Proof.

This lemma is a special case of Lemma 2.4. ∎

From now on, we consider the topology of $1$-Wasserstein spaces of Radon
measures on metric spaces that are not necessarily complete.

We first introduce the known description of $1$-Wasserstein spaces on complete
spaces. For a metric space $(Z,d)$, we denote by
$\mathbf{Me}_{+}\\!\mathbf{C}\\!\left(Z,d\right)$ the set of all $f\colon
Z\to\mathbb{R}$ such that there exist $A,B\in(0,\infty)$ and $p\in Z$ for
which $|f(x)|\leq A\cdot d(x,p)+B$ for all $x\in Z$. We do not impose the
continuity of each $f\in\mathbf{Me}_{+}\\!\mathbf{C}\\!\left(Z,d\right)$. The
symbol “$\mathbf{Me}_{+}\\!\mathbf{C}$” means “Metric + Constant”. Note that
$f\in\mathbf{Me}_{+}\\!\mathbf{C}\\!\left(Z,d\right)$ if and only if for every
$q\in Z$, there exist $\widetilde{A},\widetilde{B}\in(0,\infty)$ such that
$|f(x)|\leq\widetilde{A}\cdot d(x,q)+\widetilde{B}$ for all $x\in Z$.

######  Theorem 2.6.

Let $(R,w)$ be a complete metric space, and let
$\\{\mu_{i}\\}_{i\in\mathbb{Z}_{\geq 0}}$ be a sequence of
$\mathrm{W}_{1}\\!\left(R\right)$ and take
$\mu\in\mathrm{W}_{1}\\!\left(R\right)$. Then the following statements are
equivalent to each other.

  1. (1)

The sequence $\\{\mu_{i}\\}_{i\in\mathbb{Z}_{\geq 0}}$ converges to $\mu$ in
$\mathrm{W}_{1}\\!\left(R\right)$.

  2. (2)

If a continuous function $f\colon R\to\mathbb{R}$ belongs to
$\mathbf{Me}_{+}\\!\mathbf{C}\\!\left(R,w\right)$, then we have $\int_{R}f(x)\
d\mu_{i}\to\int_{R}f(x)\ d\mu$ as $i\to\infty$.

###### Proof.

See [32, Theorem 2.11 and Theorem 2.12]. ∎

In the next proposition, it is shown that Radon probability measure on $Z$ has
a $\sigma$-compact support, and hence is can be extended to a measure on an
extension space of $Z$. This observation builds a bridge between the
topologies of $1$-Wasserstein spaces on complete and incomplete spaces.

######  Proposition 2.7.

Let $(Z,d)$ be a metric space, and $\mu\in\mathcal{RP}\\!\left(Z\right)$. Then
there exists a set $S$ of $Z$ such that

  1. (m1)

the set $S$ is $\sigma$-compact. In particular, the set $S$ is absolutely
$F_{\sigma}$;

  2. (m2)

we have $\mu(S)=1$.

###### Proof.

Proposition 2.7 follows from the definition of Radon probability measures. ∎

Take a subset $S$ mentioned in Proposition 2.7, and take a metrizable space
$\Omega$ with $Z\subseteq\Omega$. Then $S$ is also Borel in $\Omega$ since it
is $\sigma$-compact. Then, for every Borel set $A$ of $\Omega$, the
intersection $A\cap S$ is Borel in $\Omega$. Thus it is also Borel in $S$.
Since $S$ is Borel in $Z$, we can conclude that $A\cap S$ is Borel in $Z$ (see
also [38, Proposition 3.1.9]). Thus, we can define $\widehat{\mu}$ by
$\widehat{\mu}(A)=\mu(A\cap S)$ for every Borel set $A$ of $\Omega$, and
$\widehat{\mu}$ is also Borel and Radon measure on $\Omega$. Note that
$\widehat{\mu}$ does not depend on the choice of $S$. Based on this
phenomenon, whenever we are given a metrizable space $\Omega$ such that
$Z\subseteq\Omega$, we use the same symbol $\widehat{\mu}$ to denote the
extension of $\mu$ constructed above.

######  Proposition 2.8.

Let $(Z,d)$ be a metric space and $(\widetilde{Z}^{d},{d}^{\diamondsuit})$ be
the completion of $(Z,d)$. Then we have

$\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)=\mathrm{W}_{1}\\!\left[{d}^{\diamondsuit}\right](\widehat{\alpha},\widehat{\beta}).$

###### Proof.

By the Kantorovich–Rubinstein duality (Theorem 2.2), for each
$\epsilon\in(0,\infty)$, there exists $f\in\mathbf{Lip}_{1}\left(X,d\right)$
such that $\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)-\epsilon<\int_{Z}f\
d(\alpha-\beta)$. Since $f$ is Lipschitz and
$(\widetilde{Z}^{d},{d}^{\diamondsuit})$ is the completion of $(Z,d)$, using
Cauchy sequences (see also [8, Theorem 2] and [9]), we can obtain
$F\in\mathbf{Lip}_{1}\left(\widetilde{Z}^{d},{d}^{\diamondsuit}\right)$ such
that $F|_{Z}=f$. Since $\widehat{\alpha}(\widetilde{Z}^{d}\setminus
Z)=\widehat{\beta}(\widetilde{Z}^{d}\setminus Z)=0$, using the
Kantorovich–Rubinstein duality (Theorem 2.2) again, we have

$\int_{Z}f\ d(\alpha-\beta)=\int_{G}F\
d(\widehat{\alpha}-\widehat{\beta})\leq\mathrm{W}_{1}\\!\left[{d}^{\diamondsuit}\right](\widehat{\alpha},\widehat{\beta}).$

Thus
$\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)\leq\mathrm{W}_{1}\\!\left[{d}^{\diamondsuit}\right](\widehat{\alpha},\widehat{\beta})$.
Next, we show the opposite inequality. For each $\epsilon\in(0,\infty)$, take
$\phi\in\mathbf{Lip}_{1}\left(\widetilde{Z}^{d},{d}^{\diamondsuit}\right)$
such that
$\mathrm{W}_{1}\\!\left[{d}^{\diamondsuit}\right](\widehat{\alpha},\widehat{\beta})-\epsilon\leq\int_{\widetilde{Z}^{d}}\phi\
d(\widehat{\alpha}-\widehat{\beta})$. Of course, we obtain
$\phi|_{X}\in\mathbf{Lip}_{1}\left(Z,d\right)$, and
$\int_{\widetilde{Z}^{d}}\phi\
d(\widehat{\alpha}-\widehat{\beta})=\int_{Z}\phi|_{X}\
d(\alpha-\beta)\leq\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)$. Hence
$\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)\leq\mathrm{W}_{1}\\!\left[{d}^{\diamondsuit}\right](\widehat{\alpha},\widehat{\beta})$.
Therefore, we conclude that
$\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)=\mathrm{W}_{1}\\!\left[{d}^{\diamondsuit}\right](\widehat{\alpha},\widehat{\beta})$.
∎

######  Remark 2.1.

Proposition 2.8 indicates that the extension $\widehat{\mu}$ of $\mu$ can be
obtained as follows: For a metrizable space $X$, let $\mathbf{LD}(X)$ be the
set of all finitely convex combination of the Dirac measures on $X$. We regard
it as a subspace of $\mathrm{W}_{1}\\!\left(X\right)$. Since $Z$ is dense in
$\widetilde{Z}^{d}$, we obtain an isometric embedding
$f\colon\mathbf{LD}(Z)\to\mathbf{LD}(\widetilde{Z}^{d})$ such that
$f(\boldsymbol{\delta}_{z})=\boldsymbol{\delta}_{z}$ for all $z\in Z$. In this
setting, the set $f(\mathbf{LD}(Z))$ is dense in
$\mathbf{LD}(\widetilde{Z}^{d})$. Thus we obtain an isometric embedding
$F\colon\mathrm{W}_{1}\\!\left(Z\right)\to\mathrm{W}_{1}\\!\left(\widetilde{Z}^{d}\right)$
such that $F|_{\mathbf{LD}(Z)}=f$. Then we can observe that
$F(\mu)=\widehat{\mu}$.

######  Proposition 2.9.

Let $(Z,d)$ be a metric space, take $e\in\mathfrak{C}$, and let $\mu$ be a
Radon probability measure on $Z$ such that $\int_{Z}d(x,p)\ d\mu<\infty$. If a
continuous function $f\colon Z\to\mathbb{R}$ belongs to
$\mathbf{Me}_{+}\\!\mathbf{C}\\!\left(Z,e\right)$, then, for every
$\epsilon\in(0,\infty)$, there exist Lipschitz functions
$M,W\colon\widetilde{Z}^{d}\to\mathbb{R}$ on
$(\widetilde{Z}^{d},{d}^{\diamondsuit})$ such that

  1. (1)

the functions $M$ and $W$ belong to
$\mathbf{Me}_{+}\\!\mathbf{C}\\!\left(\widetilde{Z}^{d},{d}^{\diamondsuit}\right)$;

  2. (2)

for every $x\in Z$, we have

$M(x)\leq f(x)\leq W(x);$

  3. (3)

we have

$\int_{Z}W(x)-f(x)\ d\mu(x)<\epsilon,$

and

$\int_{Z}f(x)-M(x)\ d\mu(x)<\epsilon.$

###### Proof.

With respect to this proof, the author was inspired by the proof of [31,
Subsection 8.5]. We make use of the McShane–Whitney extension (see [33] and
[43], see also [30, p.162]). That extension is sometimes effective for non-
Lipschitz functions in some sense. For each $r\in(0,\infty)$, for each
$x\in\widetilde{Z}^{d}$, we define

$W_{f,r}(x)=\sup\left\\{\,f(q)-r\cdot{d}^{\diamondsuit}(x,q)\mid q\in
Z\,\right\\}$

and

$M_{f,r}(x)=\inf\left\\{\,f(q)+r\cdot{d}^{\diamondsuit}(x,q)\mid q\in
Z\,\right\\}.$

Note that $M_{f,r}=-W_{-f,r}$. First, we shall show that for a sufficient
large $r\in(0,\infty)$, we have $W_{f,r}(x)<\infty$ and $-\infty<M_{f,r}(x)$.
Since $f\in\mathbf{Me}_{+}\\!\mathbf{C}\\!\left(Z,e\right)$, some
$A,\tilde{B}\in(0,\infty)$ and some $p\in Z$ satisfy that $|f(x)|\leq A\cdot
e(x,p)+\widetilde{B}$ for all $x\in Z$. Since
$\operatorname{\mathcal{D}}_{Z}(d,e)<\infty$, we have $|f(x)|\leq A\cdot
d(x,p)+B$ for all $x\in Z$, where
$B=\widetilde{B}+\operatorname{\mathcal{D}}_{Z}(d,e)$. Take $r\in(0,\infty)$
so that $A\leq r$. Then

$\displaystyle f(q)-r\cdot{d}^{\diamondsuit}(x,q)\leq A\cdot
d(q,p)+B-A\cdot{d}^{\diamondsuit}(x,q)$
$\displaystyle=B+A({d}^{\diamondsuit}(q,p)-{d}^{\diamondsuit}(q,x))\leq
B+A\cdot{d}^{\diamondsuit}(x,p).$

Thus, we have $W_{f,r}(x)\leq A\cdot{d}^{\diamondsuit}(x,p)+B<\infty$. Since
$M_{f,r}=-W_{-f,r}$ and $|-f|=|f|\leq A\cdot d(x,p)+B$, we also have
$M_{f,r}(x)\geq-A\cdot{d}^{\diamondsuit}(x,p)-B>-\infty$. As a result, we
obtain two functions $M_{f,r},W_{f,r}\colon\widetilde{Z}^{d}\to\mathbb{R}$
such that $M_{f,r}(x)\leq f(x)\leq W_{f,r}(x)$ for all $x\in Z$.

Due to [15, Theorem 2.1], we see that $M_{f,r}$ and $W_{f,r}$ are
$r$-Lipschitz on $(\widetilde{Z}^{d},{d}^{\diamondsuit})$, and hence
$M_{f,r},W_{f,r}\in\mathbf{Me}_{+}\\!\mathbf{C}\\!\left(\widetilde{Z}^{d},{d}^{\diamondsuit}\right)$.

