# Ultralimits of Wasserstein spaces and metric measure spaces with Ricci
curvature bounded from below

Andrew Warren  Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, PA 15213
USA; and,  Institut des Hautes Études Scientifiques
35 Route de Chartres
91440 Bures-sur-Yvette
France<EMAIL_ADDRESS>
###### ?abstractname? .

We investigate the stability of the Wasserstein distance, a metric structure
on the space of probability measures arising from the theory of optimal
transport, under metric ultralimits. We first show that if
$(X_{i},d_{i})_{i\in\mathbb{N}}$ is a sequence of metric spaces with metric
ultralimit $(\hat{X},\hat{d})$, then the $p$-Wasserstein space
$(\mathcal{P}_{p}(\hat{X}),W_{p})$ embeds isometrically in a canonical fashion
into the metric ultralimit of the sequence of $p$-Wasserstein spaces
$(\mathcal{P}_{p}(X_{i}),W_{p})$. Second, using a notion of ultralimit of
metric measure spaces modeled on the one introduced by Elek, we use the
machinery of ultralimits of Wasserstein spaces to prove that an ultralimit of
$CD(K,\infty)$ spaces is a $CD(K,\infty)$ space. This provides a new proof
that the $CD(K,\infty)$ property is stable under pointed measured Gromov
convergence. Along the way, we establish some basic results on how the Loeb
measure construction interacts with Wasserstein distances as well as integral
functionals, which may be of independent interest.

The plan of the paper is as follows. In Section 1, we review the necessary
background from optimal transport, as well as the theory of metric ultralimits
and related notions such as Loeb measures. We also present the notion of
ultralimit of metric measure spaces we will use in Section 3. In Section 2, we
show in Theorem 11 that the $p$-Wasserstein space
$(\mathcal{P}_{p}(\hat{X}),W_{p})$ embeds isometrically in a canonical fashion
into the metric ultralimit of the sequence of $p$-Wasserstein spaces
$(\mathcal{P}_{p}(X_{i}),W_{p})$, and explore some basic features of said
metric ultralimit. We also study the interaction between the Loeb measure
construction and the ultralimit of $p$-Wasserstein spaces construction (see
especially Theorem 17, Corollary 24, and Lemma 27). Finally, in Section 3 we
apply the machinery developed in Section 2, and give a proof that a metric
measure ultralimit of $CD(K,\infty)$ spaces is again a $CD(K,\infty)$ space
(Theorem 28).

##  1\. Introduction

###  1.1. Facts from optimal transport

Let $(X,d)$ be a complete, separable metric space. Given $p\in[1,\infty)$, the
$p$-Wasserstein metric on the space $\mathcal{P}_{p}(X)$ of Borel probability
measures with finite $p$th moments is defined as follows:

$W_{p}(\mu,\nu):=\inf_{\gamma\in\Pi(\mu,\nu)}\left(\int_{X\times
X}d(x,y)^{p}d\gamma(x,y)\right)^{1/p}.$

Here $\Pi(\mu,\nu)$ denotes the set of all couplings of the measures $\mu$ and
$\nu$.

The $p$-Wasserstein metrics possess numerous intriguing geometric features.
For instance:

  * •

When $(X,d)$ is compact, $W_{p}$ metrizes the weak convergence of probability
measures; when $(X,d)$ is not necessarily compact, convergence in $W_{p}$ is
equivalent to weak convergence together with convergence of $p$th moments. In
particular, $(\mathcal{P}_{p}(X),W_{p})$ enjoys a version of the Glivenko-
Cantelli theorem: if $\mu_{n}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$ is an
_empirical measure for $\mu$_, viz., the points $x_{i}$ are i.i.d. samples
from $\mu$, then $W_{p}(\mu_{n},\mu)\rightarrow 0$ with probability 1. In
particular, discrete measures are dense in $W_{p}$.

  * •

Let $(X,d)$ be a _geodesic metric space_ , that is, given any $y,z\in X$ there
exists a continuous function $f:[0,1]\rightarrow(X,d)$ with $f(0)=y$,
$f(1)=z$, and $d(f(t),f(s))=|t-s|d(y,z)$ for all $t,s\in[0,1]$ (and in this
case we say $f(t)$ is a constant speed $d$-geodesic connecting $y$ and $z$).
Then, $(\mathcal{P}_{p}(X),W_{p})$ is also a geodesic metric space [4, Theorem
2.10].

  * •

In the case $p=2$, $(\mathcal{P}_{2}(X),W_{2})$ comes equipped with a _formal
Riemannian metric structure_ [37, 36], so that numerous objects and
computations from (finite-dimensional) Riemannian geometry have analogues in
$(\mathcal{P}_{2}(X),W_{2})$.

  * •

In particular, there is a well-developed theory of _gradient flows_ in
$(\mathcal{P}_{2}(X),W_{2})$ [6], whereby numerous parabolic PDEs (such as the
Fokker-Planck equation [27], the porous medium equation [36], etc.) can be
recast as infinite-dimensional ODEs of the form $\dot{\mu}_{t}=-\nabla
F(\mu_{t})$, where $F$ is some functional on $\mathcal{P}_{2}(X)$.

  * •

In the case where $(X,d)$ is itself a connected Riemannian manifold, there is
a deep connection between geometric properties of the manifold $(X,d)$ and
those of the “manifold” $(\mathcal{P}_{2}(X),W_{2})$. For instance, it is
known [50, Theorem 1] that $(X,d)$ has a uniform Ricci curvature lower bound
of $K\in\mathbb{R}$ iff the _relative entropy_ functional
$H(\mu\mid\text{vol})=\int_{X}\frac{d\mu}{d\text{vol}}\log\frac{d\mu}{d\text{vol}}d\text{vol}$
(where vol denotes the Riemannian volume measure on $X$) is $K$-geodesically
convex on the space $(\mathcal{P}_{2}(X),W_{2})$, that is, for any curve
$\mu_{t}:[0,1]\rightarrow\mathcal{P}_{2}(X)$ which is a constant speed
$W_{2}$-geodesic connecting $\mu_{0}$ and $\mu_{1}$,

$H(\mu_{t}\mid\text{vol})\leq(1-t)H(\mu_{0}\mid\text{vol})+tH(\mu_{1}\mid\text{vol})-K\frac{t(1-t)}{2}W_{2}^{2}(\mu_{0},\mu_{1}).$

  * •

Moreover, it is sometimes possible to turn around and use the consequences of
geometric properties of a manifold $(X,d)$, in the space
$(\mathcal{P}_{2}(X),W_{2})$, as _surrogates_ for geometric properties on
$(X,d)$; doing so offers an avenue to _synthetically_ extend geometric notions
from manifolds to more general nonsmooth spaces, as long as the same geometric
reasoning goes through in the space $(\mathcal{P}_{2}(X),W_{2})$.

Towards this last point, such a strategy has been pursued successfully in the
setting of the Lott-Sturm-Villani theory of synthetic Ricci curvature [33, 44,
45] (see also the more recent review article by Ambrosio [3]), which we now
very briefly discuss.

Let $(X,d)$ be a complete separable metric measure space, and let $\lambda$ be
a $\sigma$-finite Borel measure on $(X,d)$ which is finite on bounded sets. We
say that $(X,d,\lambda)$ is a “strong $CD(K,\infty)$” space provided that the
relative entropy functional
$H(\cdot\mid\lambda):\mathcal{P}_{2}(X)\rightarrow\mathbb{R}\cup\\{\infty\\}$
is $K$-geodesically convex on the space $(\mathcal{P}_{2}(X),W_{2})$.
Likewise, we say that $(X,d,\lambda)$ is a “$CD(K,\infty)$” space provided
that for every $\mu_{0}$ and $\mu_{1}$ in $\mathcal{P}_{2}(X)$, there exists a
constant speed $W_{2}$-geodesic $(\mu_{t})_{t\in[0,1]}$ connecting $\mu_{0}$
and $\mu_{1}$, such that for all $t\in[0,1]$,

$H(\mu_{t}\mid\lambda)\leq(1-t)H(\mu_{0}\mid\lambda)+tH(\mu_{1}\mid\lambda)-K\frac{t(1-t)}{2}W_{2}^{2}(\mu_{0},\mu_{1}).$

Evidently, every strong $CD(K,\infty)$ space is also $CD(K,\infty)$, and every
$CD(K,\infty)$ space for which there exists only one constant speed
$W_{2}$-geodesic connecting any two measures in $\mathcal{P}_{2}(X)$ which are
absolutely continuous with respect to the reference measure $\lambda$, is
(trivially) a strong $CD(K,\infty)$. Moreover, it was proved in [41] that
every strong $CD(K,\infty)$ actually has this latter “unique constant speed
$W_{2}$-geodesic between measures in $\mathcal{P}_{2,ac(\lambda)}(X)$”
property. The $CD(K,\infty)$ property has numerous consequences, many of which
are surveyed in [49], but we give one important example. In the case where
$K>0$, a metric measure space which is $CD(K,\infty)$ satisfies the following
_$K$ -log-Sobolev inequality_:

$\forall\mu\in\mathcal{P}_{2}(X)\quad
H(\mu\mid\lambda)\leq\frac{1}{2K}\int_{\frac{d\mu}{d\lambda}>0}\frac{\left|\nabla^{-}\left(\frac{d\mu}{d\lambda}\right)\right|^{2}}{\frac{d\mu}{d\lambda}}d\lambda$

where $|\nabla^{-}f|(x):=\limsup_{y\rightarrow
x}\frac{\max\\{f(x)-f(y),0\\}}{d(x,y)}$. (See [49, Theorem 30.22] for a
proof.) A central question in the theory of synthetic Ricci curvature is that
of _stability_ : that is, given a “reasonable” notion of convergence of metric
measure spaces (and note that even devising a notion of convergence of metric
measure spaces is a nontrivial issue), one would like to know whether
properties like $CD(K,\infty)$ are preserved along convergent sequences.

In this article, we require a small relaxation of the “basic assumptions”
underlying the standard setup of the Wasserstein distance. Indeed, at the
beginning of this subsection, we defined the $p$-Wasserstein distances in the
setting where the underlying metric space is Polish (i.e. complete and
separable). In what follows, it will be beneficial to also be able to define
the $p$-Wasserstein distance atop a metric space which is merely complete, but
not necessarily separable. That separability can be dispensed with is non-
obvious, since pedagogical accounts of the $p$-Wasserstein distances (such as
those in [4, 43, 48]) present many arguments which seem to invoke separability
in an essential way.

However, it was observed in [34] that there is a relatively straightforward
way to adapt the $p$-Wasserstein distance to arbitrary complete metric spaces.
Instead of taking the space $\mathcal{P}_{p}(X)$ to be the space of all
_Borel_ probability measures on $X$ with finite $p$th moment, one instead
declares $\mathcal{P}_{p}(X)$ to be the space of _Radon_ probability measures
on $X$ with finite $p$th moment. (In the case where $X$ is separable, every
Borel probability measure is also Radon [11, Theorem 7.1.7], so we are
justified in using the same notation.111What’s more, even when $X$ is _not_
separable, the “restriction” to Radon probability measures is in some sense a
restriction in the mildest way possible, since the existence of a complete
metric space with a Borel probability measure that is not Radon requires the
existence of a real-valued measurable cardinal (this follows from [22, Theorem
438H]), and is therefore independent of ZFC. We thank Taras Banakh for
bringing this set-theoretic issue to our attention.) Crucially, Radon
probability measures automatically have separable support, and that this is
enough to recover a number of basic facts about $(\mathcal{P}_{p}(X),W_{p})$
familiar from the separable setting. In particular, it is shown in [34] that
_discrete_ probability measures are $W_{p}$-dense even when $X$ is merely a
complete metric space, which is a fact we will invoke repeatedly. Note,
however, that the fact that the general theory around $p$-Wasserstein
distances also “works” when considering Radon measures on complete metric
spaces had been alluded to earlier in the optimal transport literature, for
example [30, Remark 2.8].

###  1.2. Ultraproduct preliminaries

We briefly outline the theory of metric ultralimits. We do so in the language
of nonstandard analysis222Here we use the term “nonstandard analysis”
essentially to refer to a collection of specialized terminology and theorems
surrounding objects which are “constructed” using ultrafilters, ultraproducts,
et cetera. This is in contrast to, for instance, nonstandard analysis in the
style of Nelson’s _Internal Set Theory_ [35] and related approaches [28],
which conservatively extend ZFC itself.; it is possible to define metric
ultralimits without explicit recourse to e.g. the hyperreals, but in the
opinion of the author, doing so comes at a cost of conceptual clarity and
technical facility. The reader may consult [24] for general background. As
well, [16, Chapter 10] gives a concise summary of nonstandard analysis notions
tailored for metric geometry/geometric group theory applications, and — aside
from its omission of Loeb measures, which we address below — covers
essentially all of the nonstandard analysis we will use in this article. We
also adopt some of their notation.

Let $\omega$ denote a non-principal ultrafilter on $\mathbb{N}$. The
hyperreals $\mathbb{R}^{\omega}$ are the non-archimedian field defined by
taking an ultrapower of $\mathbb{R}$: namely, given two sequences of reals
$(r_{i})$ and $(r_{i}^{\prime})$, we identify $(r_{i})$ and $(r_{i}^{\prime})$
under the equivalence $\sim_{\omega}$ if and only if
$\\{i\in\mathbb{N}:r_{i}=r_{i}^{\prime}\\}\in\omega$; then,
$\mathbb{R}^{\omega}:=\mathbb{R}^{\mathbb{N}}/\sim_{\omega}$. We write
$[(r_{i})]\in\mathbb{R}^{\omega}$ for such an equivalence class, (or sometimes
$[(r_{i})_{i\in\mathbb{N}}]$ if we wish to emphasize that it is the index $i$
which is addressed by the ultrafilter $\omega$). Formally this is analogous to
the Cauchy sequence construction of $\mathbb{R}$ from $\mathbb{Q}$, except
here we are using a much finer equivalence relation. By making the injection

$\mathbb{R}\ni r\mapsto[(r)]\in\mathbb{R}^{\omega}$

(where $(r)$ is the constant sequence with each term as $r$), we see that
$\mathbb{R}$ embeds canonically into $\mathbb{R}^{\omega}$; we therefore say,
abusively, that $\mathbb{R}\subset\mathbb{R}^{\omega}$. $\mathbb{R}^{\omega}$
inherits the algebraic structure of $\mathbb{R}$ via the quotient with respect
to $\omega$; for instance, we overload $+$ by making the definition
$[(r_{i})]+[(r_{i}^{\prime})]:=[(r_{i}+r_{i}^{\prime})]$. (Of course one must
check this is well-defined.) We use the $\approx$ symbol (read as
“approximately equal”, or “infinitesimally close”) as follows:

$\displaystyle[(r_{i})]\approx[(r_{i}^{\prime})]$
$\displaystyle\iff(\forall\varepsilon\in\mathbb{R}_{+})(|[(r_{i})]-[(r_{i}^{\prime})]|<\varepsilon)$
$\displaystyle\iff(\forall\varepsilon\in\mathbb{R}_{+})\\{i\in\mathbb{N}\mid|r_{i}-r_{i}^{\prime}|<\varepsilon\\}\in\omega.$

For example, $[(10^{-i})]\approx[(0)]$ but $[(10^{-i})]\neq[(0)]$. If there
exists an $r\in\mathbb{R}$ such that $[(r_{i})]\approx r$, we say that
$[(r_{i})]$ is _nearstandard_. Such an $r$ is necessarily unique. On the
subset of nearstandard elements of $\mathbb{R}^{\omega}$, we define the
_standard part mapping $st(\cdot)$ _by $st([(r_{i})])=r$.

A turn of phrase we will use frequently in the sequel, to indicate that some
property $p$ holds on a set of indices $i\in\mathbb{N}$ that belongs to the
ultrafilter $\omega$, is that $p$ holds $\omega$-a.s.; the terminology is
justified by the fact that $\omega$ is a finitely additive probability measure
on $\mathbb{N}$.

If $(X_{i})$ is a sequence of algebraic structures of the same type, equipped
with a notion of equality, one can consider the ultraproduct $X^{\omega}$
defined by $X^{\omega}/\sim_{\omega}$, where now we say
$(x_{i})\sim_{\omega}(x_{i}^{\prime})$ iff
$\\{i\in\mathbb{N}:x_{i}=x_{i}^{\prime}\\}\in\omega$ (a.k.a. iff
$x_{i}=x_{i}^{\prime}$ $\omega$-a.s.). Of particular interest for us is the
case where each $X_{i}$ is a metric space.

Given a sequence of real-valued functions $f_{i}:X_{i}\rightarrow\mathbb{R}$,
we can consider the ultraproduct $f^{\omega}$ defined as follows: if
$x:=[(x_{i})]\in X^{\omega}$, then we make the pointwise definition

$f^{\omega}(x):=[f_{i}(x_{i})].$

Of course, $[f_{i}(x_{i})]\in\mathbb{R}^{\omega}$, so that
$f^{\omega}:X^{\omega}\rightarrow\mathbb{R}^{\omega}$. (This is indicative of
the functoriality of the ultraproduct construction.) Likewise, given a
sequence of metric spaces $(X_{i},d_{i})$, we can consider the ultraproduct of
metric spaces $(X^{\omega},d^{\omega})$, where

$d^{\omega}(x,x^{\prime}):=[d_{i}(x_{i},x_{i}^{\prime})].$

Hence the space $(X^{\omega},d^{\omega})$ is a _hyperreal_ metric space.
Analytically this space may be ill-behaved, but it benefits from the Theorem
of Łoś (stated in one (informal) form as Theorem 10.38 in [16]) which
indicates that _logically_ $(X^{\omega},d^{\omega})$ is a well-behaved limit
of the sequence $(X_{i},d_{i})$.