Using a similar method to [16], [15, Theorem 2.1 and Proposition 2.2], and
[37, Lemma 5.4], we will show that, for each $x\in Z$, the values $M_{f,r}(x)$
and $W_{f,r}(x)$ convergence to $f(x)$ as $r\to\infty$. First we consider
$W_{f,r}$. Take arbitrary $\epsilon\in(0,\infty)$. Since $f$ is continuous at
$x$, we can find $\delta\in(0,\infty)$ such that if $d(x,q)<\delta$, then
$|f(x)-f(q)|\leq\epsilon$. Take $k\in(0,\infty)$ so that

$\delta\cdot k>\sup_{a\in Z}(f(a)-A\cdot d(a,x))-f(x)\geq 0.$

and take $r\in(0,\infty)$ with $r>A+k$. Note that $\sup_{a\in Z}(f(a)-A\cdot
d(a,x))\leq B<\infty$. We shall estimate $f(q)-rd(x,q)$. Under this situation,
if $d(x,q)<\delta$, we have $f(q)-r\cdot d(q,x)\leq f(q)\leq f(x)+\epsilon$.
If $d(q,x)\geq\delta$, then we have

$\displaystyle f(q)-rd(q,p)\leq[f(q)-A\cdot d(q,p)]-k\cdot d(p,q)$
$\displaystyle\leq[f(q)-A\cdot d(q,p)]-\delta\cdot k$
$\displaystyle\leq\sup_{a\in Z}[f(a)-A\cdot d(a,p)]-(\sup_{a}[f(a)-A\cdot
d(a,p)]-f(p))=f(p).$

Thus, a sufficiently large number $r\in(0,\infty)$, we have $f(x)\leq
W_{f,r}(x)\leq f(x)+\epsilon$. This means that
$\lim_{r\to\infty}W_{f,r}(x)=f(x)$. Using $M_{f,r}=-W_{-f,r}$, applying the
previous discussion to $W_{-f,r}$ and $-f$, we also obtain
$\lim_{r\to\infty}M_{f,r}(x)=f(x)$. Namely, the map $f$ is a point-wise limit
of $M_{f,r}$ and $W_{f,r}$ on $Z$ as $r\to\infty$.

Since $M_{f,r}(x)\leq W_{f,r}(x)\leq A\cdot{d}^{\diamondsuit}(x,p)+B$ for all
$x\in\widetilde{Z}^{d}$, the functions $M_{f,r}$ and $W_{f,r}$ are integrable
on $(Z,\mu)$, and hence, by the dominated convergence theorem (or the monotone
convergence theorem), we obtain

$\lim_{r\to\infty}\int_{Z}M_{f,r}(x)\
d\mu(x)=\lim_{r\to\infty}\int_{Z}W_{f,r}(x)\ d\mu(x)=\int_{Z}f(x)\ d\mu(x).$

Thus take a sufficiently large $r\in(0,\infty)$ again, and put $M=M_{f,r}$ and
$W=W_{f,r}$, we obtain the functions mentioned in the proposition. ∎

In the next theorem, we finally reach a description of the topologies of the
Wasserstein spaces on incomplete spaces.

######  Proposition 2.10.

Let $Z$ be a metrizable space, and fix
$\mathfrak{C}\in\mathbf{M}_{<\infty}(Z)$. Then for every pair
$d,e\in\mathfrak{C}$, the metrics $\mathrm{W}_{1}\\!\left[d\right]$ and
$\mathrm{W}_{1}\\!\left[e\right]$ generate the same topology on
$\mathrm{W}_{1}\\!\left(Z\right)$.

###### Proof.

Take a measure $\mu$ in $\mathrm{W}_{1}\\!\left(Z\right)$, and a sequence
$\\{\mu_{i}\\}_{i\in\mathbb{Z}_{\geq 0}}$ of measures in
$\mathrm{W}_{1}\\!\left(Z\right)$. We will show that
$\mathrm{W}_{1}\\!\left[d\right](\mu_{i},\mu)\to 0$ is equivalent to
$\mathrm{W}_{1}\\!\left[e\right](\mu_{i},\mu)\to 0$ as $i\to\infty$. Let
$(\widetilde{Z}^{d},{d}^{\diamondsuit})$, and
$(\widetilde{Z}^{e},{e}^{\diamondsuit})$ be the completions of $(Z,d)$ and
$(Z,e)$, respectively.

We first assume that $\mathrm{W}_{1}\\!\left[d\right](\mu_{i},\mu)\to 0$ as
$i\to\infty$. Then, Proposition 2.8 implies that
$\mathrm{W}_{1}\\!\left[{d}^{\diamondsuit}\right](\widehat{\mu_{i}},\widehat{\mu})\to
0$ as $i\to\infty$. Note that the sets $\widetilde{Z}^{d}\setminus Z$ and
$\widetilde{Z}^{e}\setminus Z$ are null with respect to both $\widehat{\mu}$
and $\widehat{\mu_{i}}$ due to the constructions of $\widehat{\mu}$ and
$\widehat{\mu_{i}}$. To show
$\mathrm{W}_{1}\\!\left[{e}^{\diamondsuit}\right](\widehat{\mu_{i}},\widehat{\mu})\to$
as $i\to\infty$, take an arbitrary continuous map
$\psi\colon\widetilde{Z}^{e}\to\mathbb{R}$ belonging to
$\mathbf{Me}_{+}\\!\mathbf{C}\\!\left(\widetilde{Z}^{e},{e}^{\diamondsuit}\right)$.
Put $f=\psi|_{Z}$. For every $\epsilon\in(0,\infty)$, applying Proposition
2.9, to $f$ and $\widetilde{Z}^{d}$, for each $\epsilon\in(0,\infty)$, we
obtain $r$-Lipschitz functions $M$ and $W$ on
$(\widetilde{Z}^{d},{d}^{\diamondsuit})$. such that $M\leq f\leq W$ and
$\int_{Z}W-f\ d\mu<\epsilon$, and $\int_{Z}f-M\ d\mu<\epsilon$. Since
$M,W\in\mathbf{Me}_{+}\\!\mathbf{C}\\!\left(\widetilde{Z}^{d},{d}^{\diamondsuit}\right)$,
by
$\mathrm{W}_{1}\\!\left[{d}^{\diamondsuit}\right](\widehat{\mu_{i}},\widehat{\mu})\to
0$ as $i\to\infty$ and by Statement (2) in Theorem 2.6, we have

$\int_{\widetilde{Z}^{d}}M\ d\widehat{\mu_{i}}\to\int_{\widetilde{Z}^{d}}M\
d\widehat{\mu},$

and

$\int_{\widetilde{Z}^{d}}W\ d\widehat{\mu_{i}}\to\int_{\widetilde{Z}^{d}}W\
d\widehat{\mu},$

as $i\to\infty$. By $M\leq f\leq W$ and $\int_{Z}W-f\ d\mu<\epsilon$ and
$\int_{Z}f-M\ d\mu<\epsilon$, we can compute as follows:

$\displaystyle\int_{Z}f\ d\mu-\epsilon\leq\int_{Z}M\
d\mu=\lim_{i\to\infty}\int_{Z}M\ d\mu_{i}=\liminf_{i\to\infty}\int_{Z}M\
d\mu_{i}$ $\displaystyle\leq\liminf_{i\to\infty}\int_{Z}f\
d\mu_{i}\leq\limsup_{i\to\infty}\int_{Z}f\
d\mu_{i}\leq\limsup_{i\to\infty}\int_{Z}W\ d\mu_{i}$
$\displaystyle=\lim_{i\to\infty}\int_{Z}W\ d\mu_{i}=\int_{Z}W\
d\mu\leq\int_{Z}f\ d\mu+\epsilon.$

In particular, we have

$\int_{Z}f\ d\mu-\epsilon\leq\liminf_{i\to\infty}\int_{Z}f\
d\mu_{i}\leq\liminf_{i\to\infty}\int_{Z}f\ d\mu_{i}\leq\int_{Z}f\
d\mu+\epsilon.$

Since $\epsilon$ is arbitrary, we conclude that

$\lim_{i\to\infty}\int_{Z}f\ d\mu_{i}=\int_{Z}f\ d\mu.$

Since $\psi|_{Z}=f$ and $\widetilde{Z}^{e}\setminus Z$ is null with respect to
both each $\widehat{\mu_{i}}$ and $\widehat{\mu}$, we have

$\lim_{i\to\infty}\int_{\widetilde{Z}^{e}}\psi\
d\widehat{\mu_{i}}=\int_{\widetilde{Z}^{e}}\psi\ d\widehat{\mu}.$

Therefore, Statement (2) is true for $(\widetilde{Z}^{e},{e}^{\diamondsuit})$,
$\\{\widehat{\mu_{i}}\\}_{i\in\mathbb{Z}_{\geq 0}}$, and $\widehat{\mu}$.
Thus,
$\mathrm{W}_{1}\\!\left[{e}^{\diamondsuit}\right](\widehat{\mu_{i}},\widehat{\mu})\to$
as $i\to\infty$, and hence, by Proposition 2.8, we see that
$\mathrm{W}_{1}\\!\left[e\right](\mu_{i},\mu)\to 0$ as $i\to\infty$. By
replacing the role of $e$ and $d$, we also conclude that
$\mathrm{W}_{1}\\!\left[d\right]$ and $\mathrm{W}_{1}\\!\left[e\right]$
generate the same topology. ∎

######  Remark 2.2.

Due to the proof of Proposition 2.10, we can prove Theorem 2.6 without the
completeness of $(R,w)$, and we might be able to show the Portmanteau theorem
for $p$-Wasserstein spaces of Radon measures on arbitrary metric spaces that
are not necessarily complete.

######  Proposition 2.11.

Let $Z$ be a metrizable space, and fix a class
$\mathfrak{C}\in\mathbf{M}_{<\infty}(Z)$. Then the following statements are
true:

  1. (1)

The map
$\mathrm{W}_{1}\colon\mathrm{Met}(Z)\to\mathrm{Met}(\mathrm{W}_{1}\\!\left(Z\right))$
defined by $d\mapsto\mathrm{W}_{1}\\!\left[d\right]$ is an isometric
embedding, i.e., for every pair $d,e\in\mathrm{Met}(Z)$, we have

$\operatorname{\mathcal{D}}_{Z}(d,e)=\operatorname{\mathcal{D}}_{\mathrm{W}_{1}\\!\left(Z\right)}(\mathrm{W}_{1}\\!\left[d\right],\mathrm{W}_{1}\\!\left[e\right]).$

  2. (2)

If $d\in\mathrm{Met}(Z)$ is complete, then so is
$\mathrm{W}_{1}\\!\left[d\right]$.