The aim of the _metric ultralimit_ construction, which we now introduce, is to
produce a real metric space which inherits as much structure as possible from
$(X^{\omega},d^{\omega})$. One problem to overcome is that the space
$(X^{\omega},d^{\omega})$ will, in general, contain points which are
hyperfinite distance apart, that is,
$d^{\omega}(x,x^{\prime})\in\mathbb{R}^{\omega}\backslash\mathbb{R}$ and
$d^{\omega}(x,x^{\prime})>r$ for every $r\in\mathbb{R}$. To address this, we
instead work with pointed metric spaces $(X_{i},d_{i},e_{i})$ where $e_{i}$ is
the “origin” of $X_{i}$. Then $e:=[(e_{i})]\in X^{\omega}$ can be taken as the
origin of $X^{\omega}$. Let $X_{lim}^{\omega}:=\\{x\in
X^{\omega}:st(d^{\omega}(e,x))<\infty\\}$. Then we define the _metric
ultralimit_ of $(X_{i},d_{i},e_{i})$, denoted by $(\hat{X},\hat{e},\hat{d})$
or $(\hat{X},\hat{d})$ (if the distinguished points $e_{i}$ and $\hat{e}$ are
evident, or unimportant) by

$(\hat{X},\hat{d},\hat{e}):=(X_{\lim}^{\omega},d^{\omega})/\approx_{d^{\omega}}$

where the equivalence relation $\approx_{d^{\omega}}$ is given by
$x\approx_{d^{\omega}}x^{\prime}\iff st(d^{\omega}(x,x^{\prime}))=0$, or
equivalently $x\approx_{d^{\omega}}x^{\prime}\iff
d^{\omega}(x,x^{\prime})\approx 0$. Here, $\hat{d}$ is the natural quotient of
$d^{\omega}$ under $\approx_{d^{\omega}}$: given $x,x^{\prime}\in
X_{\lim}^{\omega}$ and associated equivalence classes
$[x]_{\approx_{d^{\omega}}}$, $[x^{\prime}]_{\approx_{d^{\omega}}}$, we define

$\hat{d}([x]_{\approx_{d^{\omega}}},[x^{\prime}]_{\approx_{d^{\omega}}}):=st(d^{\omega}(x,x^{\prime})).$

(Of course, $x=[(x_{i})_{i\in\mathbb{N}}]$ for some sequence
$(x_{i})_{i\in\mathbb{N}}$ of points each belonging to $X_{i}$; but we will
generally avoid burdensome notation like
$\left[\left[\left(x_{i}\right)_{i\in\mathbb{N}}\right]\right]_{\approx_{d^{\omega}}}$
if possible.) Lastly, $\hat{e}$ is the equivalence class containing
$[(e_{i})]$. The structure $(\hat{X},\hat{d})$ is a bona fide (real) metric
space, unlike $(X^{\omega},d^{\omega})$.

By way of analogy with the mapping $st(\cdot)$ from the nearstandard elements
of $\mathbb{R}^{\omega}$ to $\mathbb{R}$, in the sequel we also use
$st_{\hat{d}}(\cdot):(X_{\lim}^{\omega},d^{\omega})\rightarrow(\hat{X},\hat{d})$
to denote the map which takes a point $x^{\omega}=[(x_{i})]$ in
$(X_{\lim}^{\omega},d^{\omega})$ and returns
$[x^{\omega}]_{\approx_{d^{\omega}}}\in\hat{X}$; we also say that
$st_{\hat{d}}(x^{\omega})$ is the _pushdown_ of $x^{\omega}$ (with respect to
the metric $\hat{d}$). Dually, given a point $y\in\hat{X}$, we say that
$y^{\omega}=[(y_{i})]\in X_{\lim}^{\omega}$ is a _lifting_ of $y$ provided
$y^{\omega}\in st_{\hat{d}}^{-1}(y)$.

######  Example 1.

For a trivial example of a metric ultraproduct, take $\mathbb{R}$ with the
distinguished point $0$. The ultraproduct of the constant sequence
$(\mathbb{R},|\cdot|,0)$, as pointed metric spaces, is
$(\mathbb{R}^{\omega},|\cdot|^{\omega},0)$; the metric ultralimit of the
constant sequence $(\mathbb{R},|\cdot|,0)$ is therefore just
$(\mathbb{R},|\cdot|,0)$. However, it can be shown that the metric ultralimit
of the constant sequence $(\mathbb{Q},|\cdot|_{\mathbb{R}},0)$ is not
$(\mathbb{Q},|\cdot|_{\mathbb{R}},0)$ but
$(\mathbb{R},|\cdot|_{\mathbb{R}},0)$ — this is related to the fact that the
metric ultralimit is automatically a complete metric space.

For a nontrivial example, one might consider a sequence of pointed metric
measure spaces $(X_{i},e_{i},d_{i},\mu_{i})$ as well as their associated
Hilbert spaces $L^{2}(X_{i},\mu_{i})$. Then we can both study the metric
ultralimit $(\hat{X},\hat{d},\tilde{\mu})$ (what exactly the “right” limit
measure $\tilde{\mu}$ is, we address in the next subsection, when considering
ultraproducts of metric measure spaces — it turns out to be closely related to
the _Loeb measure_ associated to the ultraproduct of the measures $\mu_{i}$),
and also, if we make $L^{2}(X_{i},\mu_{i})$ into a pointed metric space with
the “origin” as the constant zero function and, of course, the metric as
$\|\cdot\|_{L^{2}(\mu_{i})}$, we can study the metric ultralimit
$(\widehat{L^{2}(X,\mu)},\widehat{\|\cdot\|_{L^{2}(\mu)}})$. The relationship
between $L^{2}(\hat{X},\tilde{\mu})$ and $\widehat{L^{2}(X,\mu)}$ is not _a
priori_ obvious, but is addressed briefly in [26] (see Theorem 5.2 therein,
for a related statement about Orlicz spaces, but note that much of their
terminology conflicts with ours); it turns out that
$L^{2}(\hat{X},\tilde{\mu})$ embeds isometrically into
$\widehat{L^{2}(X,\mu)}$ as a closed subspace. We encounter a similar
situation below in the context of Wasserstein spaces, in Theorem 11.

######  Remark.

(Relationship between metric ultralimits and Gromov-Hausdorff limits) We
emphasize that the metric ultralimit construction should be understood as a
generalization of Gromov-Hausdorff limits. Theorem 10.46 in [16] indicates (in
different notation) that, given a sequence of pointed, _proper_ metric spaces
$(X_{i},e_{i},d_{i})$ converging to a Gromov-Hausdorff limit $(X,e,d)$, then
$(X,e,d)$ and $(\hat{X},\hat{d})$ are isometric. On the other hand, the metric
ultralimit of a sequence of pointed metric spaces is guaranteed to exist even
when no Gromov-Hausdorff limit does (but depends on the choice of ultrafilter,
and in the absence of a Gromov-Hausdorff limit the different metric
ultralimits with respect to different ultrafilters need not be isometric to
each other). At the same time, the metric ultralimit remains a well-defined
object when considering non-proper (or even non-locally compact) metric
spaces, like say $(\mathcal{P}_{2}(X),W_{2})$ with $X$ a noncompact metric
space.

For the sake of motivation, we give three examples of metric ultralimits which
are of some intrinsic interest.

######  Example 2.

(Special metric ultralimits) Fix a metric space $(X,d)$.

(1) _The constant ultralimit_. Fix a point $x_{0}\in X$, and take each term of
our sequence of metric spaces to simply be $(X,x_{0},d)$. Then we can form the
space $(\hat{X},\hat{d})$ just as we would for a non-constant sequence. (This
space is also called the _nonstandard hull of the ultrapower of $X$_.) In
general, if $X$ is locally sequentially compact (i.e. the Bolzano-Weierstrass
theorem holds in the completion), then $(\hat{X},\hat{d})$ is isometric to the
completion of $(X,d)$. If however $(X,d)$ is _not_ locally sequentially
compact, then $(\hat{X},\hat{d})$ will be much bigger than the completion of
$(X,d)$. Indeed, if $(x_{i})$ is a bounded sequence in $X$ which fails to have
any convergent subsequence, then there is no constant $y$ in the completion of
$(X,d)$ such that $d(x_{i},y)\rightarrow 0$, regardless of subsequence; this
implies that, for $\varepsilon>0$ small enough, there is _no_ infinite subset
of $\mathbb{N}$ such that $\\{i\mid d(x_{i},y_{i})<\varepsilon\\}\in\omega$
(where $(y_{i})$ is a Cauchy sequence converging to $y$ in the completion of
$(X,d)$) which in turn shows that $[(x_{i})]\neq[(y_{i})]$ in
$(\hat{X},\hat{d})$. On the other hand, given any two sequences $(x_{i})$ and
$(z_{i})$ which are both bounded, have no convergent subsequence in $(X,d)$,
and are asymptotically bounded away _from each other_ (or more precisely,
there is some $\varepsilon>0$ such that $d(x_{i},z_{i})>\varepsilon$
$\omega$-a.s.), then $(x_{i})$ and $(z_{i})$ are included in distinct
equivalence classes $[(x_{i})]$ and $[(z_{i})]$ within $\hat{X}$. It is
therefore reasonable to view $(\hat{X},\hat{d})$, as a “locally sequentially
compactification” of $(X,d)$ with base point $x_{0}$.

(2) _Ultratangent space_. This is a nonsmooth generalization of the tangent
space of a point on a manifold. Given a distinguished point $x_{0}\in X$, our
sequence of pointed metric spaces is $(X,x_{0},\lambda_{i}d)$, where
$\lambda_{i}\rightarrow+\infty$ is a sequence of positive constants; the
tangent cone is the object $(\hat{X},x_{0},\hat{d})$, where

$\hat{d}([x]_{\approx_{d^{\omega}}},[x^{\prime}]_{\approx_{d^{\omega}}}):=st(\lambda^{\omega}d^{\omega}(x,x^{\prime})):=st([(\lambda_{i}d_{i}(x_{i},x_{i}^{\prime}))])$

(and where $x=[(x_{i})]$ and $x^{\prime}=[(x_{i}^{\prime})]$). For an example
of usage of the ultratangent space, we refer the reader to [2, Chapter 11]; it
is also shown in [2, Theorem 3.4.1] that the ultratangent space contains the
usual (nonsmooth) tangent space from Alexandrov geometry as a subset.

(3) _Asymptotic cone_. Instead of blowing up a metric space at a point, we
can, in Gromov’s parlance, zoom out infinitely far away. Take as a sequence of
pointed metric spaces $(X,x_{0},\lambda_{i}^{-1}d)$ where again
$\lambda_{i}\rightarrow+\infty$. Borrowing two examples from [25, Ch.2]: if
$(X,d)$ is an abelian group equipped with the word metric (like
$(\mathbb{Z}^{d},\|\cdot\|_{1})$), and (by convention) our distinguished point
is the identity element, then the metric ultralimit of
$(X,\lambda_{i}^{-1}d,e)$ (as well as the pointed Gromov-Hausdorff limit) is
isometric to $\mathbb{R}^{N}$ for some finite $N$ (in fact the rank of the
group). On the other hand, if $(X,d)$ is a $\delta$-hyperbolic space, then the
pointed Gromov-Hausdorff limit of $(X,\lambda_{i}^{-1}d,e)$ does not exist,
but the metric ultralimit of $(X,\lambda_{i}^{-1}d,e)$ is isometric to a real
tree (see e.g. [16, Definition 3.60]).

######  Remark.

Metric ultralimits, in the specific case of asymptotic cones, have become a
reasonably standard tool in metric geometry and adjacent parts of geometric
group theory [2, 13, 16, 29, 42]. In the latter area, their usage dates to
[47], which offered an alterate route, employing ultraproducts, to Gromov’s
theorem on the equivalence of polynomial growth and virtual nilpotency.
Subsequently asymptotic cones were popularized in the geometric group theory
community by [25].

We close this section by mentioning several geometric properties which are
stable under metric ultralimits. For each of these, the proof is nearly
immediate from the definitions of all the relevant objects; we give citations
to point out that these facts are standard in the literature.

  * •

If a sequence of pointed metric spaces $(X_{i},d_{i},e_{i})_{i\in\mathbb{N}}$
has the property that for $\omega$-almost all $i\in\mathbb{N}$,
$(X_{i},d_{i})$ is a geodesic metric space, then the metric ultralimit
$(\hat{X},\hat{d})$ is also a geodesic metric space [16, Lemma 10.51]. In
fact, the same still holds when the spaces $(X_{i},d_{i})$ are merely complete
length spaces [2, Corollary 3.5.2].

  * •

For each (or even $\omega$-almost all) $(X_{i},d_{i},e_{i})$, suppose that
$(X_{i},d_{i})$ is a $CAT(K_{i})$ space, and suppose that
$K:=st([(K_{i})])\in\mathbb{R}$. Then $(\hat{X},\hat{d})$ is a $CAT(K)$ space
(see [2, Proposition 8.1.6] or [16, Lemma 10.53]). In other words, synthetic
upper sectional curvature bounds in the sense of Alexandrov are stable under
metric ultralimits.

  * •

Likewise, for each (or even $\omega$-almost all) $(X_{i},d_{i},e_{i})$,
suppose that $(X_{i},d_{i})$ is a $CBB(K_{i})$ space, and suppose that
$K:=st([(K_{i})])\in\mathbb{R}$. Then $(\hat{X},\hat{d})$ is a $CBB(K)$ space
[2, Proposition 7.1.7]. In other words, synthetic lower sectional curvature
bounds in the sense of Alexandrov are stable under metric ultralimits.

On the other hand, the author is not aware of any prior direct investigation
into the stability of synthetic _Ricci curvature_ lower bounds with respect to
ultralimits, either in the sense of Lott-Sturm-Villani (which we have already
mentioned) or in the sense of Bakry-Émery (which we have not discussed, but
see e.g. [9] for an overview, and [7, 21] for the relationship of this notion
to Lott-Sturm-Villani synthetic Ricci curvature bounds).

###  1.3. Loeb measures and metric measure spaces

The theory of Loeb measures, originally introduced in [31], is developed in
many sources, for instance [1, 15, 24, 32], albeit with minor variations; our
setup is very similar to the one from [14] (which considers probability
measures on standard Borel spaces, where we consider Radon probability
measures on complete metric spaces), and only slightly more general than that
of [20] or [10] (which restrict attention to ultraproducts of discrete
measures).

Consider as before a sequence of pointed metric spaces $(X_{i},d_{i},e_{i})$.
Each $(X_{i},d_{i},e_{i})$ comes equipped with a Borel $\sigma$-algebra
$\mathcal{B}_{i}$ generated by $d_{i}$. When we pass to the ultraproduct
$(X^{\omega},d^{\omega},e^{\omega})$, we can also equip $X^{\omega}$ with the
_ultraproduct measure algebra_
$\mathcal{B}^{\omega}:=\prod_{\omega}\mathcal{B}_{i}$. Note that typically
$\mathcal{B}^{\omega}$ is _not_ closed under countable unions. Here, an
arbitrary element of $\mathcal{B}^{\omega}$ has the form $\prod_{\omega}B_{i}$
for $B_{i}\in\mathcal{B}_{i}$ (and of course, $x^{\omega}\in X^{\omega}$
belongs to $B^{\omega}$ iff $x_{i}\in B_{i}$ $\omega$-a.s). Likewise, given a
sequence of Borel measures $\mu_{i}$ on $X_{i}$, these are of course each a
function $\mu_{i}:\mathcal{B}_{i}\rightarrow[0,1]$ satisfying the usual
axioms. Consequently the ultraproduct $\mu^{\omega}$ is a set-valued function
from $\mathcal{B}^{\omega}$ to $[0,1]^{\omega}$, with
$\mu^{\omega}(B^{\omega}):=\left[\left(\mu_{i}(B_{i})\right)\right]$.

Given an ultraproduct measure algebra $\mathcal{A}^{\omega}$ on $X^{\omega}$,
let $\sigma(\mathcal{A}^{\omega})$ denote the _ $\sigma$-algebra_ generated by
$\mathcal{A}^{\omega}$. This is the smallest $\sigma$-algebra containing
$\mathcal{A}^{\omega}$. If we have a distinguished ultraproduct measure
$\mu^{\omega}$ in mind, we can also perform the following construction: the
function

$st\circ\mu^{\omega}:\mathcal{A}^{\omega}\rightarrow[0,1]$
$st\circ\mu^{\omega}(A^{\omega}):=st(\mu^{\omega}(A^{\omega}))$

turns out to be a premeasure on the space $X^{\omega}$; we complete
$st\circ\mu^{\omega}$ by invoking the Carathéodory extension theorem, and
denote the resulting measure on $\hat{X}$ by $\mu_{L}$, and denote by
$L(\mathcal{A}^{\omega})$ the set of $\mu_{L}$-measurable sets. The objects
$\mu_{L}$ and $L(\mathcal{A}^{\omega})$ are denoted the _Loeb measure_
associated to $\mu^{\omega}$, and the _Loeb $\sigma$-algebra _generated by
$\mathcal{A}^{\omega}$, respectively. We will also sometimes write
$\mu_{L}=\text{Loeb}(\mu^{\omega})$ to indicate that $\mu_{L}$ is the Loeb
measure associated to $\mu^{\omega}$.