###### Proof.

By Corollary 2.5, we see that
$\operatorname{\mathcal{D}}_{Z}(d,e)\leq\operatorname{\mathcal{D}}_{\mathrm{W}_{1}\\!\left(Z\right)}(\mathrm{W}_{1}\\!\left[d\right],\mathrm{W}_{1}\\!\left[e\right])$.
Next, we prove the opposite inequality. Take
$\alpha,\beta\in\mathrm{W}_{1}\\!\left(Z\right)$. For an arbitrary
$\epsilon\in(0,\infty)$, take $\nu\in\Pi(\alpha,\beta)$ such that

$\int_{Z\times Z}e(s,t)\
d\nu\leq\mathrm{W}_{1}\\!\left[e\right](\alpha,\beta)+\epsilon.$

In this setting, by the definition of the $1$-Wasserstein distance, we also
obtain $\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)\leq\int_{Z\times
Z}d(s,t)\ d\nu$. Then we have

$\displaystyle\mathrm{W}_{1}\\!\left[d\right](\alpha,\beta)-\mathrm{W}_{1}\\!\left[e\right](\alpha,\beta)\leq\int_{Z\times
Z}d(s,t)\ d\nu-\int_{Z\times Z}e(s,t)\ d\nu+\epsilon$
$\displaystyle=\epsilon+\int_{Z\times Z}(d(s,t)-e(s,t))\
d\nu\leq\epsilon+\int_{Z\times Z}|d(s,t)-e(s,t)|\ d\nu$
$\displaystyle\leq\epsilon+\int_{Z\times
Z}\operatorname{\mathcal{D}}_{Z}(d,e)\
d\nu=\epsilon+\operatorname{\mathcal{D}}_{Z}(d,e)$

Since $\alpha$, $\beta$, and $\epsilon$ are arbitrary, we obtain
$\operatorname{\mathcal{D}}_{\mathrm{W}_{1}\\!\left(Z\right)}(\mathrm{W}_{1}\\!\left[d\right],\mathrm{W}_{1}\\!\left[e\right])=\operatorname{\mathcal{D}}_{Z}(d,e)$.
Statement (2) follows from [32, Theorem 2.7]. ∎

####  2.4.3. Spaces of measurable functions

Fix a topological space $M$, and a Borel probability measure $\mu$ on $M$. For
a metrizable space $Z$, we denote by $\mathrm{L}^{0}\\!\left(Z\right)$ the set
of all Borel measurable functions from $M$ into $Z$. We define
$\mathrm{I}^{\mathrm{L}}_{Z}\colon Z\to\mathrm{L}^{0}\\!\left(Z\right)$
defined by $\mathrm{I}^{\mathrm{L}}_{Z}(z)(a)=z$; namely, each
$\mathrm{I}^{\mathrm{L}}_{Z}(z)$ is a constant map. For a metric
$d\in\mathrm{Met}(Z)$, we also define a metric $\mathrm{L}\\!\left[d\right]$
on the space $\mathrm{L}^{0}\\!\left(Z\right)$ by

$\mathrm{L}\\!\left[d\right](f,g)=\int_{M}d(f(x),g(x))\ d\mu(x).$

In general, the metric $\mathrm{L}\\!\left[d\right](f,g)$ can take the value
$\infty$. As is the case of the $1$-Wasserstein spaces, we define
$\mathrm{L}^{\mathfrak{C}}\\!\left(Z\right)$ by the set of all
$f\in\mathrm{L}^{0}\\!\left(Z\right)$ such that
$\mathrm{L}\\!\left[d\right](f,\mathrm{I}^{\mathrm{L}}_{Z}(a))<\infty$ for
some/any $a\in Z$. When we fix a class
$\mathfrak{C}\in\mathbf{M}_{<\infty}(Z)$, we simply write
$\mathrm{L}\\!\left(Z\right)=\mathrm{L}^{\mathfrak{C}}\\!\left(Z\right)$.

The space $(\mathrm{L}\\!\left(Z\right),\mathrm{L}\\!\left[d\right])$ is an
analogue of the ordinary $L^{1}$ spaces on the Euclidean spaces. The author
inspired by the papers [6] and [44] with respect to this construction.

Let us observe some properties of spaces of measurable functions.

######  Proposition 2.12.

Let $M$ be a topological space, and $\mu$ be a Borel probability measure on
$M$. Let $Z$ be a metrizable space, fix $w\in\mathrm{Met}(Z)$, and put
$\mathfrak{C}=\mathfrak{C}(w)\in\mathbf{M}_{<\infty}(Z)$. Then we have the
following statements:

  1. (1)

For every $d\in\mathfrak{C}$, and for every pair $x,y\in Z$, we have

$\mathrm{L}\\!\left[d\right]\left(\mathrm{I}^{\mathrm{L}}_{Z}(x),\mathrm{I}^{\mathrm{L}}_{Z}(y)\right)=d(x,y).$

  2. (2)

For every $d\in\mathfrak{C}$, and for every pair
$f,g\in\mathrm{L}^{\mathfrak{C}}\\!\left(Z\right)$, we have
$\mathrm{L}(d)(f,g)<\infty$.

  3. (3)

For every pair $d,e\in\mathfrak{C}$, we have
$\operatorname{\mathcal{D}}_{Z}(d,e)=\operatorname{\mathcal{D}}_{\mathrm{L}\\!\left(Z\right)}(\mathrm{L}(d),\mathrm{L}(e))$.

###### Proof.

Statement (1) is follows from the definitions of $\mathrm{L}\\!\left[d\right]$
and $\mathrm{I}^{\mathrm{L}}_{Z}$.

Proof of (2): We only need to show the statement in the case of
$g=\mathrm{I}^{\mathrm{L}}_{Z}(a)$ for some $a\in Z$. By the definition of
$\mathrm{L}^{\mathfrak{C}}\\!\left(Z\right)$, we have
$\mathrm{L}\\!\left[w\right](f,g)<\infty$. Then we have

$\displaystyle\mathrm{L}(d)(f,g)$ $\displaystyle=\int_{M}d(f(x),a)\ d\mu(x)$
$\displaystyle\leq\int_{M}w(f(x),a)\
d\mu(x)+\operatorname{\mathcal{D}}_{Z}(d,w)$
$\displaystyle=\mathrm{L}\\!\left[w\right](f,g)+\operatorname{\mathcal{D}}_{Z}(d,w)<\infty.$

This proves Statement (2).

Proof of (3): For every pair $f,g\in\mathrm{L}\\!\left(Z\right)$, we have

$\displaystyle\mathrm{L}(d)(f,g)-\mathrm{L}(e)(f,g)=\int_{M}d(f(t),g(t))-e(f(t),g(t))\
d\mu$ $\displaystyle\leq\int_{M}|d(f(t),g(t))-e(f(t),g(t))|\
d\mu\leq\int_{M}\operatorname{\mathcal{D}}_{Z}(d,e)\
d\mu=\operatorname{\mathcal{D}}_{M}(d,e).$

Then we obtain
$\operatorname{\mathcal{D}}_{\mathrm{L}\\!\left(Z\right)}(\mathrm{L}(d),\mathrm{L}(e))\leq\operatorname{\mathcal{D}}_{Z}(d,e)$.
Statement (1) shows that
$\operatorname{\mathcal{D}}_{Z}(d,e)\leq\operatorname{\mathcal{D}}_{\mathrm{L}\\!\left(Z\right)}(\mathrm{L}(d),\mathrm{L}(e))$,
and hence

$\operatorname{\mathcal{D}}_{Z}(d,e)=\operatorname{\mathcal{D}}_{\mathrm{L}\\!\left(Z\right)}(\mathrm{L}(d),\mathrm{L}(e)).$

This finishes the proof. ∎

We next consider the topologies of $\mathrm{L}\\!\left(Z\right)$. We need the
following observation on continuous maps on compact spaces.

######  Lemma 2.13.

Let $Y$ be a metrizable space, and take
$\mathfrak{C}\in\mathbf{M}_{<\infty}(Y)$ and $d,e\in\mathfrak{C}$. Let $K$ be
a compact metrizable space, and $q\colon K\to Y$ be a continuous map. Then,
for every $\epsilon\in(0,\infty)$, there exists $\eta\in(0,\infty)$ such that
for every $t\in K$ and for every $y\in Z$, if $d(q(t),y)<\eta$, then
$e(q(t),y)<\epsilon$.

###### Proof.

Using an idea of the proof of [28, Proposition 3], for each $s\in K$, we
denote by $R(s)$ the set of all $u\in(0,\infty)$ such that for every $y\in Z$,
if $d(q(s),y)<u$, then $e(q(s),y)+u<\epsilon$. Note that $R(s)\neq\emptyset$
since $e$ and $d$ generate the same topology on $X$. The definition of $R(s)$
seems strange, but it is effective for our lemma. We now prove that the map
$r\colon K\to(0,\infty]$ defined by $s\mapsto\sup R(s)$ is lower semi-
continuous, i.e., for every $a\in\mathbb{R}$, we shall show that
$r^{-1}((a,\infty])$ is open in $K$. Take $s\in r^{-1}((a,\infty])$. By the
definition of $r$, we can take $u\in R(s)$ such that $a<u-\delta_{0}<u\leq
r(s)$, where $\delta_{0}=2^{-1}(r(s)-a)\in(0,\infty)$. Fix
$w\in\mathrm{Met}(K)$. Since the map $q\colon K\to Y$ is continuous, we can
find $\delta_{1}\in(0,\infty)$ such that

  1. (1)

if $w(s,t)<\delta_{1}$, then we have $d(q(s),q(t))<u-\delta_{0}$,

  2. (2)

if $w(s,t)<\delta_{1}$, then we have $e(q(s),q(t))<\delta_{0}$.

Assume that $t\in U(s,\delta_{1};w)$. Let us verify $u-\delta_{0}\in R(t)$. If
$d(q(t),y)<u-\delta_{0}$, then we have

$d(q(s),y)\leq d(q(s),q(t))+d(q(t),y)<\delta_{0}+(u-\delta_{0})=u,$

and hence we also have

$\displaystyle e(q(t),y)+(u-\delta_{0})$ $\displaystyle\leq
e(q(t),q(s))+e(q(s),y)+(u-\delta_{0})$
$\displaystyle<\delta_{0}+e(q(s),y)+(u-\delta_{0})=e(q(s),y)+u<\epsilon.$

Thus $u-\delta_{0}\in R(t)$. Hence $a<r(t)$. This means that
$U(s,\delta_{1};w)\subseteq r^{-1}((a,\infty])$, i.e., the set
$r^{-1}((a,\infty])$ is open in $K$. Thus $r$ is lower semi-continuous on $K$.
Since $K$ is compact and $r$ is lower semi-continuous, the map $r$ has a
minimum $r_{\min}>0$. Put $\eta=\frac{1}{2}r_{\min}$. Then it is the number as
desired. ∎

We now show that the topology induced by $\mathrm{L}\\!\left[d\right]$ does
not depend on the choice of $d\in\mathfrak{C}$ when $M$ is Polish.

######  Proposition 2.14.

Let $M$ be a Polish space, and $\mu$ be a Borel probability measure on $M$.
Let $Z$ be a metrizable space, fix $w\in\mathrm{Met}(Z)$, put
$\mathfrak{C}=\mathfrak{C}(w)\in\mathbf{M}_{<\infty}(Z)$. Then for every pair
$d,e\in\mathfrak{C}$, the metrics $\mathrm{L}(d)$ and $\mathrm{L}(e)$ generate
the same topology on $\mathrm{L}\\!\left(Z\right)$.

###### Proof.

Fix $q\in\mathrm{L}\\!\left(Z\right)$. We will show that for every
$\epsilon\in(0,\infty)$, there exists $\delta\in(0,\infty)$ such that if
$f\in\mathrm{L}\\!\left(Z\right)$ satisfies
$\mathrm{L}\\!\left[d\right](q,f)<\delta$, then we have
$\mathrm{L}\\!\left[e\right](q,f)<\epsilon$.

Put $C=\operatorname{\mathcal{D}}_{Z}(d,e)<\infty$ and take a number
$\eta_{0}\in(0,\infty)$ such that

  1. (k1)

we have $2(\eta_{0}+C)\eta_{0}<2^{-1}\epsilon$.

Since $q\colon M\to Z$ is Borel measurable and $M$ is Polish, by [10, Vol II,
Lemma 6.10.16], the image $q(M)$ of $q$ is separable. Note that the map
$q\colon M\to q(M)$ is also Borel measurable. Thus we can apply Lusin’s
theorem [25, Theorem 17.12] to $q\colon M\to q(M)$, and for the number
$\eta_{0}$ taken above, we obtain a compact subset $K$ of $M$ such that

  1. (2)

$\mu(M\setminus K)<\eta_{0}$;

  2. (3)

the restricted function $q|_{K}\colon K\to Z$ is continuous on $K$.

Take a sufficiently small number $\eta\in(0,\infty)$ such that

  1. (4)

we have $\eta<\eta_{0}$;

  2. (5)

for every $s\in K$, and for every $y\in Z$, if $d(q(s),y)<\eta$, then we have
$e(q(s),y)<2^{-1}\epsilon$.

Remark that the property 5 follows from 3 and Lemma 2.13.

Put $\delta=\eta^{2}$ and assume that $\mathrm{L}(d)(q,f)<\delta=\eta^{2}$. We
also put

$R=\\{\,t\in M\mid\eta\leq d(q(t),f(t))\,\\}.$

Then we have $\mu(R)\leq\eta$ by Chebyshev’s inequality. Thus using (k1) and
4, we also have

$\displaystyle\int_{R}e(q(t),f(t))\
d\mu\leq\int_{R}d(q(t),f(t))+\operatorname{\mathcal{D}}_{Z}(d,e)\ d\mu\ $
$\displaystyle=\int_{R}d(q(t),f(t))\
d\mu+\int_{R}\operatorname{\mathcal{D}}_{Z}(d,e)\ d\mu$
$\displaystyle\leq\mathrm{L}(d)(q,f)+C\eta<\eta^{2}+C\eta=(\eta+C)\eta\leq(\eta_{0}+C)\eta_{0}.$

For each $s\in(M\setminus R)\cap K$, we have $d(q(s),f(s))<\eta$, and hence,
by the property 5 of $K$, we have $e(q(s),f(s))<2^{-1}\epsilon$. Thus,

$\int_{(M\setminus R)\cap K}e(q(t),f(t))\ d\mu\leq\int_{(M\setminus R)\cap
K}2^{-1}\epsilon\ d\mu\leq 2^{-1}\epsilon.$

Using $\operatorname{\mathcal{D}}_{Z}(d,e)=C$ and $e(q(t),f(t))\leq
d(q(t),f(t))+C$ for all $t\in M$, due to and (k1), 2, and 4 we obtain

$\displaystyle\int_{(M\setminus R)\setminus K}e(q(t),f(t))\
d\mu=\int_{M\setminus(R\cup K)}e(q(t),f(t))\ d\mu$
$\displaystyle\leq\int_{M\setminus(R\cup K)}d(q(t),f(t))+C\ d\mu$
$\displaystyle\leq\eta\cdot\mu(M\setminus R\cup K)+C\cdot\mu(M\setminus K)$
$\displaystyle\leq\eta\cdot\mu(M\setminus K)+C\cdot\mu(M\setminus
K)<\eta\cdot\eta_{0}+C\eta_{0}<(\eta_{0}+C)\eta_{0}.$

Combining these estimations, as a result, we also obtain

$\displaystyle\int_{M}e(q(t),f(t))\ d\mu=\int_{R}e(q(t),f(t))\
d\mu+\int_{M\setminus R}e(q(t),f(t))\ d\mu$
$\displaystyle\leq\int_{R}e(q(t),f(t))\ d\mu+\int_{(M\setminus R)\cap
K}e(q(t),f(t))\ d\mu+$ $\displaystyle\int_{(M\setminus R)\setminus
K}e(q(t),f(t))\ d\mu<(\eta_{0}+C)\eta_{0}+2^{-1}\epsilon+(\eta_{0}+C)\eta_{0}$
$\displaystyle=2(\eta_{0}+C)\eta_{0}+2^{-1}\epsilon<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon.$

That means that $\mathrm{L}(e)(q,f)<\epsilon$. Replacing the role of $d$ with
that of $e$, we conclude that $\mathrm{L}\\!\left[d\right]$ and
$\mathrm{L}\\!\left[e\right]$ generate the same topology. ∎

######  Remark 2.3.

A metric measure space $(M,\mu)$, where $M$ is Polish is roughly the same to
the unit interval with the Lebesgue measure (see [37, Lemma 4.2] and [25,
(17.41)]). Thus, our construction of $\mathrm{L}\\!\left(Z\right)$ is roughly
the space of measurable maps from the unit interval into $Z$.

In what follows, we always assume that $M=\omega_{0}$; namely, $M$ is
countable discrete space. Under this assumption, the set of all measurable
maps from $\omega_{0}$ into $Z$ is identical with the set of all maps from
$\omega_{0}$ into $Z$. In the proof of the main theorem, we only consider the
measure $\Psi$ on $\omega_{0}$ defined by

$\Psi=\sum_{s\in\omega_{0}}\frac{1}{2^{s+1}}\boldsymbol{\delta}_{s}.$

###  2.5. Osmotic constructions

In this subsection, we introduce the notion of osmotic constructions as an
abstraction from these three constructions explained in the previous
subsection.

Assume that a construction $\mathrm{F}$ of metric spaces consists of three
data for every metrizable space $Z$ and for every
$\mathfrak{C}\in\mathbf{M}_{<\infty}(Z)$:

  1. (A1)

a metrizable space $\mathrm{F}^{\mathfrak{C}}\left(Z\right)$;

  2. (A2)

a topological embedding $\mathrm{I}^{\mathrm{F}}_{Z}\colon
Z\to\mathrm{F}^{\mathfrak{C}}\left(Z\right)$;

  3. (A3)

a map $\mathrm{F}$ from $\mathfrak{C}$ into a set of metrics on
$\mathrm{F}^{\mathfrak{C}}\left(Z\right)$. Namely, each $d\in\mathfrak{C}$ is
mapped to a metric $\mathrm{F}\\!\left[d\right]$ on
$\mathrm{F}^{\mathfrak{C}}\left(Z\right)$.

In this setting, we say that $\mathrm{F}$ is _osmotic_ if the following
conditions are satisfied:

  1. (O1)

for each $d\in\mathfrak{C}$ we have

$\mathrm{F}\\!\left[d\right]\left(\mathrm{I}^{\mathrm{F}}_{Z}(x),\mathrm{I}^{\mathrm{F}}_{Z}(y)\right)=d(x,y);$

  2. (O2)

for every pair $d,e\in\mathfrak{C}$, the metrics $\mathrm{F}\\!\left[d\right]$
and $\mathrm{F}\\!\left[e\right]$ generate the same topology of
$\mathrm{F}^{\mathfrak{C}}\left(Z\right)$. Namely, we have
$\mathrm{F}\\!\left[d\right]\in\mathrm{Met}(\mathrm{F}^{\mathfrak{C}}\left(Z\right))$.
This means that the map $\mathrm{F}$ from $\mathfrak{C}$ into a set of metrics
on $\mathrm{F}^{\mathfrak{C}}\left(Z\right)$ becomes a map
$\mathfrak{C}\to\mathrm{Met}(\mathrm{F}^{\mathfrak{C}}\left(Z\right))$;

  3. (O3)

the map
$\mathrm{F}\colon\mathfrak{C}\to\mathrm{Met}(\mathrm{F}^{\mathfrak{C}}\left(Z\right))$
satisfies that, for every pair $d,e\in\mathfrak{C}$, we have

$\operatorname{\mathcal{D}}_{Z}(d,e)=\operatorname{\mathcal{D}}_{\mathrm{F}^{\mathfrak{C}}\left(Z\right)}(\mathrm{F}\\!\left[d\right],\mathrm{F}\\!\left[e\right]).$

Namely, these conditions mean that $d$ can be naturally extended into a metric
on $\mathrm{F}\\!\left(Z\right)$ as if $d\in\mathfrak{C}$ on $Z$ osmoses from
$\mathrm{I}^{\mathrm{F}}_{Z}(Z)(=Z)$ into the extension space
$\mathrm{F}\\!\left(Z\right)$.

We can summarize the results in the previous subsection using the concept of
osmotic constructions.

######  Theorem 2.15.

The construction $\$$ of $\ell^{1}$-products, the construction
$\mathrm{W}_{1}$ of $1$-Wasserstein spaces, and the construction $\mathrm{L}$
of spaces of measurable functions are osmotic.

##  3\. The Whitney–Dugundji decomposition

To show our main result, we review the classical discussion called the
Whitney–Dugundji decomposition, which is a special partition of unity on the
complement of a closed subset in consideration. Such a decomposition in the
Euclidean case was used by Whitney [43] to prove the famous Whitney extension
theorem. The general case of metric spaces was found by Dugundji [13]. See
also [1].

For the sake of convenience, in this paper, we define such a decomposition as
a quadruplet of open covering, a partition of unity, two family of points.

######  Definition 3.1.

Let $X$ be a metrizable space, and $A$ be a closed subset of $X$. Fix
$w\in\mathrm{Met}(X)$. For $k\in\mathbb{Z}_{\geq 0}$ and for a metric space
$(X,w)$, we say that a quadruplet

$\mathcal{WD}=[\mathcal{O},\\{\mathbf{y}_{O}\\}_{O\in\mathcal{O}},\\{\mathbf{a}_{O}\\}_{O\in\mathcal{O}},\\{\varphi_{O}\\}_{O\in\mathcal{O}}]$

is a _$(k,w)$ -WD collection_ if the following conditions are satisfied:

  1. (WD1)

the family $\mathcal{O}$ is a locally finite open covering of $X\setminus A$
consisting of open sets of $X\setminus A$, each $\mathbf{y}_{O}$ is a point in
$X\setminus A$, each $\mathbf{a}_{O}$ is a point in $A$, and each
$\varphi_{O}$ is a continuous function from $X$ to $[0,1]$;

  2. (WD2)

For every $O\in\mathcal{O}$, we have

$O\subset U\left(\mathbf{y}_{O},\frac{w(\mathbf{y}_{O},A)}{4^{k+1}};w\right);$

  3. (WD3)

For every $O\in\mathcal{O}$, we have

$w(\mathbf{y}_{O},\mathbf{a}_{O})\leq\frac{4^{k+1}+1}{4^{k+1}}w(\mathbf{y}_{O},A);$

  4. (WD4)

For every $O\in\mathcal{O}$, we have

$\text{{\rm supp}}(\varphi_{O})\subseteq O,$

where $\text{{\rm supp}}(\varphi_{O})$ stands for the support of $\varphi_{O}$
defined as $\\{\,x\in X\mid\varphi_{O}(x)\neq 0\,\\}$;

  5. (WD5)

The family $\\{\varphi_{O}\\}_{O\in\mathcal{O}}$ satisfies

$\sum_{O\in\mathcal{O}}\varphi_{O}(x)=\begin{cases}0&\text{if $x\in A$;}\\\
1&\text{if $x\in X\setminus A$}.\end{cases}$

Moreover, if the quadruplet satisfies the following condition, then it is
called a _strong $(k,w)$-WD collection_:

  1. (WD6)

if $x\in X\setminus A$ and $O\in\mathcal{O}$ satisfy $\varphi_{O}(x)>0$, then

$\operatorname{diam}_{w}(O)\leq 16\cdot\frac{w(x,A)}{4^{k+1}}.$

Before proving the existence of strong WD collections, we provides a
convenient estimation related to WD collections.

######  Lemma 3.1.

Let $X$ be a metrizable space, and $A$ be a closed subset of $X$. Fix
$w\in\mathrm{Met}(X)$ and $k\in\mathbb{Z}_{\geq 0}$. Assume that a quadruplet
$\mathcal{WD}=[\mathcal{O},\\{\mathbf{y}_{O}\\}_{O\in\mathcal{O}},\\{\mathbf{a}_{O}\\}_{O\in\mathcal{O}},\\{\varphi_{O}\\}_{O\in\mathcal{O}}]$
is a $(k,w)$-WD collection. Then, for every $a\in A$ and for every $x\in X$,
if $\varphi_{O}(x)>0$, then we have

$w(a,\mathbf{a}_{O})\leq 4w(a,x).$

###### Proof.

Our proof is based on that of [1, Statement 2.3]. Since $\varphi_{O}(x)>0$,
the conditions (WD4) and (WD2) implies that $x\in O$ and

$O\subset U\left(\mathbf{y}_{O},\frac{w(\mathbf{y}_{O},A)}{4^{k+1}}\right).$

Thus we have

$w(x,\mathbf{y}_{O})\leq\frac{w(\mathbf{y}_{O},A)}{4^{k+1}}.$

Using this estimation and the condition (WD3), we also have

$\displaystyle w(a,\mathbf{a}_{O})$ $\displaystyle\leq
w(a,x)+w(x,\mathbf{y}_{O})+w(\mathbf{y}_{O},\mathbf{a}_{O})$
$\displaystyle\leq
w(a,x)+\frac{1}{4^{k+1}}w(\mathbf{y}_{O},A)+\frac{1+4^{k+1}}{4^{k+1}}w(\mathbf{y}_{O},A)$
$\displaystyle=w(a,x)+\frac{4^{k+1}+2}{4^{k+1}}w(\mathbf{y}_{O},A).$

Namely,

(1) $\displaystyle w(a,\mathbf{a}_{O})\leq
w(a,x)+\frac{4^{k+1}+2}{4^{k+1}}w(\mathbf{y}_{O},A).$

Moreover, by

$w(\mathbf{y}_{O},A)\leq w(\mathbf{y}_{O},a)\leq
w(\mathbf{y}_{O},x)+w(x,a)\leq\frac{1}{4^{k+1}}w(\mathbf{y}_{O},A)+w(x,a),$

we obtain

$\left(1-\frac{1}{4^{k+1}}\right)w(\mathbf{y}_{O},A)\leq w(x,a),$

and hence, we also obtain

(2) $\displaystyle w(\mathbf{y}_{O},A)\leq\frac{4^{k+1}}{4^{k+1}-1}w(x,a).$

Therefore, combining these inequalities (1) and (2), we confirm the following
computations.