Let us provide some motivation. Suppose we wanted to develop a theory of
“ultralimit of metric spaces with distinguished functionals”: we would take
tuples $(X_{i},d_{i},e_{i},F_{i})$, and we would be interested in the natural
analogue of the ultralimit $(\hat{X},\hat{d},\hat{e})$. Certainly, when
generating the _ultraproduct_ , we already _have_ a notion of ultraproduct of
functions, so the naive way to define “ultraproduct of pointed metric spaces
with distinguished functionals” is just to also take the ultraproduct
$F^{\omega}$ of the $F_{i}$’s, and attach it to the ultraproduct of pointed
metric spaces, to get the tuple
$(X^{\omega},d^{\omega},e^{\omega},F^{\omega})$. Likewise, the naive way to
produce a function $\hat{F}:\hat{X}\rightarrow\mathbb{R}$, given
$F^{\omega}:X^{\omega}\rightarrow\mathbb{R}^{\omega}$, is to post-compose with
the standard part map, namely,

$\hat{F}(st[(x)]):=st(F^{\omega}[(x)])$

and then check whether this operation is well-defined; if so, we have our
limiting “ultralimit of pointed metric spaces with distinguished functionals”,
namely $(\hat{X},\hat{d},\hat{e},\hat{F})$.

In order to define a reasonable notion of “ultraproduct of pointed metric
_measure_ spaces”, we would like to do something like this for measures, and
push down a measure $\mu^{\omega}$ on $X^{\omega}$ onto a measure on
$\hat{X}$. But instead of applying the standard part map to points in
$X^{\omega}$, we would now apply it to sets in the measure algebra
$\mathcal{A}^{\omega}$; and then one would need to check that given a Borel
set $B$ on $(\hat{X},\hat{d})$, that $st^{-1}(B)$ is
$\mathcal{A}^{\omega}$-measurable; with this in hand, one could define a
measure $\mu$ on $\hat{X}$ by $\mu(B)=\mu^{\omega}(st^{-1}(B))$. But now the
problems arise: it turns out that $st$ is _not_ measurable from
$\mathcal{A}^{\omega}$ to the Borel sets on $\hat{X}$ — for one thing, the
latter is a $\sigma$-algebra, thus countably additive, while
$\mathcal{A}^{\omega}$ is not! Therefore, we need a _surrogate_ object for
$\mu^{\omega}$ that _can_ be pushed down onto $\hat{X}$ (at least under some
regularity conditions), and it so happens that the Loeb measure $\mu_{L}$ is
the correct surrogate object for the job. (Indeed, the Loeb measure was
devised precisely to be such a surrogate, in the original paper [31], just in
a slightly different setting than ours.)

In the sequel we require several facts connecting the measures $\mu^{\omega}$
and $\mu_{L}$, and their associated integrals and integrable functions. These
facts are all now classical in the Loeb measure literature; proofs may be
found, using various setups of the “underlying machinery” of nonstandard
analysis, [1, 8, 15, 46], for instance. In particular, proofs of the following
facts require only that the “universe” of internal objects satisfies Łoś’s
theorem together with the _countable saturation property_ , and this latter
property is automatically imposed by any non-principal ultrafilter on
$\mathbb{N}$. To provide a more specific reference for the reader, we mention
[46, Ch. 2.2], which uses the same foundational setup as ours but slightly
different terminology.

######  Fact 3.

(see [46, Prop. 2.2.11 and Cor. 2.2.12]) Let
$(X^{\omega},\mathcal{A}^{\omega},\mu^{\omega})$ be an internal measure space,
with associated Loeb measure space
$(X^{\omega},L(\mathcal{A}^{\omega}),\mu^{\omega})$. Let $A\in
L(\mathcal{A}^{\omega})$. Then:

  1. (1)

For every $\varepsilon>0$, there exist
$B^{\omega},C^{\omega}\in\mathcal{A^{\omega}}$ with $B^{\omega}\subseteq
A\subseteq C^{\omega}$ and $\mu^{\omega}(C^{\omega}\backslash
B^{\omega})<\varepsilon$.

  2. (2)

There exists an $A^{\omega}\in\mathcal{A}^{\omega}$ such that $\mu_{L}(A\Delta
A^{\omega})=0$.

######  Definition 4.

Fix an internal measure space
$(X^{\omega},\mathcal{A}^{\omega},\mu^{\omega})$. Let
$f^{\omega}:X^{\omega}\rightarrow\mathbb{R}^{\omega}$ be an internal function
which is $\mathcal{A}^{\omega}$-measurable. We say that $f^{\omega}$ is
$S$-integrable provided that
$st(f^{\omega}):X^{\omega}\rightarrow\mathbb{R}\cup\\{-\infty,\infty\\}$ is
$\mu_{L}$-integrable and

$st\left(\int_{X^{\omega}}|f^{\omega}|d\mu^{\omega}\right)=\int_{X^{\omega}}|st(f^{\omega})|d\mu_{L}.$

######  Fact 5.

(see [46, Thm. 2.2.19] for the case where $\mu^{\omega}(X^{\omega})<\infty$)
Let $f^{\omega}:X^{\omega}\rightarrow\mathbb{R}^{\omega}$ be an internal
function which is $\mathcal{A}^{\omega}$-measurable. Then, $f^{\omega}$ is
$S$-integrable iff

  1. (1)

$\int_{X^{\omega}}|f^{\omega}|d\mu^{\omega}$ is finite,

  2. (2)

if $A\in\mathcal{A}^{\omega}$ and $\mu^{\omega}(A)\approx 0$, then
$\int_{A}|f^{\omega}|d\mu^{\omega}\approx 0$; and

  3. (3)

if $A\in\mathcal{A}^{\omega}$ and $f^{\omega}(a)\approx 0$ for all $a\in A$,
then $\int_{A}|f^{\omega}|d\mu^{\omega}\approx 0$. (Note this automatically
holds if $\mu^{\omega}(X^{\omega})$ is finite.)

######  Definition 6.

Fix an internal measure space
$(X^{\omega},\mathcal{A}^{\omega},\mu^{\omega})$. Let
$g:X^{\omega}\rightarrow\mathbb{R}$. We say that
$f^{\omega}:X^{\omega}\rightarrow\mathbb{R}^{\omega}$ is an _internal lifting_
of $g$ provided that $st(f^{\omega}(x))=g(x)$ holds $\mu_{L}$-almost
everywhere.

######  Fact 7.

A function $g:X^{\omega}\rightarrow\mathbb{R}$ is $\mu_{L}$-integrable iff it
has an $S$-integrable internal lifting $f^{\omega}$. In this case,
$st\left(\int_{X^{\omega}}|f^{\omega}|d\mu^{\omega}\right)=\int_{X^{\omega}}|g|d\mu_{L}$.

###### ?proofname? .

The forward direction is provided by [46, Thm. 2.2.21]; the converse direction
is automatic from the definitions of $S$-integrability and internal lifting. ∎

We also have the following handy sufficient condition for as internal function
$f^{\omega}$ to be $S$-integrable.

######  Fact 8.

(see [46, Prop. 2.2.20]) Let $p>1$. Suppose that
$st\left(\int_{X^{\omega}}|f^{\omega}|^{p}d\mu^{\omega}\right)<\infty$. Then,
$f^{\omega}$ is $S$-integrable, and also $st\circ f^{\omega}\in
L^{p}(\mu_{L})$.

######  Remark.

Under the heuristic that “taking an ultraproduct, then taking a standard part,
is much like taking a limit” (indeed this is literally the definition of an
ultralimit, say, of real numbers), we see that the $S$-integrable functions
are precisely those for which an “ultralimit analogue” of the dominated
convergence theorem (or, more precisely, the Vitali convergence theorem)
holds. In particular, Fact 8 is related to the fact that sequences of
functions bounded in $L^{p}$ with $p>1$ are automatically uniformly integrable
(by the theorem of de la Vallée Poussin), so that the Vitali convergence
theorem is applicable.

This concludes our background discussion of Loeb measures. Before we give the
definition of a _metric-measure ultralimit_ , however, there is a snag, more
or less general to all notions of convergence for metric measure spaces, and
observed for instance in [44]. While we can use some specific construction to
_produce_ a candidate metric measure space limiting object, in a specific
instantiation, it’s desirable for the property “is a metric measure space
limit of a given sequence of metric measure spaces” to be invariant under
isomorphism of metric measure spaces. And if our candidate limit definition
has the form of “the underlying pointed metric space converge, with respect to
our favourite notion of convergence of pointed metric spaces, and also the
measures on top converge in some associated sense”, then invariance under
metric measure space isomorphism can _fail_ when the measures do not have full
support, in which case we should really only care about isometry of the
underlying measure spaces _on the supports of the measures_ (and also the
distinguished points $e$ in the two spaces!).

With all this preparatory discussion, we can now present the following
definition. Due to the restricted interests of our present work, we specialize
the definition for Radon probability measures only. We mention that our
definition is a variant on the one introduced by Elek in [19], who we believe
to be the first to consider ultralimits of metric measure spaces.

######  Definition 9.

(ultralimit of pointed metric measure spaces)_Let
$(X_{i},d_{i},e_{i},\mu_{i})$ be a sequence of tuples where
$(X_{i},d_{i},e_{i})$ are pointed metric spaces and $\mu_{i}$ is a Radon
probability measure on $X_{i}$. We say that $(X_{i},d_{i},e_{i},\mu_{i})$
converges to $(X,d,e,\mu)$ (with $\mu$ also a Radon probability measure) in
the sense of pointed metric measure ultralimits if_

  1. (1)

_The Loeb measure $\mu_{L}$ associated to $\mu^{\omega}$ pushes forward via
$\text{st}_{\hat{d}}$ to a Radon probability measure $\tilde{\mu}$ on
$(\hat{X},\hat{d},\hat{e})$, that is, $\tilde{\mu}=\mu_{L}\circ
st_{\hat{d}}^{-1}$; and_

  2. (2)

_There exists a partial isometry $\iota$ between some $X_{0}\subseteq X$ and
some $\hat{X}_{0}\subseteq\hat{X}$, such that $\iota$ maps $e$ to $\hat{e}$
and $supp(\mu)$ to $supp(\tilde{\mu})$, and for every Borel set $B\subset
supp(\tilde{\mu})$, we have that $\mu(\iota^{-1}(B))=\tilde{\mu}(B)$ (and vice
versa). _

Note that there is a certain asymmetry between the metric and measure
structures here, because, while we can _always_ produce a limiting metric
space (in the metric ultraproduct sense), it is _not_ generally the case that
condition (1) is satisfied [38], so in particular there does _not_ always
exist a pointed metric measure ultralimit of a sequence of pointed metric
measure spaces $(X_{i},d_{i},e_{i},\mu_{i})$. However, Lemma 22 below gives a
characterization of which Loeb probability measures satisfy condition (1).

#### Note.

While our work was in progress, we learned of the recent preprint [38], which
is also concerned with ultralimits of metric measure spaces. [38] considers a
notion of metric measure ultralimit which allows for a more general limiting
measure $\mu$ (not even necessarily Borel), and investigates many structural
properties of metric measure ultralimits. In particular, [38] shows that the
notion of metric measure ultralimits they consider extends the notion of
_pointed measured Gromov_(pmG) convergence from [23], and moreover our
Definition 9 is actually _equivalent_ to pmG convergence, _precisely_ because
we assume that the reference measure on the limiting space is Radon. (See in
particular [38, Theorems 8.3, 11.4, and 12.2] and our remark following Lemma
22 below.) This actually allows [38] to deduce various synthetic geometric
ultralimit stability results by indirect means, including our Theorem 28, as
they discuss in their introduction, in particular by quoting existing results
on $CD(K,\infty)$ stability under pmG convergence from works such as [23].

Since our definition of “metric-measure ultralimit” stipulates that
$\hat{\mu}$ is a Loeb measure, it would be nice to know that _any_ reasonable
measure $\mu$ on $(X,d)$ can be viewed as a Loeb measure. Similar results are
already known in the literature, for instance:

######  Theorem 10.

(Anderson [8]) Let $\nu$ be a Radon measure on a Hausdorff space
$(X,\mathcal{T})$. Then the measure space
$(X,\overline{\sigma(\mathcal{T})},\nu)$ can be represented as a Loeb measure
space, where $\overline{\sigma(\mathcal{T})}$ is the completion of the Borel
$\sigma$-algebra generated by $\mathcal{T}$.

Therefore, we aim to prove a variant of Anderson’s theorem, where $\nu$ lives
on $\hat{X}$ and its Loeb measure representative lives on $X^{\omega}$. We do
so below, in Theorem 17.

##  2\. Ultralimits of Wasserstein spaces

Given a sequence $(X_{i},d_{i},e_{i})$ of metric spaces with distinguished
points (which we do not always notate explicitly), we can form the metric
ultralimit space $(\hat{X},\hat{d})$ as discussed in the previous section. At
the same time, for each $i\in\mathbb{N}$, we can consider, for
$p\in[1,\infty)$, the (pointed) $p$-Wasserstein space atop
$(X_{i},d_{i},e_{i})$, namely $(\mathcal{P}_{p}(X_{i}),W_{p},\delta_{e_{i}})$;
as well as the $p$-Wasserstein space atop ($\hat{X},\hat{d})$, namely
$(\mathcal{P}_{p}(\hat{X}),W_{p})$ — recall, from the previous section, that
this denotes the space of Radon probability measures on the (complete) metric
space $\hat{X}$ with finite $p$th moments, equipped with the metric

$W_{p}(\mu,\nu):=\left(\inf_{\gamma\in\Pi(\mu,\nu)}\int_{\hat{X}\times\hat{X}}\hat{d}(x,y)^{p}d\gamma(x,y)\right)^{1/p}$

where $\Pi(\mu,\nu)$ is the space of couplings of the measures $\mu$ and
$\nu$.

At the same time, the sequence
$(\mathcal{P}_{p}(X_{i}),W_{p},\delta_{e_{i}})_{i\in\mathbb{N}}$ is __ also a
sequence of metric spaces (in fact a sequence of complete metric spaces,
provided each $(X_{i},d_{i})$ is complete), so we can also consider the metric
ultralimit of the sequence $(\mathcal{P}_{p}(X_{i}),W_{p})_{i\in\mathbb{N}}$,
which we denote $(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$. Let us recap the
metric ultraproduct construction, reviewed in the previous section. in this
specific instance. Explicitly, this space is constructed by first forming the
ultraproduct
$\left(\mathcal{P}_{p}(X)^{\omega},W_{P}^{\omega},[(\delta_{e_{i}})]\right)$,
where

$\mathcal{P}_{p}(X)^{\omega}:=\left\\{[(\mu_{i})_{i\in\mathbb{N}}]:\forall
i\in\mathbb{N},\mu_{i}\in\mathcal{P}_{p}(X_{i})\right\\}$

and

$W_{p}^{\omega}\left(\left[(\mu_{i})_{i\in\mathbb{N}}\right],\left[(\nu_{i})_{i\in\mathbb{N}}\right]\right):=\left[\left(W_{p}(\mu_{i},\nu_{i})\right)\right].$

We then restrict to the “limited” subset of $\mathcal{P}_{p}(X)^{\omega}$,
which is the subset of points $[(\mu_{i})]\in\mathcal{P}_{p}(X)^{\omega}$
which are a nearstandard distance from the distinguished point
$[(\delta_{e_{i}})]$, and which we denote
$\mathcal{P}_{p}(X)_{\lim}^{\omega}$:

$\mathcal{P}_{p}(X)_{\lim}^{\omega}:=\left\\{[(\mu_{i})]\in\mathcal{P}_{p}(X)^{\omega}:st\left(W_{p}^{\omega}\left([(\mu_{i})],[(\delta_{e_{i}})]\right)\right)<\infty\right\\}.$

Finally, $(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$ is formed by quotienting
$\mathcal{P}_{p}(X)_{\lim}^{\omega}$ with respect to the equivalence relation
$W_{p}^{\omega}\left([(\mu_{i})],[(\nu_{i})]\right)\approx 0$:

$\widehat{\mathcal{P}_{p}(X)}:=\mathcal{P}_{p}(X)_{\lim}^{\omega}/\approx_{W_{p}^{\omega}}.$

Points in $\widehat{\mathcal{P}_{p}(X)}$ (i.e. equivalence classes of points
$[(\mu_{i})]\in\mathcal{P}_{p}(X)_{\lim}^{\omega}$) are denoted by $\hat{\mu}$
or $st_{\hat{W}_{p}}[(\mu_{i})]$, and

$\hat{W}_{p}(\hat{\mu},\hat{\nu}):=st\left(W_{p}^{\omega}\left([(\mu_{i})],[(\nu_{i})]\right)\right)\text{
where
}st_{\hat{W}_{p}}[(\mu_{i})]=\hat{\mu},st_{\hat{W}_{p}}[(\nu_{i})]=\hat{\nu}.$

(If one prefers, the space $(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$ is
also a pointed metric space in a canonical fashion: simply add the
distinguished point $st_{\hat{W}_{p}}[(\delta_{e_{i}})]$, namely the
equivalence class containing the ultraproduct of the distinguished points
$\delta_{e_{i}}$.)

The following theorem addresses the relationship between
$(\mathcal{P}_{p}(\hat{X}),W_{p})$ and
$(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$.

######  Theorem 11.

Let $p\in[1,\infty)$. Suppose that $(X_{i},d_{i})$ is a sequence of pointed
complete metric spaces with metric ultralimit $(\hat{X},\hat{d})$. Then, there
is a canonical isometric embedding of $(\mathcal{P}_{p}(\hat{X}),W_{p})$ into
$(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$, extending the map

$(\mathcal{P}_{p}(\hat{X}),W_{p})\ni\delta_{st_{\hat{d}}[(x_{i})]}\longmapsto
st_{\hat{W}_{p}}\left[\left(\delta_{x_{i}}\right)\right]\in(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p}).$

###### ?proofname?.

Fix an $N\in\mathbb{N}$; let
$\mu=\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{x}_{j}}$ and
$\mu^{\prime}=\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{x}_{j}^{\prime}}$, where
the $\hat{x}_{j}$’s and $\hat{x}_{j}^{\prime}$’s are arbitrary points in
$\hat{X}$; clearly $\mu,\mu^{\prime}\in\mathcal{P}_{p}(\hat{X})$. For each
such point, let $[(x_{i,j})]\in X^{\omega}$ be a lifting of $\hat{x}_{j}$ (and
similarly for $[(x_{i,j}^{\prime})]$ and $\hat{x}_{j}^{\prime}$). Then we can
also consider Dirac measures on each $X_{i}$, of the form
$\mu_{i}:=\frac{1}{N}\sum_{j=1}^{N}\delta_{x_{i,j}}$ for $x_{i,j}\in X_{i}$,
and similarly
$\mu_{i}^{\prime}:=\frac{1}{N}\sum_{j=1}^{N}\delta_{x_{i,j}^{\prime}}$. Then,
we can consider
$[(\mu_{i})],[(\mu_{i}^{\prime})]\in\mathcal{P}_{p}(X)^{\omega}$ as well as
$\hat{\mu}=st_{\hat{W}_{p}}[(\mu_{i})]$ and
$\hat{\mu^{\prime}}=st_{\hat{W}_{p}}[(\mu_{i}^{\prime})]$ in
$\widehat{\mathcal{P}_{p}(X)}$; and it holds that

$\hat{W_{p}}(\hat{\mu},\hat{\mu^{\prime}}):=st(W_{p}^{\omega}([(\mu_{i})],[(\mu_{i}^{\prime})]):=st([(W_{p}(\mu_{i},\mu_{i}^{\prime}))]).$

On the other hand, we’d like to say that
$W_{p}(\mu,\mu^{\prime})=\hat{W}_{p}(\hat{\mu},\hat{\mu^{\prime}})$. Indeed,
this is actually sufficient to prove the theorem: consider a Cauchy sequence
$(\mu_{k})$ in $(\mathcal{P}_{p}(\hat{X}),W_{p})$, converging to $\nu$, such
that each $\mu_{k}$ is an average of Dirac masses. Then (possibly by
representing $\mu_{k}$ and $\mu_{k^{\prime}}$ as averages of an artificially
larger, equal number of Dirac masses; the least common multiple suffices) we
have that
$W_{p}(\mu_{k},\mu_{k^{\prime}})=\hat{W}_{p}(\hat{\mu}_{k},\hat{\mu}_{k^{\prime}})$,
which implies that $(\hat{\mu}_{k})$ is a Cauchy sequence in
$(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$. Since
$(\mathcal{P}_{p}(\hat{X}),W_{p})$ and
$(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$ are both complete metric spaces,
it follows that we can extend the map

$\mathcal{P}_{p}(\hat{X})\ni\mu\mapsto\hat{\mu}\in\widehat{\mathcal{P}_{p}(X)}$

from an isometry on the set of Dirac clouds in $\mathcal{P}_{p}(\hat{X})$, to
an isometry on all of $\mathcal{P}_{p}(\hat{X})$, by density. In particular,
the image of $\mathcal{P}_{p}(\hat{X})$ inside $\widehat{\mathcal{P}_{p}(X)}$
under this embedding is automatically $\hat{W}_{p}$-closed.

However, it can be readily seen that the fact that
$W_{p}(\mu,\mu^{\prime})=\hat{W}_{p}(\hat{\mu},\hat{\mu^{\prime}})$ follows
from Birkhoff’s theorem on convex polytopes (cf. Proposition 2.1 in [39]).
That is,

$W_{p}^{p}(\mu,\mu^{\prime})=\frac{1}{N}\sum_{j=1}^{N}\hat{d}^{p}(\hat{x}_{j},\hat{x}_{\sigma(j)}^{\prime})$

where $\sigma:\\{N\\}\rightarrow\\{N\\}$ is some permutation. But by
definition,
$\hat{d}(\hat{x}_{j},\hat{x}_{\sigma(j)}^{\prime})=st((d^{\omega}([(x_{i,j})],[(x_{i,\sigma(j)}^{\prime})])$.
At the same time, for each $i\in\mathbb{N}$,

$W_{p}^{p}(\mu_{i},\mu_{i}^{\prime})=\frac{1}{N}\sum_{j=1}^{N}d_{i}^{p}(x_{i,j},x_{i,\sigma_{i}(j)}^{\prime})$

where $\sigma_{i}:\\{N\\}\rightarrow\\{N\\}$ is again a permutation. It then
follows from Łoś’s theorem (really just careful manipulation of our definition
of an ultraproduct of functions) that

$\left(W_{p}^{\omega}([(\mu_{i})],[(\mu_{i}^{\prime})])\right)^{p}=\frac{1}{N}\sum_{j=1}^{N}\left(d^{\omega}(x^{\omega},x_{\sigma^{\omega}(j)}^{\prime\omega})\right)^{p}$

where $\sigma^{\omega}$ is the ultraproduct of the $\sigma_{i}$’s (but since
$\\{N\\}$ is a finite set, $\sigma^{\omega}$ is just some permutation on $N$
elements; in particular, it is the permutation amongst the $\sigma_{i}$’s that
has $\omega$ measure 1). It follows, from the definition of $\hat{d}$ and
$\hat{W}_{2}$, that

$\hat{W}_{p}^{p}(\hat{\mu},\hat{\mu}^{\prime})=\frac{1}{N}\sum_{j=1}^{N}\hat{d}^{p}(\hat{x}_{j},\hat{x}_{\sigma^{\omega}(j)}^{\prime}).$

Moreover, it also holds by Łoś’s theorem that

$\left(W_{p}^{\omega}([(\mu_{i})],[(\mu_{i}^{\prime})])\right)^{p}\leq\frac{1}{N}\sum_{j=1}^{N}\left(d^{\omega}(x^{\omega},x_{\tilde{\sigma}(j)}^{\prime\omega})\right)^{p}$

for any $\tilde{\sigma}\neq\sigma^{\omega}$, which implies that

$\hat{W}_{p}^{p}(\hat{\mu},\hat{\mu}^{\prime})\leq\frac{1}{N}\sum_{j=1}^{N}\hat{d}^{p}(\hat{x}_{j},\hat{x}_{\tilde{\sigma}(j)}^{\prime})$

for any $\tilde{\sigma}\neq\sigma^{\omega}$. In particular,

$\hat{W}_{p}^{p}(\hat{\mu},\hat{\mu}^{\prime})\leq\frac{1}{N}\sum_{j=1}^{N}\hat{d}^{p}(\hat{x}_{j},\hat{x}_{\sigma(j)}^{\prime})=W_{p}^{p}(\mu,\mu^{\prime}).$

But we _also_ know that

$\frac{1}{N}\sum_{j=1}^{N}\hat{d}^{p}(\hat{x}_{j},\hat{x}_{\tilde{\sigma}(j)}^{\prime})\geq
W_{p}^{p}(\mu,\mu^{\prime})$

for any $\tilde{\sigma}\neq\sigma$. Plugging in $\sigma^{\omega}$ for
$\tilde{\sigma}$ in the expression above, we conclude
$\hat{W}_{p}^{p}(\hat{\mu},\hat{\mu}^{\prime})\geq
W_{p}^{p}(\mu,\mu^{\prime})$. Hence
$\hat{W}_{p}^{p}(\hat{\mu},\hat{\mu}^{\prime})=W_{p}^{p}(\mu,\mu^{\prime})$ as
desired. ∎

######  Definition 12.

Let
$\iota:(\mathcal{P}_{p}(\hat{X}),W_{p})\hookrightarrow(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$
denote the isometric embedding constructed in the proof of the previous
theorem: namely, if $\mu\in\mathcal{P}_{p}(\hat{X})$ is a discrete measure of
the form $\mu=\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{x}_{j}}$ where
$\hat{x}_{j}=st_{\hat{d}}[(x_{j,i})]$, then

$\iota\left(\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{x}_{j}}\right)=st_{\hat{W}_{p}}\left[\left(\frac{1}{N}\sum_{j=1}^{N}\delta_{x_{j,i}}\right)_{i\in\mathbb{N}}\right].$

The map $\iota$ is then extended to all of $(\mathcal{P}_{p}(\hat{X}),W_{p})$
by density.

We have shown that
$(\mathcal{P}_{p}(\hat{X}),W_{p})\hookrightarrow(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$,
but when are the two spaces isometric? The following proposition indicates a
sufficient condition. We remark that it is also possible to prove the
proposition indirectly, by appealing to existing results on the equivalence
with the Gromov-Hausdorff limit. Nevertheless, we give such a direct,
“intrinsic”, argument below, since doing so turns out to be more informative
for us in the sequel.

Recall that a metric space $(X,d)$ is _totally bounded_ if, for every
$\varepsilon>0$, there exists some integer $k(\varepsilon)$ such that $X$ can
be covered with $k$ many $d$-balls of radius $\varepsilon$. Call
$k(\varepsilon)$ the $\varepsilon$-covering number of $X$; in other words,
$(X,d)$ is totally bounded if it has finite $\varepsilon$-covering number for
every $\varepsilon>0$.

More generally, we can talk about the $\varepsilon$-covering number of a
subset of $X$, or even a family of sets belonging to different spaces. In
particular, if $(F_{i})_{i\in I}$ is a family of sets, where each $F_{i}$ is
contained in a metric space $(X_{i},d_{i})$, we say that $(F_{i})_{i\in I}$ is
_uniformly totally bounded_ if there is a single $k(\varepsilon)$ which
simultaneously is an $\varepsilon$-covering number for all $F_{i}$’s, for
every $\varepsilon>0$.

For example, compact sets in metric spaces are totally bounded (conversely,
closed totally bounded sets are compact), and balls in infinite-dimensional
Banach spaces are not totally bounded. Note that totally bounded sets
automatically have finite diameter. They are also automatically separable:
take a sequence of small quantities $\varepsilon_{n}>0$ converging to zero;
then the set

$\bigcup_{n\in\mathbb{N}}\\{x\in X:x\text{ is a center of an
$\varepsilon_{n}$-ball in an $\varepsilon_{n}$-covering of }X\\}$

is dense, and can be taken to be countable.

######  Proposition 13.

Let $(X_{i},d_{i},e_{i})$ be a sequence of pointed metric spaces with metric
ultralimit $(\hat{X},\hat{d})$. Let $F$ be a compact set in $\hat{X}$ and let
$(F_{i})_{i\in\mathbb{N}}$ be a sequence of sets, each contained in $X_{i}$,
such that $st_{\hat{d}}(F^{\omega})=F$. Then, it automatically holds that the
sequence $(F_{i})_{i\in\mathbb{N}}$ is uniformly totally bounded
$\omega$-a.s.; and, the isometric image w.r.t. $\iota$ of the subspace
$(\mathcal{P}_{p}(F),W_{p})$ of $(\mathcal{P}_{p}(\hat{X}),W_{p})$ is
identical to $(\widehat{\mathcal{P}_{p}(F)},\hat{W}_{p})$, the metric
ultralimit of the spaces $(\mathcal{P}_{p}(F_{i}),W_{p})$.

###### ?proofname?.

$F\subseteq\hat{X}$ is compact iff it is both closed and totally bounded, the
latter meaning that for every $\varepsilon>0$ there exists some integer
$k(\varepsilon)$ such that $\hat{X}$ can be covered with $k$ many
$\hat{d}$-balls of radius $\varepsilon$ (call $k(\varepsilon)$ the
$\varepsilon$-covering number of $F$). That is, for every $\varepsilon>0$
there exist $\hat{x}_{1},\ldots,\hat{x}_{k(\varepsilon)}$ such that

$(\forall\hat{x}\in
F)\qquad\bigvee_{j=1}^{k(\varepsilon)}\hat{d}(\hat{x}_{j},\hat{x})<\varepsilon$

so we can pass to arbitrary liftings of the distinguished points
$\hat{x}_{1},\ldots,\hat{x}_{k(\varepsilon)}$ of the form
$x_{1}^{\omega},\ldots,x_{k(\varepsilon)}^{\omega}$, all belonging to
$st_{\hat{d}}^{-1}(F)$; in fact, we can always select
$x_{1}^{\omega},\ldots,x_{k(\varepsilon)}^{\omega}$ to belong to
$F^{\omega}:=[(F_{i})]$. Lifting all other points $\hat{x}\in F$, we see that

$(\forall x^{\omega}\in
st_{\hat{d}}^{-1}(F))\qquad\bigvee_{j=1}^{k(\varepsilon)}d^{\omega}(x_{j}^{\omega},x^{\omega})<\varepsilon$

so in particular

$(\forall x^{\omega}\in
F^{\omega})\qquad\bigvee_{j=1}^{k(\varepsilon)}d^{\omega}(x_{j}^{\omega},x^{\omega})<\varepsilon.$

Setting $x_{j}^{\omega}=[(x_{j,i})]$ for each
$j\in\\{1,\ldots,k(\varepsilon)\\}$, this means that for $\omega$-almost all
$i\in\mathbb{N}$,

$(\forall x_{i}\in
F_{i})\qquad\bigvee_{j=1}^{k(\varepsilon)}d_{i}(x_{j,i},x_{i})<\varepsilon.$

That is, $k(\varepsilon)$ is an $\varepsilon$-covering number for
$\omega$-almost all $F_{i}$. Quantifying over all $\varepsilon>0$, this shows
that $\omega$-almost all of the $F_{i}$’s are necessarily totally bounded,
with uniform covering number $k(\varepsilon)$.

Now, observe the metric ultraproduct of the spaces
$(F_{i}\cup\\{e_{i}\\},d_{i},e_{i})_{i\in\mathbb{N}}$ is equal to
$(\hat{F}\cup\\{\hat{e}\\},\hat{d},\hat{e})$; in what follows, we assume that
$e_{i}\in F_{i}$ $\omega$-almost surely. (There is no loss of generality,
since the ambient space $(\hat{X},\hat{d})$ is indifferent to changes of the
distinguished point that do not change which set in $X^{\omega}$ is
$X_{lim}^{\omega}$, and here we are only moving $\hat{e}$ a finite distance,
say $dist(F,\hat{e})$, hence $e^{\omega}$ a finite distance, hence
$X_{\lim}^{\omega}$ is unchanged.) At the same time, let $D>diam(F)$. Observe
that

$(\forall x^{\omega},y^{\omega}\in st_{\hat{d}}^{-1}(F))\qquad
d^{\omega}(x^{\omega},y^{\omega})<D$

This implies (since $\hat{e}\in K$, hence $e^{\omega}\in K^{\omega}$) that

$(\forall x^{\omega}\in F^{\omega})\qquad d^{\omega}(x^{\omega},e^{\omega})<D$

which in turn means that in this case, $F^{\omega}=F_{\lim}^{\omega}$, the set
of points in $F^{\omega}$ which are a finite distance from $e^{\omega}$.

Now, an inspection of [12, Proposition 1.1] indicates __3 33This is a crude
use of the error estimate from [12]: we don’t care, here, about random i.i.d.
samples, rather, we just want _any_ Dirac cloud with uniform weights
approximating a given measure. Specifically, the reason why a uniform
approximation is necessary is that, in the absence of such, it may be the case
that an arbitrary element of $\mathcal{P}_{p}(X)^{\omega}$ could only be
approximated by a “Dirac cloud” with _hyperfinitely_ many atoms, or by a Dirac
cloud where some of the weights are infinitesimal. In this case our proof
breaks down. We also remark that the paper [12] assumes that the underlying
space is Polish. This is not a problem for us, since we have restricted to a
set $F$ which is compact, hence separable. that, since the $F_{i}$’s have
$\omega$-a.s. diameter less than $D$, and some $\omega$-a.s. uniform
$\varepsilon$-covering number: for each $\delta>0$ there exists a _uniform
$N(\delta)$_ (depending only on the covering number function $k(\varepsilon)$
and $D$)__ such that for each $i\in I$ and each
$\nu_{i}\in\mathcal{P}_{p}(F_{i})$, there exists a Dirac cloud with uniform
weights
$\mu_{\delta,i}:=\frac{1}{N(\delta)}\sum_{\ell=1}^{N(\delta)}\delta_{x_{\ell,i}}$
with $W_{p}(\mu_{i},\nu_{i})<\delta$.

Therefore, let $\hat{\nu}\in(\widehat{\mathcal{P}_{p}(F)},\hat{W}_{p})$ with
representative $[(\nu_{i})]\in(\mathcal{P}_{p}(F)^{\omega},W_{p}^{\omega})$,
where the $\nu_{i}$’s are otherwise arbitrary; and given $\delta>0$, let
$\hat{\mu}_{\delta}$ denote the image of
$[(\mu_{\delta,i})]\in\mathcal{P}_{p}(F)_{\lim}^{\omega}$ in
$\widehat{\mathcal{P}_{p}(F)}$, where the $\mu_{\delta,i}$’s are as just
described. Likewise let $\mu_{\delta}\in\mathcal{P}_{p}(F)$ denote
$\frac{1}{N(\delta)}\sum_{\ell=1}^{N}\delta_{\hat{x}_{\ell}}$, where
$\hat{x}_{\ell}=st_{\hat{d}}[(x_{\ell,i})]$ for each $\ell=1,\ldots,N$. In
Theorem 11, we showed that the isometric embedding $\iota$ maps $\mu_{\delta}$
to $\hat{\mu}_{\delta}$.