$\displaystyle w(a,\mathbf{a}_{O})\leq
w(a,x)+\frac{4^{k+1}+2}{4^{k+1}}w(\mathbf{y}_{O},A)\leq
w(a,x)+\frac{4^{k+1}+2}{4^{k+1}-1}w(a,x)$ $\displaystyle=\frac{2\cdot
4^{k+1}+1}{4^{k+1}-1}w(a,x)=\frac{2+4^{-k-1}}{1-4^{-k-1}}w(x,a)\leq\frac{2+1}{3/4}\leq
4w(x,a).$

This finishes the proof. ∎

Now we prove existence of the $(k,w)$-WD collections.

######  Theorem 3.2.

Let $X$ be a metrizable space, and $A$ be a closed subset of $X$. Fix
$w\in\mathrm{Met}(X)$ and $k\in\mathbb{Z}_{\geq 0}$. Then there exists a
strong $(k,w)$-WD collection.

###### Proof.

For each $i\in\mathbb{Z}$, we define an open set $V_{i}$ of $X\setminus A$ by

$V_{i}=\\{\,x\in X\setminus A\mid 2^{i-1}<w(x,A)<2^{i+1}\,\\}.$

Note that if $2\leq|i-j|$, then $V_{i}\cap V_{j}=\emptyset$. Put

$\mathcal{V}_{i}=\\{\,V_{i}\cap U\left(x,w(x,A)4^{-k-1};w\right)\mid x\in
V_{i}\,\\}$

Then each $\mathcal{V}_{i}$ is an open covering of $V_{i}$. Put
$\mathcal{W}=\bigcup_{i\in\mathbb{Z}_{\geq 0}}\mathcal{V}_{i}$. Since $X$ is
paracompact (see [40, Corollary 1] and [36]), we can take a locally finite
partition of unity $\\{\psi_{a}\\}_{a\in I}$ subordinated to $\mathcal{W}$
(see [34, Proposition 2]). By taking finite sums if necessary, we may assume
that if $a,b\in I$ satisfy $a\neq b$, then we have $\text{{\rm
supp}}(\psi_{a})\neq\text{{\rm supp}}(\psi_{b})$. Put
$\mathcal{O}=\\{\,\text{{\rm supp}}(\psi_{a})\mid a\in I\,\\}$ and
$\varphi_{O}=\psi_{a}$, where $\psi_{a}$ is a unique member such that
$\text{{\rm supp}}(\psi_{a})=O$. Then, the family $\mathcal{O}$ is a locally
finite open covering of $X\setminus A$, and it refines $\mathcal{W}$.

For each $O\in\mathcal{O}$, choose $\mathbf{y}_{O}\in X\setminus A$ so that
$O\subset U(\mathbf{y}_{O},w(\mathbf{y}_{O},4^{-k-1}))$. Note that if
$O\subseteq V_{i}$, then there exists $j\in\mathbb{Z}$ such that $|i-j|\leq 1$
and $\mathbf{y}_{O}\in V_{j}$. We also choose $\mathbf{a}_{O}\in A$ so that we
have

$w(\mathbf{y}_{O},\mathbf{a}_{O})\leq\frac{4^{k+1}+1}{4^{k+1}}w(\mathbf{y}_{O},A).$

In this setting, if $\varphi_{O}(x)>0$ and $O\in\mathcal{W}_{i}$, then there
exists $j\in\mathbb{Z}$ with $|i-j|\leq 1$ we have $x\in V_{j}\cap
U(\mathbf{y}_{O},w(\mathbf{y}_{O},A)4^{-k-1};w)$ and $\mathbf{y}_{O}\in
V_{j}$. By the construction, we observe that the quadruplet

$[\mathcal{O},\\{\mathbf{y}_{O}\\}_{O\in\mathcal{O}},\\{\mathbf{a}_{O}\\}_{O\in\mathcal{O}},\\{\varphi_{O}\\}_{O\in\mathcal{O}}]$

is a $(k,w)$-WD collection.

We next confirm that the quadruplet satisfies the condition (WD6). Assume that
$x\in X\setminus A$ and $O\in\mathcal{O}$ satisfy $\varphi_{O}(x)>0$. Take
$i\in\mathbb{Z}$ such that $O\in V_{i}$, and take $j\in\mathbb{Z}$ such that
$\mathbf{y}_{O}\in V_{j}$. In this case, we have $|i-j|\leq 1$. Under this
setting, we obtain

$\displaystyle\operatorname{diam}_{w}O\leq 2\cdot
4^{-k-1}w(\mathbf{y}_{O},A)\leq 2\cdot 4^{-k-1}\cdot 2^{j+1}\leq 2\cdot
4^{-k-1}\cdot 2^{i+2}$ $\displaystyle=2^{4}\cdot 4^{-k-1}2^{i-1}\leq
2^{4}\cdot 4^{-k-1}w(x,A)=16\cdot w(x,A)/4^{k+1}$

This means that the quadruplet is a strong $(k,w)$-WD collection. ∎

##  4\. Proof of the main result

The whole of this section is devoted to the proof of Theorem 1.1.

Throughout this section, let $X$ be a metrizable space, and $A$ be a closed
subset of $X$.

First we fix $\mathfrak{C}\in\mathbf{M}_{<\infty}(A)$, and take
$m\in\mathrm{Met}(A)$ such that $\mathfrak{C}=\mathfrak{C}(m)$. Fix
$w\in\mathrm{Met}(X)$ with $w|_{A^{2}}=m$. If $X$ is completely metrizable, we
choose $m$ as a complete metric, which can be done by [24, Theorem 1.4], and
we also choose $w$ as a complete metric on $X$ (this is a variant of
Hausdorf’s metric extension theorem. see [7], [3], and [41]).

For each $s\in\omega_{0}$, fix a strong $(s,w)$-WD collection

$\mathcal{WD}_{s}=[\mathcal{O}_{s},\\{\mathbf{y}_{O,s}\\}_{O\in\mathcal{O}_{s}},\\{\mathbf{a}_{O,s}\\}_{O\in\mathcal{O}_{s}},\\{\varphi_{O,s}\\}_{O\in\mathcal{O}_{s}}]$

with respect to $X$ and $A$.

Using a partition of unity (with respect to “$\sum$”), we shall construct a
partition of unity with respect to “$\sup$”.

######  Lemma 4.1.

For each $s\in\omega_{0}$, there exists a family
$\\{\sigma_{O,s}\\}_{O\in\mathcal{O}_{s}}$ such that

  1. (1)

each $\sigma_{O,s}$ is a continuous function from $X$ to $[0,1]$;

  2. (2)

we have $\text{{\rm supp}}(\sigma_{O,s})=\text{{\rm supp}}(\varphi_{O,s})$ for
all $O\in\mathcal{O}_{s}$;

  3. (3)

for every $x\in X$, there exists $O\in\mathcal{O}_{s}$ such that
$\sigma_{O,k}(x)=1$.

###### Proof.

Define $\Phi\colon X\setminus A\to(0,1]$ by
$\Phi(x)=\sup_{O\in\mathcal{O}_{s}}\varphi_{O,s}(x)$. We also define
$\sigma_{O,s}\colon X\to[0,1]$ by

$\sigma_{O,s}(x)=\frac{2}{\Phi(x)}\cdot\min\left\\{\varphi_{O,s}(x),\frac{\Phi(x)}{2}\right\\}.$

Then it is a map as required. ∎

In what follows, we fix a family $\\{\sigma_{O,s}\\}_{O\in\mathcal{O}_{s}}$
stated in Lemma 4.1.

Consider $\coprod_{s\in\omega_{0}}\mathcal{O}_{s}\times\\{s\\}$, and put
$S=\mathbf{L}_{fin}^{\infty}(\coprod_{s\in\omega_{0}}\mathcal{O}_{s}\times\\{s\\})$
(see Subsection 2.3). Recall that $\mathbf{D}_{L^{\infty}}$ stands for the
supremum metric on this space
$S=\mathbf{L}_{fin}^{\infty}(\coprod_{s\in\omega_{0}}\mathcal{O}_{s}\times\\{s\\})$.
Namely, the space $(S,\mathbf{D}_{L^{\infty}})$ is a space of families
$\\{f(O,s)\\}_{(O,s)\in\coprod_{s\in\omega_{0}}\mathcal{O}_{s}\times\\{s\\}}$
of real numbers indexed by the set
$\coprod_{s\in\omega_{0}}\mathcal{O}_{s}\times\\{s\\}$. We denote by $0\in S$
the constant map taking the value $0\in\mathbb{R}$.

For each $s\in\omega_{0}$, fix a continuous map
$\gamma_{s}\colon[0,\infty)\to[0,1]$ such that
$\gamma_{s}([0,2^{-s}])=\\{0\\}$,
$\gamma_{s}((2^{-s},2^{-s+1}))\subseteq(0,1)$, and
$\gamma_{s}([2^{-s+1},\infty))=\\{1\\}$. We also define $\xi_{s}\colon
X\to[0,1]$ by

$\xi_{s}(x)=\gamma_{s}(w(x,A)).$

For $x\in X$, and $s\in\omega_{0}$, we define a map
$\mathbf{h}(x,s)\colon\coprod_{s\in\omega_{0}}\mathcal{O}_{s}\times\\{s\\}\to\mathbb{R}$
by

$\mathbf{h}(x,s)(O,i)=\begin{cases}\xi_{s}(x)\cdot\sigma_{O,s}(x)&\text{if
$s=i$;}\\\ 0&\text{if $s\neq i$.}\end{cases}$

Note that each $\mathbf{h}(x,s)$ belongs to $S$. Let us see basic properties
of $\mathbf{h}(x,s)$.

######  Lemma 4.2.

The following statement are true:

  1. (1)

Fix $s\in\omega_{0}$. Then the map $\mathbf{h}_{s}\colon X\to S$ defined by
$x\mapsto\mathbf{h}(x,s)$ is continuous.

  2. (2)

If $x\in X$ and $N\in\omega_{0}$ satisfy
$\mathbf{D}_{L^{\infty}}(\mathbf{h}(x,N),0)<1$, then we have $w(x,A)\leq
2^{-N+1}$.

  3. (3)

If $x,y\in X$ and $N\in\omega_{0}$ satisfy $\xi_{N}(y)=\xi_{N}(y)=1$ and
$\mathbf{D}_{L^{\infty}}(\mathbf{h}(x,N),\mathbf{h}(y,N))<1$, then there
exists $O\in\mathcal{O}_{N}$ such that $\sigma_{O,N}(x)>0$ and
$\sigma_{O,N}(y)>0$. In particular, we have $x,y\in O$, and $w(x,y)\leq
16\cdot 4^{-N-1}w(x,A)$.

###### Proof.

Statement (1) follows from the fact that each $\sigma_{O,s}$ is continuous. We
shall show (2). If $x\in A$, we have $w(x,A)=0\leq 2^{-N}$. If $x\in
X\setminus A$, then the assumption implies that
$\max_{O\in\mathcal{O}_{N}}\xi_{N}(x)\sigma_{O,N}(x)<1$. Take
$P\in\mathcal{O}_{N}$ with $\sigma_{P,N}(x)=1$ (see Lemma 4.1). Thus we have
$\xi_{N}(x)<1$. By the definition of $\xi_{N}$, we conclude that $w(x,A)\leq
2^{-N+1}$. Now we show (3). In this setting, we see that $x,y\in A$ and

$\sup_{x\in X}|\sigma_{O,N}(x)-\sigma_{O,N}(y)|<1.$

Take $O\in\mathcal{O}_{s}$ with $\sigma_{O,N}(x)=1$. Then $\sigma_{O,N}(y)>0$.
Since $\text{{\rm supp}}(\sigma_{O,N})=\text{{\rm supp}}(\varphi_{O,N})$, the
conditions (WD4) and (WD6) show (3). ∎

Next, we consider
$(\mathrm{W}_{1}\\!\left(A\right),\mathrm{W}_{1}\\!\left[m\right])$ and we
represent

$\$\mathrm{W}_{1}\\!\left(A\right)=\$\\!\left(\mathrm{W}_{1}\\!\left(A\right)\right)=\$\\!\left(\mathrm{W}_{1}\\!\left(A\right),S\right)=\mathrm{W}_{1}\\!\left(A\right)\times
S,$

and

$\$\mathrm{W}_{1}\\!\left[d\right]=\$\\!\left[\mathrm{W}_{1}\\!\left[d\right]\right]=\mathrm{W}_{1}\\!\left[d\right]\times_{\ell^{1}}\mathbf{D}_{L^{\infty}}.$

for every $d\in\mathfrak{C}$. In the proof of the main theorem, we only
consider the measure $\Psi$ on $\omega_{0}$ defined by

$\Psi=\sum_{s\in\omega_{0}}\frac{1}{2^{s+1}}\boldsymbol{\delta}_{s}.$

Using the measure space $(\omega_{0},\Psi)$, we construct
$\mathrm{L}\\!\left(\$\mathrm{W}_{1}\\!\left(A\right)\right)$. Using the zero
element $0\in S$, we also construct a topological embedding
$\mathrm{I}^{\mathrm{W}_{1}}_{A}$. Then we can obtain an topological embedding
$I\colon A\to\mathrm{L}\\!\left(\$\mathrm{W}_{1}\\!\left(A\right)\right)$ as a
composition
$\mathrm{I}^{\mathrm{L}}_{\$\mathrm{W}_{1}\\!\left(A\right)}\circ\mathrm{I}^{\$}_{\mathrm{W}_{1}\\!\left(A\right)}\circ\mathrm{I}^{\mathrm{W}_{1}}_{A}$.
Namely, $I(a)$ is a constant map
$\omega_{0}\to\$\mathrm{W}_{1}\\!\left(A\right)$ such that
$I(a)(s)=(\boldsymbol{\delta}_{a},0)$, where $0\in S$. Now, using $w$, we
construct a topological embedding $\mathbf{J}\colon
X\to\mathrm{L}\\!\left(\$\mathrm{W}_{1}\\!\left(A\right)\right)$ such that
$\mathbf{J}|_{A}=I$.

For $x\in X$, and for $s\in\omega_{0}$ we define
$\mathbf{H}(x,s)\in\mathrm{W}_{1}\\!\left(A\right)$ by

$\mathbf{H}(x,s)=\begin{cases}\sum_{O\in\mathcal{O}_{s}}\varphi_{O,s}(x)\cdot\boldsymbol{\delta}_{a_{O,s}}&\text{if
$x\in X\setminus A$}\\\ \boldsymbol{\delta}_{x}&\text{if $x\in A$}\end{cases}$

######  Lemma 4.3.

Fix $s\in\omega_{0}$. Then the map $\mathbf{H}_{s}\colon
X\to\mathrm{W}_{1}\\!\left(A\right)$ defined by $x\mapsto\mathbf{H}(x,s)$ is
continuous.

###### Proof.

The lemma follows from Lemma 2.3 and the local finiteness of $\\{\text{{\rm
supp}}(\varphi_{O,s})\\}_{O\in\mathcal{O}_{s}}$. ∎

For $x\in X$, we put $\mathbf{J}_{s}(x)=(\mathbf{H}(x,s),\mathbf{h}(x,s))$,
and define
$\mathbf{J}(x)\colon\omega_{0}\to\$\mathrm{W}_{1}\\!\left(A\right)=\mathrm{W}_{1}\\!\left(A\right)\times
S$ by $\mathbf{J}(x)(s)=\mathbf{J}_{s}(x)$. Then $\mathbf{J}$ is a map from
$X$ into $\mathrm{L}^{0}\\!\left(\$\mathrm{W}_{1}\\!\left(A\right)\right)$.

In the next lemma, we shall see that $\mathbf{J}$ is actually a map into the
space $\mathrm{L}\\!\left(\$\mathrm{W}_{1}\\!\left(A\right)\right)$.

######  Lemma 4.4.

For every $x\in X$, we have
$\mathbf{J}(x)\in\mathrm{L}\\!\left(\$\mathrm{W}_{1}\\!\left(A\right)\right)$.
Namely, for every $x\in X$ and for every $y\in A$, we have

$\mathrm{L}\\!\left[\$\mathrm{W}_{1}\\!\left[m\right]\right](\mathbf{J}(x),\mathbf{J}(y))\leq
4w(x,y)+1<\infty.$

In particular, if $w$ is bounded, then so is
$\mathrm{L}\\!\left[\$\mathrm{W}_{1}\\!\left[m\right]\right]$.

###### Proof.

For every $s\in\omega_{0}$, Lemmas 2.3 and 3.1 show that

$\displaystyle\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(x),\mathbf{J}_{s}(y))\leq\sum_{O\in\mathcal{O}_{s}}\varphi_{O}\cdot
m(\mathbf{a}_{O,s},y)+1\leq 4w(x,y)+1.$

Thus, we have

$\displaystyle\mathrm{L}\\!\left[\$\mathrm{W}_{1}\\!\left[d\right]\right](\mathbf{J}(x),\mathbf{J}(y))\leq\int_{\omega_{0}}4w(x,y)+1\
d\Psi=4w(x,y)+1<\infty.$

This completes the proof. ∎

By the help of Lemma 4.4, in what follows, we often represent

$\mathrm{L}\$\mathrm{W}_{1}\\!\left(A\right)=\mathrm{L}\\!\left(\$\mathrm{W}_{1}\\!\left(A\right)\right),$

and

$\mathrm{L}\$\mathrm{W}_{1}\\!\left[d\right]=\mathrm{L}\\!\left[\$\mathrm{W}_{1}\\!\left[d\right]\right],$

for $d\in\mathfrak{C}$.

######  Proposition 4.5.

For every $d\in\mathfrak{C}$, and for every pair $x,y\in A$, we have

$\mathrm{L}\$\mathrm{W}_{1}\\!\left[d\right](\mathbf{J}(x),\mathbf{J}(y))=d(x,y)$

###### Proof.

The proposition is deduced from the fact that these constructions are osmotic
(Theorem 2.15). ∎

Next, let us prove that $\mathbf{J}\colon
X\to\mathrm{L}\$\mathrm{W}_{1}\\!\left(A\right)$ is a topological embedding.

######  Lemma 4.6.

The map $\mathbf{J}\colon X\to\mathrm{L}\$\mathrm{W}_{1}\\!\left(A\right)$ is
continuous.

###### Proof.

Take an arbitrary point $p\in X$ and $\epsilon\in(0,\infty)$. Take a
sufficiently large number $N$ such that $2^{-N}\leq\alpha\epsilon$, where
$\alpha\in(0,\infty)$ satisfies that

($\alpha$1)
$\displaystyle\alpha\leq\min\left\\{\frac{1}{2}\cdot\frac{1}{20w(p,A)+1},\frac{1}{4}\right\\}.$

We divide the proof into two cases.

Case 1. [$p\in X\setminus A$]: Take $b\in A$ such that $w(p,b)\leq 2\cdot
w(p,A)$. We take a sufficiently small number $\eta\in(0,\infty)$ so that the
next statements are true:

  1. (a1)

We have $\eta\leq w(p,A)$,

  2. (a2)

For every $z\in U(p,\eta;w)$, we have $w(z,A)\leq w(p,A)+\eta$. As a result,
we observe that $w(b,z)\leq w(b,p)+w(p,z)\leq 2w(p,A)+\eta\leq 3w(p,A)$.

  3. (a3)

For every $z\in U(p,\eta;w)$, and for every $s\in\\{0,\dots,N\\}$, we have

$\mathbf{D}_{L^{\infty}}(\mathbf{h}(z,s),\mathbf{h}(p,s))\leq\frac{\epsilon}{4}.$

  4. (a4)

For every $z\in U(p,\eta;w)$, and for every $s\in\\{0,\dots,N\\}$, we have

$\sum_{O\in\mathcal{O}_{s}}\left|\varphi_{O,s}(z)-\varphi_{O,s}(p)\right|\leq\frac{1}{8w(p,A)+1}\cdot\frac{\epsilon}{4}.$

First, we provide an upper estimation of
$\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))$ for
every $s\in\\{0,\dots,N\\}$.

For every $s\in\\{0,\dots,N\\}$, Lemma 3.1 implies that

$m(\mathbf{a}_{O,s},b)\leq 4w(p,b)\leq 8\cdot w(p,A).$

Then, for every $s\in\\{0,\dots,N\\}$, Lemma 2.3 and (a4) show that

$\displaystyle\mathrm{W}_{1}\\!\left[w\right](\mathbf{H}(z,s),\mathbf{H}(p,s))\leq\sum_{O\in\mathcal{O}_{s}}|\varphi_{O,s}(z)-\varphi_{O,s}(p)|m(b,\mathbf{a}_{O,s})$
$\displaystyle\leq 8\cdot
w(p,A)\sum_{O\in\mathcal{O}_{s}}|\varphi_{O,s}(z)-\varphi_{O,s}(p)|$
$\displaystyle\leq\frac{1}{8w(p,A)+1}\cdot\frac{\epsilon}{4}\cdot
8w(p,A)\leq\frac{\epsilon}{4}.$

Next, we estimate
$\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))$. By
the argument discussed above, and by (a3), for every $s\in\\{0,\dots,N\\}$, we
have

$\displaystyle\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))$
$\displaystyle=\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(x,s),\mathbf{H}(p,s))+\mathbf{D}_{L^{\infty}}(\mathbf{h}(x,s),\mathbf{h}(p,s))\leq\frac{\epsilon}{4}+\frac{\epsilon}{4}=\frac{\epsilon}{2}.$

From now on, we estimate
$\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))$ for
$s\in\omega_{0}$ with $N\leq s$. For every $s\in\omega_{0}$ with $s\geq N$,
using Lemma 2.3 again, and using Lemma (a2), we have

$\displaystyle\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(z,s),\mathbf{H}(p,s))\leq\sum_{O\in\mathcal{O}_{s}}\left|\varphi_{O,s}(z)-\varphi_{O,s}(p)\right|w(\mathbf{a}_{O,s},b)$
$\displaystyle\leq\sum_{O\in\mathcal{O}_{s}}\left(\varphi_{O,s}(z)w(\mathbf{a}_{O,s},b)+\varphi_{O,s}(p)w(\mathbf{a}_{O,s},b)\right)=$
$\displaystyle\sum_{O\in\mathcal{O}_{s},\varphi_{O,s}(z)>0}\varphi_{O,s}(z)w(\mathbf{a}_{O,s},b)+\sum_{O\in\mathcal{O}_{s},\varphi_{O,s}(p)>0}\varphi_{O,s}(z)w(\mathbf{a}_{O,s},b)$
$\displaystyle\leq
4w(b,z)\cdot\sum_{O\in\mathcal{O}_{s},\varphi_{O,s}(z)>0}\varphi_{O,s}(z)+4w(b,p)\cdot\sum_{O\in\mathcal{O}_{s},\varphi_{O,s}(p)>0}\varphi_{O,s}(z)$
$\displaystyle\leq 4w(b,z)+4w(b,p)\leq 4\cdot 3w(p,A)+4\cdot 2w(p,A)\leq
20w(p,A).$

Since $\mathbf{D}_{L^{\infty}}(\mathbf{h}(z,s),\mathbf{h}(p,s))\leq 1$ is
always true, for every $s\in\omega_{0}$ with $N\leq s$, we also have

$\displaystyle\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))\leq\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(z,s),\mathbf{H}(p,s))+1$
$\displaystyle\leq 20w(x,A)+1.$

Therefore, combining the estimations obtained above, we see:

$\displaystyle\mathrm{L}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}(z),\mathbf{J}(p))=\int_{\omega_{0}}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))\
d\Psi(s)$
$\displaystyle=\sum_{s=0}^{\infty}\frac{1}{2^{s+1}}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))$
$\displaystyle\leq\sum_{s=N}^{\infty}\frac{\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))}{2^{s+1}}+\sum_{s=0}^{N}\frac{\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))}{2^{s+1}}$
$\displaystyle\leq\sum_{s=N}^{\infty}\frac{20w(x,A)+1}{2^{s+1}}+\sum_{s=0}^{N}\frac{2^{-1}\epsilon}{2^{s+1}}\leq\frac{2(20w(x,A)+1)}{2^{N+1}}+\frac{\epsilon}{2}$
$\displaystyle\leq(20w(x,A)+1)2^{-N}+\frac{\epsilon}{2}\leq\alpha(20w(x,A)+1)\epsilon+\frac{\epsilon}{2}$
$\displaystyle\leq\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon.$

This means that $J$ is continuous at $p$.