We now select a decreasing sequence of small quantities $\delta_{k}>0$: for
each $i\in I$, let $(\mu_{i,\delta_{k}})$ denote a Cauchy sequence of Dirac
clouds in $(\mathcal{P}_{p}(F_{i}),W_{p})$ with $N(\delta)$ atoms converging
to $\nu_{i}$, such that $W_{p}(\mu_{i,k},\nu_{i})<\delta_{k}$. Then
$\hat{\mu}_{\delta_{k}}$ is a Cauchy sequence in
$(\widehat{\mathcal{P}_{p}(F)},\hat{W}_{p})$, converging to $\hat{\nu}$, with
the same property. But $(\mu_{\delta_{k}})$ is also a Cauchy sequence in
$(\mathcal{P}_{p}(F),W_{p})$ (call its limit $\nu$); therefore, under the
isometric embedding described in Definition 12, that $\iota\nu=\hat{\nu}$. But
since $\hat{\nu}$ was chosen arbitrarily, we conclude that the embedding is
actually a surjection. ∎

######  Remark.

In the setting of the previous proposition, there is an obvious canonical
injection of $(\widehat{\mathcal{P}_{p}(F)},\hat{W}_{p})$ into
$(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$, as follows: the space
$(\mathcal{P}(F)_{\lim}^{\omega},W_{p}^{\omega})$ is the space of internal
probability measures supported on $F^{\omega}$ with nearstandard $p$th
moments, equipped with hyperreal metric $W_{p}^{\omega}$, and this space is
contained as a subset inside $\mathcal{P}(X)_{\lim}^{\omega},W_{p}^{\omega})$.
The injection of $(\widehat{\mathcal{P}_{p}(F)},\hat{W}_{p})$ into
$(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$ is then induced according to the
following diagram:

$\textstyle{(\mathcal{P}(F)_{\lim}^{\omega},W_{p}^{\omega})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st_{\hat{W}_{p}}}$$\textstyle{\subseteq}$$\textstyle{\mathcal{P}(X)_{\lim}^{\omega},W_{p}^{\omega})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st_{\hat{W}_{p}}}$$\textstyle{(\widehat{\mathcal{P}_{p}(F)},\hat{W}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})}$

Up to this point, we have used very little of the detailed structure of the
distances $W_{p}^{\omega}$ and $\hat{W}_{p}$, and this is a good point to say
a bit more. Interpreted literally, the “fine structure” of the space
$\widehat{\mathcal{P}_{p}(X)}$ is as follows. First, a point in
$\mathcal{P}_{p}(X)^{\omega}$ is an ultraproduct of probability measures. In
other words, if $(X_{i},d_{i})$ is equipped with the Borel $\sigma$-algebra
$\mathcal{B}_{i}$ , we have
$\mu^{\omega}:\mathcal{B}^{\omega}\rightarrow[0,1]^{\omega}$ is an internal
function which is finitely additive, but not necessarily countably additive.
Second,
$\mu^{\omega}=[(\mu_{i})]\approx_{W_{p}^{\omega}}\nu^{\omega}=[(\nu_{i})]$,
and thus $\mu^{\omega}$ and $\nu^{\omega}$ get mapped to the same point in
$\widehat{\mathcal{P}_{p}(X)}$, iff there exists a sequence of couplings
$(\gamma_{i})_{i\in\mathbb{N}}$ where $\gamma_{i}\in\Pi(\mu_{i},\nu_{i})$,
such that

$\left[\left(\left(\int_{X_{i}\times
X_{i}}\left(d_{i}(x,y)\right)^{p}d\gamma_{i}(x,y)\right)^{1/p}\right)\right]\approx
0.$

In fact, it will be conceptually convenient to make the following definition.

######  Definition 14.

(Space of internal couplings) Given,
$\mu^{\omega},\nu^{\omega}\in\mathcal{P}_{p}(X)^{\omega}$, define
$\Pi^{\omega}(\mu^{\omega},\nu^{\omega})$, the space of _internal couplings_
of $\mu^{\omega}$ and $\nu^{\omega}$, as follows:

$[(\gamma_{i})]:=\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})\iff\gamma_{i}\in\Pi(\mu_{i,}\nu_{i})\text{
$\omega$-almost surely.}$

With this definition in hand, we see that
$W_{2}^{\omega}(\mu^{\omega},\nu^{\omega})\approx 0$ iff there exists an
internal coupling $\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})$
such that $\int_{X^{\omega}\times
X^{\omega}}(d^{\omega}(x,y)^{p}d\gamma^{\omega}(x,y)\approx 0$.

On the other hand, while a point in $\widehat{\mathcal{P}_{p}(X)}$ carries the
structure of an equivalency class of ultraproducts of measures that are all
$W_{2}^{\omega}$-infinitesimally close to each other, if we wanted to view a
point in $\widehat{\mathcal{P}_{p}(X)}$ _as a measure_ it is not entirely
obvious how to do this (except for the portion of
$\widehat{\mathcal{P}_{p}(X)}$ that we can identify with
$\mathcal{P}_{p}(\hat{X})$!). This is especially awkward since it is desirable
to pass measure-theoretic data between points in $\mathcal{P}_{p}(\hat{X})$
and their liftings in $\mathcal{P}_{p}(X)^{\omega}$ (or, what is much the
same, sequences of measures in the sequence of spaces
$(\mathcal{P}_{p}(X_{i}))_{i\in\mathbb{N}}$), and superficially, such data
must “go through” the space $\widehat{\mathcal{P}_{p}(X)}$ (cf. Figure
2.1(b)). It turns out that, to some extent, one can use Loeb measures as a
“workaround”, as will be explained in the remainder of this section.

|
---|---
$\textstyle{\left(X^{\omega},d^{\omega}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supseteq}$$\textstyle{\left(X_{\lim}^{\omega},d^{\omega}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{/\approx}$$\textstyle{\left(X_{i},d_{i}\right)_{i\in\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{(\hat{X},\hat{d})}$
(a)

---
$\textstyle{\left(\mathcal{P}_{p}(X)^{\omega},W_{p}^{\omega}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supseteq}$$\textstyle{\left(\mathcal{P}_{p}(X)_{\lim}^{\omega},W_{p}^{\omega}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{/\approx}$$\scriptstyle{L}$$\textstyle{\left(\mathcal{P}_{p}(X_{i}),W_{p}\right)_{i\in\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{\left(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p}\right)}$$\textstyle{\left(\mathcal{P}_{p,L}(X)_{\lim}^{\omega},W_{p,L}\right)}$$\textstyle{\left(\mathcal{P}_{p}(\hat{X}),W_{p}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$
(b)

?figurename? 2.1. Cartoons in the style of commutative diagrams, depicting
generic proof strategies when working with the metric ultraproduct
construction. (A) The arrow
$\stackrel{{\scriptstyle\omega}}{{\longrightarrow}}$ represents the
ultrafilter quotient map which takes a sequence
$(x_{i})_{i\in\mathbb{N}}\in(X_{i},d_{i})_{i\in\mathbb{N}}$ and returns a
point $x^{\omega}\in(X^{\omega},d^{\omega})$; this arrow is meant to suggest
the use of Łoś’s theorem to deduce facts about the ultraproduct
$(X^{\omega},d^{\omega})$ from the sequence of metric spaces
$(X_{i},d_{i})_{i\in\mathbb{N}}$. When the spaces $(X_{i},d_{i})$ are moreover
_pointed_ metric spaces, it makes sense to talk about the _limited_ subset
$(X_{\lim}^{\omega},d^{\omega})$ of $(X^{\omega},d^{\omega})$, as discussed in
the introduction; from $(X_{\lim}^{\omega},d^{\omega})$, we create the metric
ultralimit, $(\hat{X},\hat{d})$, by quotienting using the equivalence class
$x^{\omega}\sim y^{\omega}\iff d^{\omega}(x^{\omega},y^{\omega})\approx 0$. By
analytic study of this quotient map, we ultimately deduce facts about
$(\hat{X},\hat{d})$; this is represented by the dashed arrow
$\dashrightarrow$. (B) Here we consider ultraproducts of Wasserstein spaces.
Since Wasserstein spaces are themselves metric spaces, the situation is in
many respects similar. However, there is an additional complication, in that,
as shown in Theorem 11, the limiting space we are ultimately interested in,
namely $(\mathcal{P}_{p}(\hat{X}),W_{p})$, is not generally the metric
ultralimit of $(\mathcal{P}_{p}(X_{i}),W_{p})_{i\in\mathbb{N}}$, but rather
merely embeds isometrically (as indicated by the arrow
$\stackrel{{\scriptstyle\iota}}{{\hookrightarrow}}$) into the metric
ultralimit $(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$. Furthermore, in the
rightmost column in the diagram we include the space
$(\mathcal{P}_{p,L}(X)_{\lim}^{\omega},W_{p,L})$, which is the space of Loeb
measures associated to points in $\mathcal{P}_{p}(X)_{\lim}^{\omega}$, with a
“distance” $W_{p,L}$ defined in terms of couplings $\gamma_{L}$ which are
themselves Loeb measures, see Definition 20. Note that the space
$(\mathcal{P}_{p,L}(X)_{\lim}^{\omega},W_{p,L})$ is a space of measures with
potentially pathological “metric”, whereas
$(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$ is a metric space but whose
points do not generally have an obvious measure structure. The dashed arrow
from $(\mathcal{P}_{p}(\hat{X}),W_{p})$ to
$(\mathcal{P}_{p,L}(X)_{\lim}^{\omega},W_{p,L})$ is justified by Theorem 17.

We recall the notion of a _lifting_ from $(\hat{X},\hat{d})$ to
$(X_{\lim}^{\omega},d^{\omega})$, from the introduction. Likewise, we say that
$\mu^{\omega}\in(\mathcal{P}(X)_{\lim}^{\omega},W_{p}^{\omega})$ is a lifting
of $\mu\in(\mathcal{P}_{p}(\hat{X}),W_{p})$ provided that
$\iota\mu=st_{\hat{W}_{p}}(\mu^{\omega})$, where $\iota$ is the isometric
embedding of $(\mathcal{P}_{p}(\hat{X}),W_{p})$ into
$(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$ and $st_{\hat{W}_{p}}(\cdot)$ is
the quotient map from $(\mathcal{P}(X)_{\lim}^{\omega},W_{p}^{\omega})$ onto
$(\widehat{\mathcal{P}_{p}(X)},\hat{W}_{p})$. Dually, we say that
$\mu=\iota^{-1}st_{\hat{W}_{p}}(\mu^{\omega})$ is the _pushdown_ of
$\mu^{\omega}$ onto $(\mathcal{P}_{p}(\hat{X}),W_{p})$ provided $\mu^{\omega}$
is in the domain of $st_{\hat{W}_{p}}(\cdot)$ and
$st_{\hat{W}_{p}}(\mu^{\omega})$ is in the range of $\iota$.

At the same time, given $\mu^{\omega}\in(X_{\lim}^{\omega},d^{\omega})$, we
can construct the Loeb measure associated to $\mu^{\omega}$, denoted by
$\mu_{L}$, as discussed in Section 1. Note that $\mu_{L}$ is a (real-valued)
probability measure on $(X_{\lim}^{\omega},d^{\omega})$. Using the map
$st_{\hat{d}}:X_{\lim}^{\omega}\rightarrow\hat{X}$, we might also consider the
pushforward of $\mu_{L}$, namely $\mu_{L}\circ st_{\hat{d}}^{-1}$, which is a
probability measure on $(\hat{X},\hat{d})$, at least _provided that
$st_{\hat{d}}$ is measurable_. It has already been observed in [19] that as a
map between $X_{\lim}^{\omega}$ (with the Loeb $\sigma$-algebra
$\sigma(\mathcal{B}^{\omega})$) and $\hat{X}$ (with the Borel $\sigma$-algebra
generated by $\hat{d}$), measurability of $st_{\hat{d}}$ can _fail_ when
$\hat{X}$ is not separable; consequently, checking that $st_{\hat{d}}$ is
measurable _when restricted to the supports of $\mu_{L}$ and $\mu$ _is
necessary.

######  Lemma 15.

Suppose that $A\subset(\hat{X},\hat{d})$ is separable and closed. Let
$\widehat{\mathcal{B}}$ denote the Borel $\sigma$-algebra generated by
$\hat{d}$ on $\hat{X}$. Then
$st_{\hat{d}}:X_{\lim}^{\omega}\rightarrow\hat{X}$ is measurable with respect
to the $\sigma$-algebras $\sigma(\mathcal{B}^{\omega})$ and
$\widehat{\mathcal{B}}\cap A$.

###### ?proofname?.

The argument is a minor extension of [19, Lemma 3.1]. Indeed, let
$\\{x_{j}\\}_{i=1}^{\infty}$ be a countable dense set in $A$. Then for all
$n\in\mathbb{N}$, $A\subseteq\bigcup_{j}B(x_{j},1/n)$, so

$A\subseteq\bigcap_{n=1}^{\infty}\bigcup_{j=1}^{\infty}B(x_{j},1/n).$

On the other hand, $\overline{\\{x_{j}\\}_{j=1}^{\infty}}\subseteq A$ since
$A$ is closed. It follows that

$A=\bigcap_{n=1}^{\infty}\bigcup_{j=1}^{\infty}B(x_{j},1/n).$

Consequently, $st^{-1}(A)\in\sigma(\mathcal{B}^{\omega})$ by [19, Lemma 3.1].
Moreover, $st^{-1}(A\cap B(x,\varepsilon))\in\sigma(\mathcal{B}^{\omega})$ for
every ball $B(x,\varepsilon)$ in $\hat{X}$, since balls have
$\sigma(\mathcal{B}^{\omega})$-measurable preimages with respect to
$st(\cdot)$, again by [19, Lemma 3.1]. It follows (from the $\pi$-$\lambda$
theorem) that every set in $\widehat{\mathcal{B}}\cap A$ has a
$st(\cdot)$-preimage which is in $\sigma(\mathcal{B}^{\omega})$, since
$\widehat{\mathcal{B}}\cap A$ is countably generated by sets of the form
$B(x_{j},1/n)\cap A$. ∎

######  Remark.

Suppose $\mu$ is a Radon probability measure on $(\hat{X},\hat{d})$. Then
$supp(\mu)$ is closed and separable, so the preceding lemma shows that we can
always work with $st_{\hat{d}}^{-1}$ if we restrict to the support of some
Radon measure of interest.

In the sequel, we will sometimes want to consider the _lifting_ $\mu^{\omega}$
of a given $\mu\in(\mathcal{P}_{p}(\hat{X}),W_{p})$, but also sometimes work
with a Loeb measure $\mu_{L}$ whose $st_{\hat{d}}$-pushforward is $\mu$;
better still, if $\mu_{L}$ is the Loeb measure associated to some
$\mu^{\omega}$ which is a lifting of $\mu$. In what follows, we show that this
is all doable.

######  Lemma 16.

Suppose that $(\mu_{n})_{n\in\mathbb{N}}\in\mathcal{P}_{p}(\hat{X})$ is a
$W_{p}$-Cauchy sequence with limit $\mu$. For each $\mu_{n}$, fix a sequence
of measures $(\mu_{n,i})$ in $\mathcal{P}_{p}(X_{i})$, such that
$\iota^{-1}\circ st_{\hat{W}_{p}}[(\mu_{n,i})]=\mu_{n}$. Then the following
hold:

  1. (1)

For every $\varepsilon>0$, there exists an $N(\varepsilon)$ such that for all
$n,m\geq N(\varepsilon)$, and $\omega$-almost all $i\in\mathbb{N}$,
$W_{p,i}(\mu_{n,i},\mu_{m,i})<\varepsilon.$ In fact $N(\varepsilon)$ is the
modulus of convergence of $(\mu_{n})$ in $(\mathcal{P}_{p}(\hat{X}),W_{2})$.

  2. (2)

(Diagonal sequence) Let $N(\varepsilon)$ be the modulus of convergence from
(1). Then, $[(\mu_{N(1/i),i})]\in\mathcal{P}_{p}(X)^{\omega}$ is a lifting of
$\mu$. Moreover, so is any sequence $[(\mu_{\Phi(i),i})]$ where $\Phi(i)\geq
N(1/i)$ for $\omega$-almost all $i\in\mathbb{N}$.

###### ?proofname?.

(1) First, it is clear from construction that
$W_{p}(\mu_{n},\mu_{m})<\varepsilon$ iff the same holds for any liftings
$\mu_{n}^{\omega}=[(\mu_{n,i})]$ and $\mu_{m}^{\omega}=[(\mu_{m,i})]$ in
$\mathcal{P}_{p}(X)^{\omega}$. And this in turn, holds iff (1) holds.

(2) In what follows, we take the function $N(\varepsilon)$ from (1) to be
nondecreasing as $\varepsilon\rightarrow 0$, without loss of generality.