Case 2. [$p\in A$]: Take a sufficient small number $\eta\in(0,\infty)$ so that
the next statements are satisfied:

  1. (b1)

We have $\eta\leq 2$;

  2. (b2)

For every point $z\in U(p,\eta;w)$, we have $w(z,A)\leq 4^{-1}2^{-N}$;

  3. (b3)

For every point $z\in U(p,\eta;w)$, and for every $s\in\\{0,\dots,N\\}$, we
have

$\mathbf{D}_{L^{\infty}}(\mathbf{h}(z,s),\mathbf{h}(p,s))\leq 4^{-1}\epsilon;$

Under this conditions, we obtain:

  1. (c1)

For every point $z\in U(p,\eta;w)$, and for every $s\in\\{0,\dots,N\\}$, we
also have

$\displaystyle\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(z,s),\mathbf{H}(p,s))=\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(z,s),\boldsymbol{\delta}_{p})$
$\displaystyle=\sum_{O\in\mathcal{O}_{s}}\varphi_{O,s}\cdot
w(\mathbf{a}_{O,s},p)\leq 4w(z,A)\leq 2^{-N};$

  2. (c2)

For every point $z\in U(p,\eta;w)$, if $z\not\in A$, then for every
$s\in\mathbb{Z}_{\geq 0}$ we have

$\displaystyle\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))\leq\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(x,s),\mathbf{H}(p,s))+1$
$\displaystyle=\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(x,s),\boldsymbol{\delta}_{p})+1=\sum_{O\in\mathcal{O}_{s}}\varphi_{O,s}\cdot
w(\mathbf{a}_{O,i},p)+1$ $\displaystyle\leq 4w(z,A)+1\leq 2^{-N}+1\leq 2;$

  3. (c3)

For every point $z\in U(p,\eta;w)$, if $z\in A$, then we have
$w(z,p)=m(z,p)\leq\eta\leq 2$.

Then, due to (b3) and (c1), for every $s\in\\{0,\dots,N\\}$, we have

$\displaystyle\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))$
$\displaystyle=\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(z,s),\mathbf{H}(p,s))+\mathbf{D}_{L^{\infty}}(\mathbf{h}(z,s),\mathbf{h}(p,s))$
$\displaystyle<2^{-N}+4^{-1}\epsilon\leq(\alpha+4^{-1})\epsilon.$

According to (c2) and (c3), for $s\in\omega_{0}$ with $N\leq s$, we also have

$\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))\leq
2.$

Thus we obtain

$\displaystyle\mathrm{L}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}(z),\mathbf{J}(p))=\int_{\omega_{0}}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))\
d\Psi(s)$
$\displaystyle=\sum_{s=0}^{\infty}2^{-(s+1)}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{s}(z),\mathbf{J}_{s}(p))\leq\sum_{s=N}^{\infty}\frac{2}{2^{s+1}}+\sum_{n=0}^{N}\frac{(\alpha+4^{-1})\epsilon}{2^{s+1}}$
$\displaystyle\leq 2\cdot 2^{-N}+(\alpha+4^{-1})\epsilon\leq
2\alpha\epsilon+(\alpha+4^{-1})\epsilon\leq\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon.$

This means that $\mathbf{J}$ is continuous at $p\in A$.

In any case, we confirm that $\mathbf{J}$ is continuous. Hence this completes
the proof. ∎

######  Proposition 4.7.

The map $\mathbf{J}\colon X\to\mathrm{L}\$\mathrm{W}_{1}\\!\left(A\right)$ is
homeomorphic.

###### Proof.

Take an arbitrary point $p\in X$, and an arbitrary number
$\epsilon\in(0,\infty)$. Now we will find a sufficiently small
$\eta\in(0,\infty)$ so that if we have
$\mathrm{L}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}(p),\mathbf{J}(z))<\eta$,
then $w(z,p)\leq\epsilon$. Put $V_{\eta}=\\{\,z\in
X\mid\mathrm{L}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}(p),\mathbf{J}(z))<\eta\,\\}$.

Case 1. [$p\in X\setminus A$]: Take a sufficiently large $N\in\omega_{0}$ so
that $2^{-N+1}<w(p,A)$ and $16\cdot 4^{-N-1}w(p,A)\leq\epsilon$. We also take
a sufficiently small number $\eta$ so that $2^{N}\eta<1$ and every $z\in
V_{\eta}$ satisfies $\xi_{N}(z)=\xi_{N}(p)=1$ and $2^{-N+1}<w(z,A)$. Then, by
the definition of the metric $\mathrm{L}\$\mathrm{W}_{1}\\!\left[m\right]$,
for every $z\in V_{\eta}$, we have

$\mathbf{D}_{L^{\infty}}(\mathbf{h}(z,N),\mathbf{h}(p,N))\leq 2^{N}\eta<1.$

Thus, Statement (3) in Lemma 4.2 implies that

$w(z,p)\leq 16\cdot 4^{-N-1}w(p,A)\leq\epsilon.$

This is the inequality that we want to prove.

Case 2. [$p\in A$]: Take a sufficiently large number $N\in\mathbb{Z}_{\geq 0}$
so that $71\cdot 2^{-N}\leq 2^{-1}\epsilon$. We also take a sufficiently small
number $\eta\in(0,\infty)$ so that $2^{N}\eta<1$ and $\eta<2^{-1}\epsilon$. If
$z\in V_{\eta}$ belongs to $A$, then
$\mathrm{L}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}(p),\mathbf{J}(z))<\eta$
implies that $w(z,p)\leq\eta$ since we already know
$\mathrm{L}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}(x),\mathbf{J}(y))=m(x,y)$
whenever $x,y\in A$ (see Proposition 4.5). Thus, we may assume that $z\in
X\setminus A$. In this setting, by the definition of the metric
$\mathrm{L}\$\mathrm{W}_{1}\\!\left[m\right]$, the point $z\in V_{\eta}$
satisfies

$\mathbf{D}_{L^{\infty}}(\mathbf{h}(z,N),\mathbf{h}(p,N))=\mathbf{D}_{L^{\infty}}(\mathbf{h}(z,N),0)\leq
2^{N}\eta<1.$

Hence Statement (2) in Lemma 4.2 implies that $w(z,A)\leq 2^{-N+1}$. Put
$\mathbf{Ind}=\\{\,O\in\mathcal{O}_{N}\mid\varphi_{O,N}(z)>0\,\\}$. We obtain

$\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(p,N),\mathbf{H}(z,N))=\sum_{O\in\mathbf{Ind}}\varphi_{O,N}\cdot
w(\mathbf{a}_{O,N},p)$

Take $P\in\mathbf{Ind}$ such that
$w(\mathbf{a}_{P,N},p)=\min_{O\in\mathbf{Ind}}w(\mathbf{a}_{O,N},p)$. Thus we
have

$\displaystyle\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(p,N),\mathbf{H}(z,N))$
$\displaystyle=\sum_{O\in\mathbf{Ind}}\varphi_{O,N}\cdot
w(\mathbf{a}_{O,N},p)$ $\displaystyle\geq
w(\mathbf{a}_{P,N},p)\cdot\sum_{O\in\mathbf{Ind}}\varphi_{O,N}=w(\mathbf{a}_{P,N},p),$

and then this estimation indicates

$w(\mathbf{a}_{P,N},p)\leq\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{N}(p),\mathbf{J}_{N}(z))\leq\mathrm{L}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}_{N}(p),\mathbf{J}_{N}(z)).$

Thus $w(\mathbf{a}_{P,N},p)\leq\eta$. Since $\varphi_{O,N}(z)>0$ for all
$O\in\mathbf{Ind}$, using the condition (WD6), we also have

$w(z,\mathbf{y}_{P,N})\leq 16\cdot 4^{-N-1}w(z,A)\leq 16w(z,A)\leq 32\cdot
2^{-N}.$

The condition (WD3) also implies that

$\displaystyle
w(\mathbf{y}_{P,N},\mathbf{a}_{P,N})\leq(1+4^{-N-1})w(\mathbf{y}_{P,N},A)$
$\displaystyle\leq(1+4^{-N-1})(w(\mathbf{y}_{P,N},z)+w(z,A))$
$\displaystyle\leq(1+4^{-N-1})(1+8\cdot 4^{-N-1})w(z,A)\leq 2\cdot 9w(z,A)$
$\displaystyle\leq 18\cdot w(z,A)\leq 39\cdot 2^{-N}.$

As a result, we see the following estimation:

$\displaystyle w(p,z)$ $\displaystyle\leq
w(p,\mathbf{a}_{P,N})+w(\mathbf{a}_{P,N},\mathbf{y}_{P,N})+w(\mathbf{y}_{P,N},z)$
$\displaystyle\leq\eta+39\cdot 2^{-N}+32\cdot 2^{-N}=\eta+71\cdot 2^{-N}\leq
2^{-1}\epsilon+2^{-1}\epsilon=\epsilon.$

In any case, we have $w(p,z)\leq\epsilon$. Since $p$ and $\epsilon$ are
arbitrary, we conclude that $\mathbf{J}$ is homeomorphic. This finishes the
proof. ∎

We next consider the topologies on $\mathbf{J}(X)$ induced by
$\mathrm{L}\$\mathrm{W}_{1}\\!\left[d\right]$.

######  Lemma 4.8.

The following statements are true:

  1. (1)

For every $d\in\mathfrak{C}$ and for every pair $x,y\in X$, we have

$\mathrm{L}\$\mathrm{W}_{1}\\!\left[d\right](\mathbf{J}(x),\mathbf{J}(y))<\infty.$

  2. (2)

For every $d\in\mathfrak{C}$, the metric
$\mathrm{L}\$\mathrm{W}_{1}\\!\left[d\right]$ generate the same topology on
$\mathbf{J}(X)$ as $w$.

###### Proof.

Statement (1) follows from Lemma 4.4 and Statements (2) in Proposition 2.12.
Statement (2) is deduced from Statement (3) in Proposition 2.1, Proposition
2.10, and Proposition 2.14. ∎

Recall that if $X$ is completely metrizable, then we choose $m$ and $w$ as
complete metrics on $A$ and $X$, respectively.

######  Proposition 4.9.

If $X$ is completely metrizable, then for every complete metric $d$ in
$\mathfrak{C}$, the metric subspace
$(\mathbf{J}(X),\mathrm{L}\$\mathrm{W}_{1}\\!\left[d\right])$ of
$\mathrm{L}\$\mathrm{W}_{1}\\!\left(A\right)$ is complete. In particular, the
map $\mathbf{J}\colon X\to\mathrm{L}\$\mathrm{W}_{1}\\!\left(A\right)$ is a
closed map.

###### Proof.

Assume that a sequence $\\{\mathbf{J}(x_{i})\\}_{i\in\mathbb{Z}_{\geq 0}}$ is
Cauchy with respect to $\mathrm{L}\$\mathrm{W}_{1}\\!\left[d\right]$. We will
show that $\\{x_{i}\\}_{i\in\mathbb{Z}_{\geq 0}}$ has a subsequence possessing
a limit $p\in X$ with respect to the topology of $X$. If
$\\{x_{i}\\}_{i\in\mathbb{Z}_{\geq 0}}$ has a subsequence contained in $A$,
then we can find $p\in X$ such that $x_{i}\to p$ as $i\to\infty$ since $(A,m)$
is complete. Thus, we may assume that $x_{i}\in X\setminus A$ for every
$i\in\mathbb{Z}_{\geq 0}$ by taking a subsequence if necessary. We divide the
proof into two cases.