Consider the diagonal sequence $(\mu_{N(1/i),i})$ of measures. Fix
$\varepsilon>0$ arbitrary. Let $i_{0}$ be sufficiently large that
$\varepsilon>1/i_{0}$. Then, note that if $i\geq i_{0}$, it holds that if
$n,m\geq N(1/i)$, then also $n,m\geq N(1/i_{0})$. So since

$\\{i\in\mathbb{N}\mid\forall n,m\geq
N(1/i_{0}),W_{p,i}(\mu_{n,i},\mu_{m,i})<1/i_{0}\\}\in\omega$

it holds that

$\\{i\in\mathbb{N}\mid i\geq i_{0}\\}\cap\\{i\in\mathbb{N}\mid\forall n,m\geq
N(1/i_{0}),W_{p,i}(\mu_{n,i},\mu_{m,i})<1/i_{0}\\}\in\omega$

and since

$\\{i\in\mathbb{N}\mid i\geq i_{0}\wedge\forall n,m\geq
N(1/i_{0}),W_{p,i}(\mu_{n,i},\mu_{m,i})<1/i_{0}\\}\subseteq\\\
\\{i\in\mathbb{N}\mid i\geq i_{0}\wedge\forall n\geq N(1/i_{0})\forall m\geq
N(1/i),W_{p,i}(\mu_{n,i},\mu_{m,i})<1/i_{0}\\}$

it also holds, for $\omega$-almost all $i\in\mathbb{N}$, that
$W_{p,i}(\mu_{n,i},\mu_{m,i})<1/i_{0}$ whenever $n\geq N(1/i_{0})$ and $m\geq
N(1/i)$. In particular, it holds $\omega$-almost surely that

$\forall n\geq N(1/i_{0}),W_{p,i}(\mu_{n,i},\mu_{N(1/i),i})<1/i_{0}.$

Consequently, by construction, we have that

$\forall n\geq
N(1/i_{0}),W_{p}^{\omega}([(\mu_{n,i})],[(\mu_{N(1/i),i})])<1/i_{0}.$

Now, let $\mu$ be the limit of $(\mu_{n})$ in
$(\mathcal{P}_{p}(\hat{X}),W_{p})$, and let
$[(\mu_{-,i})]\in\mathcal{P}(X)^{\omega}$ be any lifting of $\mu$. Note that
since $N(\varepsilon)$ is the modulus of convergence for $(\mu_{n})$, we have
that

$\forall n\geq
N(1/i_{0}),W_{p}^{\omega}([(\mu_{n,i})],[(\mu_{-,i})])<1/i_{0}.$

Consequently, by the triangle inequality in
$(\mathcal{P}_{p}(X)^{\omega},W_{p}^{\omega})$,

$W_{p}^{\omega}([(\mu_{-,i})],[(\mu_{N(1/i),i})])<2/i_{0}.$

But our choice of $\varepsilon>0$ (and thus $i_{0}$) was arbitrary, hence
$W_{p}^{\omega}([(\mu_{-,i})],[(\mu_{N(1/i),i})])$ is smaller than every
positive standard real, hence
$W_{p}^{\omega}([(\mu_{-,i})],[(\mu_{N(1/i),i})])\approx 0$. Thus,
$[(\mu_{N(1/i),i)})]$ is _also_ a lifting of $\mu$.

The exact same reasoning also allows us to pick a different $n_{i}\geq N(1/i)$
for each $i$, so the remaining claim in the statement of (2) holds. ∎

######  Theorem 17.

Let $p\in[1,\infty)$. Let $\mu$ be a Radon measure on $\hat{X}$, with
$\mu\in\mathcal{P}_{p}(\hat{X})$. Then there exists a Loeb measure $\mu_{L}$
on $X_{\lim}^{\omega}$ whose $st_{\hat{d}}$-pushforward is $\mu$. Moreover, we
can take $\mu_{L}$ to be the Loeb measure associated to a lifting
$\mu^{\omega}$ of $\mu$ which is hyperfinite (i.e. an ultraproduct of discrete
measures).

###### ?proofname?.

Let $(\mu_{n})$ be a sequence of discrete probability measures converging in
$W_{p}$ to $\mu\in\mathcal{P}_{p}(\hat{X})$, such that $supp(\mu_{n})\subseteq
supp(\mu)$, and such that each $\mu_{n}$ has the form

$\mu_{n}=\frac{1}{k_{n}}\sum_{j=1}^{k_{n}}c_{j,n}\delta_{\hat{x}_{j,n}};\quad\hat{x}_{j,n}\in
supp(\mu).$

For example, one might take $\mu_{n}$ to be the $n$th empirical measure for
$\mu$ (in this case $c_{j,n}=1$). We let $N(\varepsilon)$ denote the modulus
of convergence for $(\mu_{n})_{n\in\mathbb{N}}$ in
$(\mathcal{P}_{p}(\hat{X}),W_{p})$. For each $\hat{x}_{j,n}$, pick a sequence
$(x_{j,n,i})$ of points in $(X_{i},d_{i})$ so that
$st_{\hat{d}}[(x_{j,n,i})]=\hat{x}_{j,n}$. Define the lifting of $\mu_{n}$ to
an internal measure supported within $X_{\lim}^{\omega}$ like so:

$\tilde{\mu}_{n}=\frac{1}{k_{n}}\sum_{j=1}^{k_{n}}c_{j,n}\delta_{[(x_{j,n,i})]}:=\left[\left(\frac{1}{k_{n}}\sum_{j=1}^{k_{n}}c_{j,n}\delta_{x_{j,n,i}}\right)_{i\in\mathbb{N}}\right]:=[(\tilde{\mu}_{n,i})].$

Let $\mathcal{B}^{\omega}$ denote the internal measure algebra of internal
Borel sets. For trivial reasons, $\tilde{\mu}_{n}$ extends directly to
$\sigma(\mathcal{B}^{\omega})$ from $\mathcal{B}^{\omega}$, and only assigns
standard real measures to sets, so $\tilde{\mu}_{n}$ may be easily confused
with its associated Loeb measure $\tilde{\mu}_{n,L}$ (but we resist abusive
identification).

Let $B$ be any Borel set in $\hat{X}\cap supp(\mu)$. It is clear that
$\mu_{n}(B)=\tilde{\mu}_{n,L}(st^{-1}(B))$, simply because the locations of
the constituent atoms of $\tilde{\mu}_{n}$ (and thus $\tilde{\mu}_{n,L}$)
correspond exactly to those of $\mu_{n}$, up to an arbitrary selection of a
point in each pre-image $st_{\hat{d}}^{-1}\\{\hat{x}_{j,n}\\}$. At the same
time, $\mu_{n}(B)\rightarrow\mu(B)$ for every continuity set
$B\subset\hat{X}\cap supp(\mu)$ (that is, a Borel set in $\hat{X}\cap
supp(\mu)$ with $\mu(\partial B)=0$); this follows from the fact that the
topology induced by $W_{p}$ is stronger than that of the weak convergence of
probability measures. In particular, for each such $B$, the sequence
$\\{\tilde{\mu}_{n,L}(st^{-1}(B))\\}_{n\in\mathbb{N}}$ is Cauchy, since
$\mu_{n}(B)=\tilde{\mu}_{n,L}(st^{-1}(B))$ for each $n\in\mathbb{N}$.

It would be nice to proceed as follows: passing to a nonstandard extension of
the sequence of objects $\tilde{\mu}_{n}$, we find that
$\tilde{\mu}_{N}(st^{-1}(B))\approx\tilde{\mu}_{M}(st^{-1}(B))$ for all
strictly hyperfinite $N,M$, and moreover
$\tilde{\mu}_{N}(st^{-1}(B))\approx\mu(B)$; and thus it suffices to take the
Loeb measure associated to any such internal measure $\tilde{\mu}_{N}$ as a
representative of $\mu$. However, we have not developed the theory of
nonstandard extensions in this article, and so in the interest of self-
containment, we instead reason by way of the diagonal lifting construction
from the previous lemma.

To wit, let $B(x_{0},\delta)\cap supp(\mu)$ be an open ball in the restriction
of $(\hat{X},\hat{d})$ to the support of $\mu$ which is a continuity set for
$\mu$ (so in particular, $\mu(\partial B(x_{0},\delta))=0$). Let

$\Psi_{(x_{0},\delta)}(\varepsilon):=\underset{N\in\mathbb{N}}{\text{argmin}}\left\\{(\forall
n,m\geq N)|\mu_{n}(B(x_{0},\delta)\cap supp(\mu))-\mu_{m}(B(x_{0},\delta)\cap
supp(\mu))|<\varepsilon\right\\}.$

In other words, $\Psi_{(x_{0},\delta)}(\varepsilon)$ is a modulus of
convergence for the sequence $(\mu_{n}(B(x_{0},\delta)\cap
supp(\mu)))_{n\in\mathbb{N}}$. From part (2) of Lemma 16, we know that if
$\Phi:\mathbb{N}\rightarrow\mathbb{N}$ is greater, $\omega$-a.s., than
$(\max\\{N(1/i),\Psi_{(x_{0},\delta)}(1/i)\\})_{i\in\mathbb{N}}$, it then it
holds that $[(\mu_{\Phi(i)})]$ is a lifting of $\mu$. At the same time,
letting $\mu_{\Phi,L}:=\text{Loeb}[(\mu_{\Phi})]$, we claim that
$\mu_{\Phi,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu)))=\mu(B(x_{0},\delta)\cap supp(\mu))$.

Let us demonstrate why the claim holds. Fix $\varepsilon>0$. Then, if
$n,m\geq\Psi_{(x_{0},\delta)}(\varepsilon)$, it holds that

$|\mu_{n}(B(x_{0},\delta)\cap supp(\mu))-\mu_{m}(B(x_{0},\delta)\cap
supp(\mu))|<\varepsilon$

and hence

$|\tilde{\mu}_{n,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu)))-\tilde{\mu}_{m,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu)))|<\varepsilon.$

Since for each $n\in\mathbb{N}$, $\mu_{n}$ is supported on a finite number of
points, it follows that there is some minimum distance between the boundary
$\partial(B(x_{0},\delta)\cap supp(\mu))$ and any atom in $\mu_{n}$.
Consequently, if $\eta:=[(\eta_{i})]$ is any positive infinitesimal, then
(since $st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu))=st_{\hat{d}}^{-1}(B(x_{0},\delta))\cap st_{\hat{d}}^{-1}supp(\mu)$)

$\tilde{\mu}_{n,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu)))=\tilde{\mu}_{n,L}(B(x_{0}^{\omega},\delta+\eta)\cap
st_{\hat{d}}^{-1}supp(\mu));$
$\tilde{\mu}_{m,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu)))=\tilde{\mu}_{m,L}(B(x_{0}^{\omega},\delta+\eta)\cap
st_{\hat{d}}^{-1}(supp(\mu))$

where $x_{0}^{\omega}$ is any lifting of $x_{0}$, and

$B(x_{0}^{\omega},\delta+\eta):=\\{x^{\omega}\in X^{\omega}\mid
d^{\omega}(x_{0}^{\omega},x^{\omega})<\delta+\eta\\}.$

Note that $B(x_{0}^{\omega},\delta+\eta)$ is the ultraproduct of the sequence
of open balls $B(x_{0,i},\delta+\eta_{i})\subseteq X_{i}$.

Let $S^{\omega}:=[(S_{i})]$ be any internal measurable set containing
$st_{\hat{d}}^{-1}supp(\mu)$. We deduce (using the fact that $\tilde{\mu}_{n}$
has no mass outside of $st_{\hat{d}}^{-1}supp(\mu)$) that

$(\forall
n,m\geq\Phi_{(x_{0},\delta)}(\varepsilon))|\tilde{\mu}_{n}(B(x_{0}^{\omega},\delta+\eta)\cap
S^{\omega})-\tilde{\mu}_{m}(B(x_{0}^{\omega},\delta+\eta)\cap
S^{\omega})|<\varepsilon.$

In turn, this holds iff

$\\{i\in\mathbb{N}\mid(\forall
n,m\geq\Phi_{(x_{0},\delta)}(\varepsilon))|\tilde{\mu}_{n,i}(B(x_{0,i},\delta+\eta_{i})\cap
S_{i})-\tilde{\mu}_{m,i}(B(x_{0,i},\delta+\eta_{i})\cap
S_{i})|<\varepsilon\\}\in\omega.$

It follows, by the same reasoning as in the proof of part 2 of Lemma 16, that
if $\Phi(i)\geq\Phi_{(x_{0},\delta)}(1/i)$, it holds that

$\\{i\in\mathbb{N}\mid
n\geq\Phi_{(x_{0},\delta)}(\varepsilon)|\tilde{\mu}_{\Phi(i),i}(B(x_{0,i},\delta+\eta_{i})\cap
S_{i})-\tilde{\mu}_{n,i}(B(x_{0,i},\delta+\eta_{i})\cap
S_{i})|<\varepsilon\\}\in\omega$

and therefore,

$|\tilde{\mu}_{\Phi}(B(x_{0}^{\omega},\delta+\eta)\cap
S^{\omega})-\tilde{\mu}_{n}(B(x_{0}^{\omega},\delta+\eta)\cap
S^{\omega})|<\varepsilon.$

It follows that

$|\tilde{\mu}_{\Phi}(B(x_{0}^{\omega},\delta+\eta)\cap
S^{\omega})-\tilde{\mu}_{n,L}(B(x_{0}^{\omega},\delta+\eta)\cap
S^{\omega})|<\varepsilon$

and hence

$|\tilde{\mu}_{\Phi}(B(x_{0}^{\omega},\delta+\eta)\cap
S^{\omega})-\tilde{\mu}_{n,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu)))|<\varepsilon.$

Sending $\varepsilon\rightarrow 0$, we deduce that for any infinitesimal
$\eta>0$,

$\tilde{\mu}_{\Phi}(B(x_{0}^{\omega},\delta+\eta)\cap
S^{\omega})\approx\lim_{n\rightarrow\infty}\tilde{\mu}_{n,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu))).$

Now, we can consider $\tilde{\mu}_{\Phi}(B(x_{0}^{\omega},\delta+\eta)\cap
S^{\omega})$ as an internal function of $\eta$, that is, from
$\mathbb{R}_{+}^{\omega}$ to $\mathbb{R}_{+}^{\omega}$; it follows from the
overspill principle [24, Chapter 11; viz. Theorem 11.9.1] that for any
$\varepsilon_{0}\in\mathbb{R}_{+}$, there exists some
$\eta_{0}\in\mathbb{R}_{+}$ such that for all positive hyperreal
$\eta<\eta_{0}$,

$|\tilde{\mu}_{\Phi}(B(x_{0}^{\omega},\delta+\eta)\cap
S^{\omega})-\lim_{n\rightarrow\infty}\tilde{\mu}_{n,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu)))|<\varepsilon_{0}$

as well, and hence

$|\tilde{\mu}_{\Phi,L}(B(x_{0}^{\omega},\delta+\eta)\cap
S^{\omega})-\lim_{n\rightarrow\infty}\tilde{\mu}_{n,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu)))|<\varepsilon_{0}.$

Pick $\eta_{1}\approx 0$ and $\eta_{2}=\frac{\eta_{0}}{2}$; using the fact
that

$\left(B(x_{0}^{\omega},\delta+\eta_{1})\cap
S^{\omega}\right)\subset\left(st_{\hat{d}}^{-1}(\bar{B}(x_{0},\delta))\cap
S^{\omega}\right)\subset\left(B(x_{0}^{\omega},\delta+\eta_{2})\cap
S^{\omega}\right)$

we deduce that

$|\tilde{\mu}_{\Phi,L}(st_{\hat{d}}^{-1}(\bar{B}(x_{0},\delta))\cap
S^{\omega})-\lim_{n\rightarrow\infty}\tilde{\mu}_{n,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu)))|<\varepsilon_{0}.$

Since $\varepsilon_{0}>0$ was arbitrary, we deduce that

$\tilde{\mu}_{\Phi,L}(st_{\hat{d}}^{-1}(\bar{B}(x_{0},\delta))\cap
S^{\omega})=\lim_{n\rightarrow\infty}\tilde{\mu}_{n,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu)))$

But since $\mu(\delta B(x_{0},\delta))=0$, it holds that

$\lim_{n\rightarrow\infty}\tilde{\mu}_{n,L}(st_{\hat{d}}^{-1}(B(x_{0},\delta)\cap
supp(\mu)))=\lim_{n\rightarrow\infty}\tilde{\mu}_{n,L}(st_{\hat{d}}^{-1}(\bar{B}(x_{0},\delta)\cap
supp(\mu))).$

Finally, we select $S^{\omega}$ so that
$\tilde{\mu}_{\Phi,L}(st_{\hat{d}}^{-1}supp(\mu)\Delta S^{\omega})=0$. Hence,

$\displaystyle\tilde{\mu}_{\Phi,L}(st_{\hat{d}}^{-1}(\bar{B}(x_{0},\delta))\cap
S^{\omega})$
$\displaystyle=\tilde{\mu}_{\Phi,L}(st_{\hat{d}}^{-1}(\bar{B}(x_{0},\delta))\cap
st_{\hat{d}}^{-1}supp(\mu))$
$\displaystyle=\tilde{\mu}_{\Phi,L}(st_{\hat{d}}^{-1}(\bar{B}(x_{0},\delta)\cap
supp(\mu)))$

and so $\tilde{\mu}_{\Phi,L}\circ
st_{\hat{d}}^{-1}(B(x_{0},\delta))=\mu(B(x_{0},\delta))$, and the claim is
proved.

We finish the proof by considering a countable family of continuity sets for
$\mu$ of the form $B(x_{0},\delta)\cap supp(\mu)$ which generate the
$\sigma$-algebra of Borel sets in the support of $\mu$; such a countable
family exists since $\mu$ has separable support. Selecting a function
$\tilde{\Phi}(i)$ which is $\omega$-almost surely larger than $N(\varepsilon)$
as well as the convergence modulus $\Phi_{B(x_{0},\delta)\cap supp(\mu)}$ for
_every_ $B(x_{0},\delta)$ in the countable family (for example, take a
diagonalization of all these convergence moduli), we conclude that
$\tilde{\mu}_{\tilde{\Phi},L}\circ st_{\hat{d}}^{-1}=\mu$. ∎

In fact, it is possible to modify the preceding argument to get a
superficially stronger result, namely:

######  Corollary 18.

Let $\mu$ be an $\sigma$-finite Radon measure on $\hat{X}$ (that is, $\mu$ can
be written as a countable sum of Radon probability measures). Then, there
exists an internal measure $\mu^{\omega}$ on $X^{\omega}$ whose Loeb measure
$\mu_{L}$ pushes onto $\mu$.

###### ?proofname?.

(sketch) To drop the assumption that $\mu\in\mathcal{P}_{p}(\hat{X})$, we
simply replace the metric structure of $W_{p}$ with one that is well-defined
on all of $\mathcal{P}(\hat{X})$ and has the property that discrete measures
are dense, such as the Lévy-Prokhorov metric $d_{LP}$; that is, one considers
the spaces $(\mathcal{P}(\hat{X}),d_{LP})$ and
$(\mathcal{P}(X)^{\omega},d_{LP}^{\omega})$. The argument from the preceding
lemma and theorem then goes through otherwise unchanged, and allows us to
deduce a version of the previous theorem for any Radon probability measure on
$\hat{X}$. The case of $\sigma$-finite Radon measures on $\hat{X}$ then
follows immediately by globalization. ∎

At this point, is worth briefly touching on “what goes wrong” for the space
$(\widehat{\mathcal{P}_{p}(X)},\widehat{W}_{p})$ to be strictly bigger than
$(\mathcal{P}_{p}(\hat{X}),W_{p})$ when $\hat{X}$ is not compact, since we are
able to do so now. Consider, in particular, the case where $(X_{i},d_{i})$ is
a constant sequence where each $(X_{i},d_{i})$ is $\mathbb{R}^{d}$ with the
Euclidean norm. The _concentration-compactness_ phenomenon is that, if we have
a sequence of (Radon) measures on $\mathbb{R}^{d}$, one of three things can
happen (up to a subsequence): the sequence is _tight_ , the sequence _diffuses
towards infinity_ , or only a _fraction_ of the mass is tight and the
remainder of the mass has more complicated behavior. In the case where a
sequence of measures $(\mu_{i})$ is asymptotically diffuse, or even a finite
fraction of the mass diffuses towards infinity,, we have, in the ultraproduct,
that $W_{p}^{\omega}(\delta_{e}^{\omega},\mu^{\omega})$ is a strictly
hyperfinite hyperreal quantity (because the $p$th moment is a strictly
hyperreal quantity), and so $(\mu_{i}^{\omega})$ is excluded from the domain
when we construct $\widehat{\mathcal{P}_{p}(X)}$. However, even when
$W_{p}^{\omega}(\delta_{e_{i}}^{\omega},\mu_{i}^{\omega})$ is finite, it is
still possible that
$st_{\hat{W}_{p}}(\mu^{\omega})\in\widehat{\mathcal{P}_{p}(X)}$ does not lie
in (the embedded image of) $(\mathcal{P}_{p}(\hat{X}),W_{p})$, as the next
example illustrates.

######  Example 19.

(Prototypical instance showing that
$(\mathcal{P}_{2}(\hat{X}),W_{2})\subsetneq(\widehat{\mathcal{P}_{2}(X)},\hat{W}_{2})$).
Consider the case where, in the construction of the ultralimit
$(\hat{X},\hat{d})$, we take $X_{i}=\mathbb{R}$ for all $i\in\mathbb{N}$, with
the usual metric. Note that in this situation, $\hat{X}=\mathbb{R}$. Obviously
part (3) of the preceding theorem does not apply to this case since
$\mathbb{R}$ is not compact. We consider the (ultraproducts of) measures
$\delta_{0}$ and $(1-N^{-1})\delta_{0}+N^{-1}\delta_{N^{1/2}}$ in
$\mathcal{P}(\mathbb{R})^{\omega}$, with
$N=[(N_{i})]\in\mathbb{N}^{\omega}\backslash\mathbb{N}$. Then,

$W_{2}^{\omega}(\delta_{0},(1-N^{-1})\delta_{0}+N^{-1}\delta_{N^{1/2}})=\left(N^{-1}d^{2}(0,N^{1/2})\right)^{1/2}=1.$

More generally, the measure $(1-N^{-1})\delta_{0}+N^{-1}\delta_{N^{1/2}}$ is a
finite distance away from _any_ measure in $\mathcal{P}(\mathbb{R})^{\omega}$
which is supported inside $\mathbb{R}_{\lim}^{\omega}$. Indeed, let
$\mu^{\omega}=[(\mu_{i})]$ be supported within $[-R,R]$ where
$R\in\mathbb{R}$; note that in this case, $\mu_{i}$ is supported within
$[-R,R]$ $\omega$-almost surely. Then, by definition.

$W_{2}^{\omega}(\mu^{\omega},(1-N^{-1})\delta_{0}+N^{-1}\delta_{N^{1/2}})=\left[\left(\left(\inf_{\gamma_{i}\in\Pi(\mu_{i},(1-N_{i}^{-1})\delta_{0}+N_{i}^{-1}\delta_{N_{i}^{1/2}}}\int|x-y|^{2}d\gamma_{i}(x,y)\right)^{1/2}\right)_{i\in\mathbb{N}}\right].$

Observe that for each $i\in\mathbb{N}$, the atom
$N_{i}^{-1}\delta_{N_{i}^{1/2}}$ must be transported into the region $[-R,R]$,
at cost bounded between $N_{i}^{-1/2}|N_{i}^{1/2}+R|$ and
$N_{i}^{-1/2}|N_{i}^{1/2}-R|$. But $|N_{i}|\rightarrow\infty$, so in
particular,

$W_{2}^{\omega}(\mu^{\omega},(1-N^{-1})\delta_{0}+N^{-1}\delta_{N^{1/2}})\geq\left[\left(\min\left\\{N_{i}^{-1/2}|N_{i}^{1/2}+R|,N_{i}^{-1/2}|N_{i}^{1/2}-R|\right\\}\right)_{i\in\mathbb{N}}\right]\approx
1.$

On the other hand, since
$W_{2}^{\omega}(\delta_{0},(1-N^{-1})\delta_{0}+N^{-1}\delta_{N^{1/2}})=1$, we
know that
$(1-N^{-1})\delta_{0}+N^{-1}\delta_{N^{1/2}}\in(\mathcal{P}(\mathbb{R})_{\lim}^{\omega},W_{2}^{\omega})$.
At the same time, given any $\nu\in(\mathcal{P}_{2}(\mathbb{R}),W_{2})$ with
compact support (say inside $[-R,R]$), we can then consider a lifting
$\nu^{\omega}$ of $\nu$ which (in accordance with Proposition 13) is supported
inside a totally bounded set which pushes down to $[-R,R]$, hence also has
bounded diameter. This implies that

$W_{2}^{\omega}(\nu^{\omega},(1-N^{-1})\delta_{0}+N^{-1}\delta_{N^{1/2}})\gtrapprox
1.$

Hence, (letting $\hat{\nu}=\iota\nu=st_{\hat{W}_{2}}(\nu^{\omega})$)

$\hat{W}_{2}(\hat{\nu},\widehat{(1-N^{-1})\delta_{0}+N^{-1}\delta_{N^{1/2}}})\geq
1.$

Since measures with compact support are dense inside
$(\mathcal{P}_{2}(\mathbb{R}),W_{2})$, it follows, by density, that for any
$\nu$ in $(\mathcal{P}_{2}(\mathbb{R}),W_{2})$,

$\hat{W}_{2}(\hat{\nu},\widehat{(1-N^{-1})\delta_{0}+N^{-1}\delta_{N^{1/2}}})\geq
1$

as well.

######  Remark.

There are two other reasons why it may be the case that
$\hat{\mu}\in\widehat{\mathcal{P}_{p}(X)}\backslash\iota\mathcal{P}_{p}(\hat{X})$.
The first is that, given a $\mu^{\omega}\in\mathcal{P}_{p}(X)^{\omega}$ and
its associated Loeb measure $\mu_{L}$, then the $st_{\hat{d}}$-pushforward of
$\mu_{L}$, because of failure of the measurability of $st_{\hat{d}}$ on the
support of $\mu_{L}$. If this occurs for an _entire_ equivalency class
$\hat{\mu}$ in $\widehat{\mathcal{P}_{p}(X)}$, then no element in that
equivalency class is a lifting of a measure $\mu\in\mathcal{P}_{p}(\hat{X})$,
so in particular $\hat{\mu}\notin\iota\mathcal{P}_{p}(\hat{X})$. In principle,
it is also possible that we _do_ have $st_{\hat{d}}$-measurability without
separable support on $\hat{X}$; in this case, $\mu_{L}$ still pushes forward
to a measure on $\hat{X}$ but since the support is not separable, $\mu$ is not
Radon, hence excluded from $\mathcal{P}_{p}(\hat{X})$ also. (Note however that
this occurrence is independent of ZFC! Indeed, the existence of a non-Radon
probability measure on a metric space require a large cardinal axiom, as
discussed above.)

######  Definition 20.

Let $\mu^{\omega},\nu^{\omega}\in\mathcal{P}_{p}(X)_{\lim}^{\omega}$, and let
$\mu_{L}$ and $\nu_{L}$ be their associated Loeb measures. Let

$\Pi_{L}(\mu_{L},\nu_{L}):=\\{\gamma_{L}\in\mathcal{P}(X_{\lim}^{\omega}\times
X_{\lim}^{\omega})\mid\\\
\exists\mu^{\prime\omega},\nu^{\prime\omega}\in\mathcal{P}_{p}(X)^{\omega}\exists\gamma^{\omega}\in\Pi^{\omega}(\mu^{\prime\omega},\nu^{\prime\omega})\left[\gamma_{L}=\text{Loeb}(\gamma^{\omega})\wedge\mu_{L}^{\prime}=\mu_{L}\wedge\nu_{L}^{\prime}=\nu_{L}\right]\\}.$

In other words, $\Pi_{L}(\mu_{L},\nu_{L})$ is the space of couplings between
$\mu_{L}$ and $\nu_{L}$ that arise as the Loeb measures of internal couplings
between internal measures $\mu^{\prime\omega}$ and $\nu^{\prime\omega}$ with
the same associated Loeb measures as $\mu^{\omega}$ and $\nu^{\omega}$.
Likewise, define

$W_{p,L}(\mu_{L},\nu_{L}):=\left(\inf_{\gamma_{L}\in\Pi_{L}(\mu_{L},\nu_{L})}\int_{X_{\lim}^{\omega}\times
X_{\lim}^{\omega}}st\left(\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}\right)d\gamma_{L}(x^{\omega},y^{\omega})\right)^{1/p}.$

######  Remark.

Let $B^{\omega}\in\mathcal{B}^{\omega}$ be any internal Borel set on
$X^{\omega}$ contained within $X_{\lim}^{\omega}$. Then, $1_{B^{\omega}\times
X^{\omega}}$ is an $S$-integrable function w.r.t. the measure
$\gamma^{\omega}$. Hence,

$\int 1_{B^{\omega}\times X^{\omega}}d\gamma_{L}=st\left(\int
1_{B^{\omega}\times
X^{\omega}}d\gamma^{\omega}\right)=st\left(\mu^{\omega}(B^{\omega})\right)=\mu_{L}(B^{\omega}).$

It follows that the first marginal of $\gamma_{L}$ agrees with $\mu_{L}$ on
all internally measurable sets, hence on all of the Loeb measurable sets. The
same reasoning applies to the second marginal also. Hence, $\gamma_{L}$ is
indeed a coupling between $\mu_{L}$ and $\nu_{L}$; in particular, if
$\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})$ then also
$\gamma_{L}\in\Pi_{L}(\mu_{L},\nu_{L})$.

#### Warning.

The space of Loeb probability measures on $X_{\lim}^{\omega}$, equipped with
$W_{p,L}$, is not obviously a well-behaved space. For one thing, $W_{p,L}$
does not separate points: if we pick two Dirac measures whose atoms are
infinitesimally far apart, these are two different Loeb measures but their
$W_{p,L}$ distance is zero. Yet, the space is certainly not just a
“pseudometric analogue” of the space $(\mathcal{P}_{p}(\hat{X}),W_{p})$, since
there may be Loeb measures on $X_{\lim}^{\omega}$ that do not push forward to
$\hat{X}$. For this reason, we will use $W_{p,L}$ only as an intermediate
calculating device, and avoid working with it directly.

######  Proposition 21.

With the same notation as the previous definition,

$W_{p,L}(\mu_{L},\nu_{L})\lessapprox
W_{p}^{\omega}(\mu^{\omega},\nu^{\omega}).$

###### ?proofname?.

Let $\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})$. Since the
function $\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}$ is
$S$-integrable on $X_{\lim}^{\omega}$, it follows that

$st\left(\int_{X^{\omega}\times
X^{\omega}}\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}d\gamma^{\omega}(x^{\omega},y^{\omega})\right)=\int_{X_{\lim}^{\omega}\times
X_{\lim}^{\omega}}st\left(\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}\right)d\gamma_{L}(x^{\omega},y^{\omega}).$

By quantifying over all
$\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})$, we see that

$\inf_{\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})}st\left(\int_{X^{\omega}\times
X^{\omega}}\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}d\gamma^{\omega}(x^{\omega},y^{\omega})\right)\\\
=\inf_{\gamma_{L}:\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})}\int_{X_{\lim}^{\omega}\times
X_{\lim}^{\omega}}st\left(\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}\right)d\gamma_{L}(x^{\omega},y^{\omega}).$

Since (from the preceding remark)
$\\{\gamma_{L}:\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})\\}\subseteq\Pi_{L}(\mu_{L},\nu_{L})$,
it follows that

$\inf_{\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})}st\left(\int_{X^{\omega}\times
X^{\omega}}\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}d\gamma^{\omega}(x^{\omega},y^{\omega})\right)\\\
\geq\inf_{\gamma_{L}\in\Pi_{L}(\mu_{L},\nu_{L})}\int_{X_{\lim}^{\omega}\times
X_{\lim}^{\omega}}st\left(\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}\right)d\gamma_{L}(x^{\omega},y^{\omega}):=W_{p,L}^{p}(\mu_{L},\nu_{L}).$

On the other hand, for each
$\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})$, it holds (by
definition of $st$) that

$st\left(\int_{X^{\omega}\times
X^{\omega}}\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}d\gamma^{\omega}(x^{\omega},y^{\omega})\right)\approx\int_{X^{\omega}\times
X^{\omega}}\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}d\gamma^{\omega}(x^{\omega},y^{\omega})$

which implies that that

$\displaystyle\inf_{\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})}st\left(\int_{X^{\omega}\times
X^{\omega}}\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}d\gamma^{\omega}(x^{\omega},y^{\omega})\right)$
$\displaystyle\approx\inf_{\gamma^{\omega}\in\Pi^{\omega}(\mu^{\omega},\nu^{\omega})}\int_{X^{\omega}\times
X^{\omega}}\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}d\gamma^{\omega}(x^{\omega},y^{\omega})$
$\displaystyle=\left(W_{p}^{\omega}(\mu^{\omega},\nu^{\omega})\right)^{p}.$

Hence, $W_{p,L}(\mu_{L},\nu_{L})\lessapprox
W_{p}^{\omega}(\mu^{\omega},\nu^{\omega})$. ∎

######  Lemma 22.

Let $\mu_{L}$ be a Loeb probability measure on $X_{\lim}^{\omega}$. Then the
$st_{\hat{d}}$-pushforward of $\mu_{L}$ is well-defined and belongs to
$\mathcal{P}(\hat{X})$ iff for every $\varepsilon>0$, there exists a compact
$K_{\varepsilon}\subseteq\hat{X}$ such that
$\mu_{L}(st_{\hat{d}}^{-1}(K_{\varepsilon}))>1-\varepsilon$.

######  Remark.

Combining this lemma with Proposition 13, we see that
$[(\mu_{i})]=\mu^{\omega}\in\mathcal{P}^{\omega}(X^{\omega})$ has a Loeb
measure $\mu_{L}$ whose $st_{\hat{d}}$-pushforward is well-defined and belongs
to $\mathcal{P}(\hat{X})$, if and only if:
$\mu^{\omega}\in\mathcal{P}^{\omega}(X_{\text{lim}}^{\omega})$, and
$\omega$-almost surely, for every $\varepsilon>0$ the probability measures
$\mu_{i}$ are concentrated on sets $F_{\varepsilon,i}$ which are uniformly
totally bounded (meaning $\mu_{i}(F_{\varepsilon,i})>1-\varepsilon$). This
provides an explicit criterion for the satisfaction of condition (1) in
Definition 9 above. (Compare also [38, Theorem 8.3].)

###### ?proofname?.

One direction is obvious: if there exists a Radon measure $\mu$ such that
$\mu_{L}\circ st_{\hat{d}}^{-1}=\mu$, then we just use the tightness of $\mu$
to produce a $K_{\varepsilon}$ such that $\mu(K_{\varepsilon})>1-\varepsilon$,
hence $\mu_{L}(st_{\hat{d}}^{-1}(K_{\varepsilon}))>1-\varepsilon$ also.