Case 1. [$\inf_{i\in\mathbb{Z}_{\geq 0}}w(x_{i},A)>0$]: In this case, we can
find $n\in\mathbb{Z}_{\geq 0}$ such that
$2^{-n+1}\leq\inf_{i\in\mathbb{Z}_{\geq 0}}w(x_{i},A)$. Then there exists
$K\in\mathbb{Z}_{\geq 0}$ such that for every $i\in\mathbb{Z}_{\geq 0}$ with
$K\leq i$, we have
$\mathbf{D}_{L^{\infty}}(\mathbf{h}(x_{i},n),\mathbf{h}(x_{K},n))<1$. Then
Statement (3) in Lemma 4.2 shows that $w(x_{i},x_{K})\leq 16\cdot
4^{-n-1}w(x_{K},A)$. In this case, we have

$w(x_{i},A)\leq w(x_{K},A)+w(x_{i},x_{K})\leq(1+16\cdot 4^{-n-1})w(x_{K},A).$

Thus, we obtain $\sup_{i\in\mathbb{Z}_{\geq 0}}w(x_{i},A)<\infty$. Put
$C=\sup_{i\in\mathbb{Z}_{\geq 0}}w(x_{i},A)$. Then for every
$\epsilon\in(0,\infty)$, we can find $N\in\omega_{0}$ such that $16\cdot
4^{-N-1}C\leq\epsilon$. In this setting, for sufficiently large numbers $i,j$,
we have $\xi_{N}(x_{i})=\xi_{N}(x_{j})=1$ and
$\mathbf{D}_{L^{\infty}}(\mathbf{h}(x_{i},N),\mathbf{h}(x_{j},N))<1$. Thus
Statement (3) in Lemma 4.2 shows that

$w(x_{i},x_{j})\leq 16\cdot 4^{-N-1}w(x_{i},A)\leq 16\cdot
4^{-N-1}C\leq\epsilon.$

Therefore the sequence $\\{x_{i}\\}_{i\in\mathbb{Z}_{\geq 0}}$ is Cauchy in
$(X,w)$. Hence, using the completeness of $w$, we can find a limit $p\in X$ of
$\\{x_{i}\\}_{i\in\mathbb{Z}_{\geq 0}}$.

Case 2. [$\inf_{i\in\mathbb{Z}_{\geq 0}}w(x_{i},A)=0$]: We may assume that
$w(x_{i},A)\to 0$ as $i\to\infty$ by taking a subsequence if necessary. Since

$\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(x_{i},0),\mathbf{H}(x_{i},0))\leq\mathrm{L}\$\mathrm{W}_{1}\\!\left[m\right](\mathbf{J}(x_{i}),\mathbf{J}(x_{j})),$

the sequence $\\{\mathbf{H}(x_{i},0)\\}_{i\in\mathbb{Z}_{\geq 0}}$ is Cauchy
in $(\mathrm{W}_{1}\\!\left(A\right),\mathrm{W}_{1}\\!\left[d\right])$. Thus
we can find $\theta\in\mathrm{W}_{1}\\!\left(A\right)$ such that
$\mathbf{H}(x_{i},0)\to\theta$ as $i\to\infty$ since the space
$(\mathrm{W}_{1}\\!\left(A\right),\mathrm{W}_{1}\\!\left[d\right])$ is
complete (see (2) in Proposition 2.11). This implies that the sequence
$\\{\mathbf{H}(x_{i},0)\\}_{i\in\mathbb{Z}_{\geq 0}}$ is also Cauchy in
$(\mathrm{W}_{1}\\!\left(A\right),\mathrm{W}_{1}\\!\left[m\right])$. For each
$i\in\mathbb{Z}_{\geq 0}$, take $b_{i}\in A$ such that $w(x_{i},b_{i})\leq
2w(x_{i},A)$. Then Lemma 2.4 shows that

$\mathrm{W}_{1}\\!\left[m\right](\boldsymbol{\delta}_{b_{i}},\mathbf{H}(x_{i},0))\leq
w(x_{i},b_{i})\leq 2w(x_{i},A).$

Thus, for sufficiently large numbers $i,j\in\mathbb{Z}_{\geq 0}$, we have

$\displaystyle
w(b_{i},b_{j})=\mathrm{W}_{1}\\!\left[m\right](\boldsymbol{\delta}_{b_{i}},\boldsymbol{\delta}_{b_{j}})\leq$
$\displaystyle\mathrm{W}_{1}\\!\left[m\right](\boldsymbol{\delta}_{b_{i}},\mathbf{H}(x_{i},0))+\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(x_{i},0),\mathbf{H}(x_{j},0))+$
$\displaystyle\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(x_{j},0),\boldsymbol{\delta}_{b_{j}})$
$\displaystyle\leq
2w(x_{i},A)+\mathrm{W}_{1}\\!\left[m\right](\mathbf{H}(x_{i},0),\mathbf{H}(x_{j},0))+2w(x_{j},A)\to
0,$

as $i,j\to\infty$. This means that the sequence
$\\{b_{i}\\}_{i\in\mathbb{Z}_{\geq 0}}$ is Cauchy in $(A,m)$. Hence there
exists $p\in A$ such that $b_{i}\to p$ as $i\to\infty$. Since
$w(b_{i},x_{i})\leq 2w(x_{i},A)\to 0$ as $i\to\infty$, we also see that
$x_{i}\to p$ as $i\to\infty$.

In any case, we can find a subsequence of $\\{x_{i}\\}_{i\in\mathbb{Z}_{\geq
0}}$ possessig a limit $p\in X$. Due to the continuity of $\mathbf{J}$ (Lemma
4.6), the Cauchy sequence $\\{\mathbf{J}(x_{i})\\}_{i\in\mathbb{Z}_{\geq 0}}$
has a limit $\mathbf{J}(p)$. This finishes the proof. ∎

######  Remark 4.1.

Some readers may feel that the proof of Proposition 4.9 is strange because it
does not use the essential information of
$\mathrm{L}\$\mathrm{W}_{1}\\!\left[d\right]$, or because the whole space
$\mathrm{L}\$\mathrm{W}_{1}\\!\left(A\right)$ is not complete since neither is
$S$. However, the embedding $\mathbf{J}$ is constructed using $w$ and the
sequence of $(k,w)$-WD collections, which contains all information of the
topology of $X\setminus A$. Indeed, the family
$\bigcup_{s\in\omega_{0}}\mathcal{O}_{s}$ becomes an open basis on $X\setminus
A$. Therefore, we can draw out the information of the topology and the
completeness from the embedding $\mathbf{J}$.

To construct $E$ stated in Theorem 1.1, it should be noted that we can
represent
$\mathrm{Met}(A)=\coprod_{\mathfrak{C}\in\mathbf{M}_{<\infty}(X)}\mathfrak{C}$.
For each $\mathbf{M}_{<\infty}(X)$, choose $m_{\mathfrak{C}}\in\mathfrak{C}$
and $w_{\mathfrak{C}}\in\mathrm{Met}(X)$ such that
$w_{\mathfrak{C}}|_{A^{2}}=m_{\mathfrak{C}}$. Using these $m_{\mathfrak{C}}$
and $w_{\mathfrak{C}}$, and using the above discussion, we obtain an isometric
operator $\mathbf{J}^{\mathfrak{C}}\colon\mathfrak{C}\to\mathrm{Met}(X)$. Then
we define a map $E\colon\mathrm{Met}(A)\to\mathrm{Met}(X)$ by

$E(d)(x,y)=\mathrm{L}\$\mathrm{W}_{1}\\!\left[d\right]\left(\mathbf{J}^{\mathfrak{C}(d)}(x),\mathbf{J}^{\mathfrak{C}(d)}(y)\right).$

Proposition 4.7 shows that $E(d)$ actually belongs to $\mathrm{Met}(X)$ for
all $d\in\mathfrak{C}$.

Now, let us complete the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

First, Proposition 4.5 and Proposition 4.9 proves Statements (1) and (4),
respectively. Statement (3) follows from Lemma 4.4. Now we shall show
Statement (2). Take $d,e\in\mathrm{Met}(A)$. If
$\mathfrak{C}(d)\neq\mathfrak{C}(d)$, then
$\operatorname{\mathcal{D}}_{A}(d,e)=\infty$. Hence we have
$\operatorname{\mathcal{D}}_{X}(E(d),E(e))=\infty=\operatorname{\mathcal{D}}_{A}(d,e)$.
Thus, we only need to show that for a fixed class
$\mathfrak{C}\in\mathbf{M}_{<\infty}(A)$, and for every pair
$d,e\in\mathfrak{C}$, we have

$\operatorname{\mathcal{D}}_{A}(d,e)=\operatorname{\mathcal{D}}_{X}(E(d),E(e)).$

Theorem 2.15 indicates that

$\displaystyle\operatorname{\mathcal{D}}_{A}(d,e)\leq\operatorname{\mathcal{D}}_{X}(E(d),E(e))=\operatorname{\mathcal{D}}_{\mathbf{J}(X)}(\mathrm{L}\$\mathrm{W}_{1}\\!\left[d\right],\mathrm{L}\$\mathrm{W}_{1}\\!\left[e\right])$
$\displaystyle\leq\operatorname{\mathcal{D}}_{\mathrm{L}\$\mathrm{W}_{1}\\!\left(A\right)}(\mathrm{L}\$\mathrm{W}_{1}\\!\left[d\right],\mathrm{L}\$\mathrm{W}_{1}\\!\left[e\right])$
$\displaystyle=\operatorname{\mathcal{D}}_{\$\mathrm{W}_{1}\\!\left(A\right)}(\$\mathrm{W}_{1}\\!\left[d\right],\$\mathrm{W}_{1}\\!\left[e\right])=\operatorname{\mathcal{D}}_{\$\mathrm{W}_{1}\\!\left(A\right)}(\$\mathrm{W}_{1}\\!\left[d\right],\$\mathrm{W}_{1}\\!\left[e\right])$
$\displaystyle=\operatorname{\mathcal{D}}_{\mathrm{W}_{1}\\!\left(A\right)}(\mathrm{W}_{1}\\!\left[d\right],\mathrm{W}_{1}\\!\left[e\right])=\operatorname{\mathcal{D}}_{A}(d,e).$

Therefore we conclude that
$\operatorname{\mathcal{D}}_{A}(d,e)=\operatorname{\mathcal{D}}_{X}(E(d),E(e))$.
This verifies Statement (2).

Next we consider the “furthermore” part of the theorem. Since
$\mathrm{Met}(X)$ is dense in $\mathrm{CPM}(X)$ ([24, Theorem 1.3]), the
isometric embedding $E\colon\mathrm{Met}(A)\to\mathrm{Met}(X)$ can be
(uniquely) extended to an isometric embedding
$F\colon\mathrm{CPM}(A)\to\mathrm{CPM}(X)$ using Cauchy sequences (see also
[8, Theorem 2] and [9]). This finishes the whole of the proof of Theorem 1.1.
∎

##  5\. Questions

######  Question 5.1.

Let $X$ be a metrizable space, and $A$ be a closed subset of $X$. Can we
obtain a linear isometric operator extending metrics. Namely, does there
exists an extensor $G\colon\mathrm{Met}(A)\to\mathrm{Met}(X)$ satisfying the
conclusions in Theorem 1.1 and the following additional condition?

  * •

For every pair $d,e\in\mathrm{Met}(A)$, and $s,t\in(0,\infty)$, we have

$G(s\cdot d+t\cdot e)=s\cdot G(d)+t\cdot G(e).$

The author thinks that this questions is true if we can prove that every pair
$d,e\in\mathrm{Met}(A)$ satisfies that
$\mathrm{W}_{1}\\!\left[d+e\right]=\mathrm{W}_{1}\\!\left[d\right]+\mathrm{W}_{1}\\!\left[e\right]$;
however, it seems to be quite difficult. Thus, we need another osmotic
construction preserving linear combinations of metrics instead of
$\mathrm{W}_{1}$.

######  Question 5.2.

Let $X$ be a metrizable space, and $A$ be a closed subset of $X$. Assume that
$A$ and $X$ have the same topological dimensions. For each
$d\in\mathrm{Met}(A)$, does there exist a metric $D\in\mathrm{Met}(X)$ such
that the fractal dimensions of $d$ and $D$ are identical? Moreover, can we
obtain an isometric extensor of metrics preserving fractal dimensions?

A metric on a set $Z$ is said to be _proper_ if all bounded set in $(Z,d)$ is
compact. For a metrizable space $X$, we denote by $\mathrm{PrMet}(X)$ the set
of all $d\in\mathrm{Met}(X)$ that is proper.

######  Question 5.3.

Let $X$ be a second-countable locally compact Hausdorff space, and $A$ be a
closed subset of $X$. Does there exist an extensor
$F\colon\mathrm{Met}(A)\to\mathrm{Met}(X)$ satisfying the conclusions of
Theorem 1.1 and the additional condition that
$F(\mathrm{PrMet}(A))\subseteq\mathrm{PrMet}(X)$?

######  Question 5.4.

Can we investigate the topology and comeager subsets of the space of proper
metrics equipped with the supremum metric?

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