On the other hand, fix an $\varepsilon>0$ and consider the $K_{\varepsilon}$
such that $\mu_{L}(st_{\hat{d}}^{-1}(K_{\varepsilon}))>1-\varepsilon$. By
Lemma 15, we know that $st_{\hat{d}}$ restricted to $K_{\varepsilon}$ is
measurable, because $K_{\varepsilon}$ is compact (hence separable). In other
words, we can push forward $\mu_{L}$ onto $\hat{X}$ on all but at most
$\varepsilon$ of the support of $\mu_{L}$. Selecting a sequence of
$\varepsilon_{n}$’s converging to zero, we observe that
$\mu_{L}(st_{\hat{d}}^{-1}(\cup_{n}K_{\varepsilon_{n}}))=1$, and since
$\cup_{n}K_{\varepsilon_{n}}$ is separable, we have that $st_{\hat{d}}$ is
measurable on a large enough codomain to push forward all the mass of
$\mu_{L}$ (and also that the support of $\mu_{L}\circ st_{\hat{d}}^{-1}$ is
separable). Hence $\mu_{L}\circ st_{\hat{d}}^{-1}$ is a (Radon) probability
measure on $\hat{X}$ as desired. ∎

######  Proposition 23.

Let $\mu,\nu\in\mathcal{P}_{p}(\hat{X})$. Let $\mu_{L}$ and $\nu_{L}$ be Loeb
measures such that $\mu=\mu_{L}\circ st_{\hat{d}}^{-1}$ and $\nu=\nu_{L}\circ
st_{\hat{d}}^{-1}$. Then,

$W_{p}(\mu,\nu)\leq W_{p,L}(\mu_{L},\nu_{L}).$

###### ?proofname?.

Let $\gamma_{L}\in\Pi_{L}(\mu_{L},\nu_{L})$. We claim that the
$st_{\hat{d}}$-pushforward of $\gamma_{L}$ onto $\hat{X}\times\hat{X}$ is in
fact a (Radon) coupling $\gamma\in\Pi(\mu,\nu)$. This claim is sufficient to
prove the proposition, because if so, change-of-variable w.r.t. the map
$st_{\hat{d}}$ implies

$\int_{\hat{X}\times\hat{X}}\hat{d}(\hat{x},\hat{y})^{p}d\gamma(\hat{x},\hat{y})=\int_{X_{\lim}^{\omega}\times
X_{\lim}^{\omega}}st\left(\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}\right)d\gamma_{L}(x^{\omega},y^{\omega})$

(where we have used the fact that
$\hat{d}(\hat{x},\hat{y})^{p}=\left(\hat{d}(x^{\omega},y^{\omega})\right)^{p}=st\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}$
whenever $st_{\hat{d}}(x^{\omega})=\hat{x}$ and
$st_{\hat{d}}(y^{\omega})=\hat{y}$). Then, quantifying over all
$\gamma_{L}\in\Pi_{L}(\mu_{L},\nu_{L})$ shows that

$\inf_{\gamma\in\Pi(\mu,\nu)}\int_{\hat{X}\times\hat{X}}\hat{d}(\hat{x},\hat{y})^{p}d\gamma(\hat{x},\hat{y})\leq\inf_{\Pi_{L}(\mu_{L},\nu_{L})}\int_{X_{\lim}^{\omega}\times
X_{\lim}^{\omega}}st\left(\left(d^{\omega}(x^{\omega},y^{\omega})\right)^{p}\right)d\gamma_{L}(x^{\omega},y^{\omega})$

as desired.

We make use of the previous lemma. Since $\mu=\mu_{L}\circ st_{\hat{d}}^{-1}$
and $\nu=\nu_{L}\circ st_{\hat{d}}^{-1}$, we can find compact sets $C_{\mu}$
and $C_{\nu}$ in $\hat{X}$ such that
$\mu_{L}(st_{\hat{d}}^{-1}C_{\mu})>1-\varepsilon/2$ and similarly
$\nu_{L}(st_{\hat{d}}^{-1}C_{\nu})>1-\varepsilon/2$. This implies that
$\gamma_{L}(st_{\hat{d}}^{-1}C_{\mu}\times
st_{\hat{d}}^{-1}C_{\nu})>1-\varepsilon$. By nearly identical reasoning to the
proof of the previous lemma, we observe that $st_{\hat{d}}\times st_{\hat{d}}$
is automatically measurable when the range is restricted to the compact set
$C_{\mu}\times C_{\nu}$. And so, by exhausting the support of $\gamma_{L}$
using preimages of compact sets, we see that $st_{\hat{d}}\times st_{\hat{d}}$
is measurable on a large enough codomain to push forward $\gamma_{L}$ to a
Radon probability measure on $\hat{X}\times\hat{X}$, which we denote by
$\gamma$.

Lastly, we observe that the pushforward of $\gamma_{L}$ is in fact a coupling
between $\mu$ and $\nu$; this is a standard argument but we include it for
completeness. Indeed, let $B\subseteq\hat{X}$ be any Borel set: then,
$\gamma(B\times\hat{X})=\gamma_{L}(st_{\hat{d}}^{-1}(B)\times
X_{\lim}^{\omega})$. Since the first marginal of $\gamma_{L}$ is $\mu_{L}$,
this implies that $\gamma(B\times\hat{X})=\mu_{L}(st_{\hat{d}}^{-1}(B))$.
Hence the first marginal of $\gamma$ is identical to $\mu_{L}\circ
st_{\hat{d}}^{-1}=\mu$. By identical reasoning, the second marginal of
$\gamma$ is $\nu$, as desired. ∎

######  Corollary 24.

(1) Let $\mu^{\omega}\approx_{W_{p}^{\omega}}\nu^{\omega}$, with associated
Loeb measures $\mu_{L}$ and $\nu_{L}$. Then, if the pushforwards $\mu_{L}\circ
st_{\hat{d}}^{-1}$ and $\nu_{L}\circ st_{\hat{d}}^{-1}$ are both well-defined
and belong to $\mathcal{P}_{p}(\hat{X})$, they are equal.

(2) Let $\mu^{\omega}\in(\mathcal{P}_{p}(X)_{\lim}^{\omega},W_{p}^{\omega})$
be a lifting of $\mu\in(\mathcal{P}_{p}(\hat{X}),W_{p})$. Let $\mu_{L}$ be the
Loeb measure associated to $\mu^{\omega}$. Then, if $\mu_{L}\circ
st_{\hat{d}}^{-1}$ is well-defined and belongs to
$(\mathcal{P}_{p}(\hat{X}),W_{p})$, it holds that $\mu=\mu_{L}\circ
st_{\hat{d}}^{-1}$.

######  Remark.

Compare Corollary 24 with the recent result [18, Theorem 4.3] (the same result
appears, under stricter hypotheses, as [17, Theorem 9]), which occurs in a
different setting than ours but nonetheless has a similar flavor. (Duanmu et
al. are able to avoid pathological phenomena similar to our Example 25 by
restricting to the case where the underlying space is bounded and
$\sigma$-compact.)

###### ?proofname?.

(1) Combining Propositions 21 and 23, we see

$W_{p}(\mu_{L}\circ st_{\hat{d}}^{-1},\nu_{L}\circ st_{\hat{d}}^{-1})\leq
W_{p,L}(\mu_{L},\nu_{L})\lessapprox
W_{p}^{\omega}(\mu^{\omega},\nu^{\omega}).$

But $W_{p}^{\omega}(\mu^{\omega},\nu^{\omega})\approx 0$, and
$W_{p}(\mu_{L}\circ st_{\hat{d}}^{-1},\nu_{L}\circ st_{\hat{d}}^{-1})$ is a
positive real, so $W_{p}(\mu_{L}\circ st_{\hat{d}}^{-1},\nu_{L}\circ
st_{\hat{d}}^{-1})=0$.

(2) Let $\tilde{\mu}^{\omega}$ be a lifting of $\mu$ such that
$\tilde{\mu}_{L}\circ st_{\hat{d}}^{-1}=\mu$; we know that such a
$\tilde{\mu}^{\omega}$ exists thanks to Theorem 17. Since both $\mu^{\omega}$
and $\tilde{\mu}^{\omega}$ are liftings of $\mu$, it follows that
$\mu^{\omega}\approx_{W_{p}^{\omega}}\tilde{\mu}^{\omega}$. Therefore, part
(1) shows that $\mu_{L}\circ st_{\hat{d}}^{-1}=\tilde{\mu}_{L}\circ
st_{\hat{d}}^{-1}=\mu$. ∎

One might ask if a certain converse to Corollary 24 holds — namely, if we have
two different Loeb measures $\mu_{L}$ and $\mu_{L}^{\prime}$ that both push
forward to some $\mu\in\mathcal{P}_{p}(\hat{X})$, is it the case that the
internal measures $\mu^{\omega}$ and $\mu^{\prime\omega}$ which generated
$\mu_{L}$ and $\mu_{L}^{\prime}$ must be infinitesimal $W_{p}^{\omega}$
distance apart? The following shows that this is not the case.

######  Example 25.

Let $\mu\in(\mathcal{P}_{2}(\hat{X}),W_{2})$. Suppose that $\mu_{L}$ is a Loeb
measure on $X^{\omega}$ that pushes forward to $\mu$, and that $\mu^{\omega}$
is an internal measure in $(\widehat{\mathcal{P}_{2}(X)},\hat{W}_{2})$. Then,
it is _not_ necessarily the case that $\mu^{\omega}$ is a
$\hat{W}_{2}$-lifting of $\mu$. Indeed, take, as in Example 19, the internal
measure $(1-N^{-1})\delta_{0}+N^{-1}\delta_{N^{1/2}}$ with
$N\in\mathbb{N}^{\omega}\backslash\mathbb{N}$. We know that this measure is
not a $\hat{W}_{2}$-lifting of $\delta_{0}$, but its associated Loeb measure
is $\delta_{0}$ (with underlying space $\mathbb{R}^{\omega}$), which pushes
forward to $\delta_{0}$ (with underlying space $\mathbb{R}$).

The source of the issue may be roughly stated as follows: going from internal
probability measures to Loeb probability measures is compatible with total
variation, which only controls the $p$-Wasserstein distance for probability
measures which are both contained in a fixed set of bounded diameter. Stated a
bit more explicitly:

######  Lemma 26.

(1) Let $\mu^{\omega},\nu^{\omega}\in\mathcal{P}(X)^{\omega}$. Then
$\mu_{L}=\nu_{L}$ iff $TV^{\omega}(\mu^{\omega},\nu^{\omega})\approx 0$.

(2) Suppose that $\mu^{\omega}$ and $\nu^{\omega}$ are both supported within
an internal subset of $X^{\omega}$ with diameter at most
$D\in\mathbb{R}^{\omega}$. Then, for all $p\in[1,\infty)$,
$W_{p}^{\omega}(\mu^{\omega},\nu^{\omega})\leq D\cdot
TV^{\omega}(\mu^{\omega},\nu^{\omega})$.

###### ?proofname?.

(1) Suppose that $TV^{\omega}(\mu^{\omega},\nu^{\omega})\approx 0$. Then
$\mu^{\omega}(B^{\omega})\approx\nu^{\omega}(B^{\omega})$ for every internal
Borel set $B^{\omega}$, so
$st\circ\mu^{\omega}(B^{\omega})=st\circ\nu^{\omega}(B^{\omega})$ for every
internal Borel set $B^{\omega}$. Since the Loeb measure is the completion of
the premeasure $st\circ\mu^{\omega}$, this shows that $\mu_{L}=\nu_{L}$.

Conversely, suppose $TV^{\omega}(\mu^{\omega},\nu^{\omega})\not\approx 0$.
Then, there exists an internal Borel set $B^{\omega}$ and a standard
$\delta>0$ such that
$|\mu^{\omega}(B^{\omega})-\nu^{\omega}(B^{\omega})|>\delta$, hence
$st\circ\mu^{\omega}(B^{\omega})\neq st\circ\nu^{\omega}(B^{\omega})$, and
hence $\mu_{L}\neq\nu_{L}$.

(2) This follows immediately from Łoś’s theorem applied to the inequality
$W_{p}(\mu,\nu)\leq D\cdot TV(\mu,\nu)$, which holds on general metric spaces.
∎

The two parts of the preceding lemma tell us that if $\mu_{L}=\nu_{L}$ and
both $\mu^{\omega}$ and $\nu^{\omega}$ have support which has bounded
diameter, then $W_{p}^{\omega}(\mu^{\omega},\nu^{\omega})\approx 0$. However
this is actually _not_ enough to deduce that $\mu^{\omega}$ (and hence also
$\nu^{\omega}$) must be a lifting of $\mu_{L}\circ st_{\hat{d}}^{-1}$, even if
this pushforward is well-defined. Nonetheless, it _does_ turn out to be true
that if $\mu^{\omega}$ has bounded support and $\mu_{L}\circ
st_{\hat{d}}^{-1}$ is well-defined (and thus belongs to
$\mathcal{P}_{p}(\hat{X})$), then indeed $\mu^{\omega}$ is automatically a
lifting of $\mu_{L}\circ st_{\hat{d}}^{-1}$, as the next lemma shows; in this
way, we have a partial converse to part (2) of Corollary 24.

######  Lemma 27.

Let $p\in[1,\infty)$. Let $\mu^{\omega}\in\mathcal{P}_{p}(X)_{\lim}^{\omega}$,
and suppose, in addition, that $\mu^{\omega}$ is supported within
$B(e^{\omega},D)$ for some $D\in\mathbb{R}$, that is,
$\mu^{\omega}(X^{\omega}\backslash B(e^{\omega},D))=0$. Then, if the
$st_{\hat{d}}$-pushforward of $\mu_{L}$ is well-defined and belongs to
$(\mathcal{P}_{p}(\hat{X}),W_{p})$, it holds that $\mu^{\omega}$ is a
$\hat{W}_{p}$-lifting of $\mu_{L}\circ st_{\hat{d}}^{-1}$.

###### ?proofname?.

We proceed by way of a _semi-discrete transport_ argument. In particular, it
turns out to be valuable to reason using transport _maps_ rather than merely
reasoning using transport plans.

Let $\mu:=\mu_{L}\circ st_{\hat{d}}^{-1}\in\mathcal{P}_{p}(\hat{X})$. Let
$(X_{1},X_{2},\ldots)$ be a sequence of i.i.d. random variables with
distribution $\mu$, and let
$\hat{\mu}_{n}=\frac{1}{n}\sum_{j=1}^{n}\delta_{x_{j}}$ be the $n$th (random)
empirical measure for $\mu$ (where $x_{j}=X_{j}$ for each $j=1,\ldots,n$).
Note that $W_{2}(\mu,\hat{\mu}_{n})\rightarrow 0$ with probability $1$; at the
same time, notice that for _any_ sequence of empirical measures
$(\hat{\mu}_{n})$, we have that

$\displaystyle W_{p}^{p}(\mu,\hat{\mu}_{n})$
$\displaystyle=\min_{\gamma\in\Pi(\mu,\tilde{\mu}_{n})}\int_{\text{supp}(\mu)\times\text{supp}(\hat{\mu}_{n})}\hat{d}^{p}(x,y)d\gamma(x,y)$
$\displaystyle\geq\int_{\text{supp}(\mu)\times\text{supp}(\hat{\mu}_{n})}\left(\min_{\tilde{y}\in\text{supp}(\mu_{n})}\hat{d}^{p}(x,\tilde{y})\right)d\gamma(x,y)$
$\displaystyle=\int_{\text{supp}(\mu)}\left(\min_{\tilde{y}\in\text{supp}(\mu_{n})}\hat{d}^{p}(x,\tilde{y})\right)d\mu.$

Let $\tau_{n}$ denote the map which sends $x\in supp(\mu)$ to
$\text{argmin}_{\tilde{y}\in\text{supp}(\hat{\mu}_{n})}\hat{d}^{p}(x,\tilde{y})$;
in the case where the argmin is not unique, we simply chose a minimizer
arbitrarily in such a way that $\tau_{n}$ is measurable (and this is easy to
do since $\hat{\mu}_{n}$ has finite support, so for instance one can map $x$
lexicographically in the case of a tie, i.e. we index the points in
$\text{supp}(\hat{\mu}_{n})$ and map $x$ to the point with the lowest index
that instantiates the minimum). Note that

$\int_{\text{supp}(\mu)}\left(\min_{\tilde{y}\in\text{supp}(\hat{\mu}_{n})}\hat{d}^{p}(x,\tilde{y})\right)d\mu=\int_{\text{supp}(\mu)}\left(\hat{d}^{p}(x,\tau_{n}(x)\right)d\mu.$

At the same time, if we consider the pushforward measure
$\mu_{n}:=\mu\circ\tau_{n}^{-1}$, we see that $\mu_{n}$ has support contained
within $\text{supp}(\hat{\mu}_{n})$, and

$W_{p}^{p}(\mu,\mu_{n})\leq\int_{\text{supp}(\mu)}\left(\hat{d}^{p}(x,\tau_{n}(x)\right)d\mu\leq
W_{p}^{p}(\mu,\hat{\mu}_{n}).$

Note also that, given any point $y_{j,n}\in\text{supp}(\mu_{n})$, we have that
$\mu_{n}(y_{j,n})=\mu(\tau_{n}^{-1}(y_{j,n}))$, so we can write $\mu_{n}$ as a
sum of point masses like so:

$\mu_{n}=\sum_{j=1}^{n}\mu(\tau_{n}^{-1}(y_{j,n}))\delta_{y_{j,n}}.$

In what follows, we condition on the event where
$W_{p}(\mu,\hat{\mu}_{n})\rightarrow 0$; this implies that
$W_{p}(\mu,\mu_{n})\rightarrow 0$.

Let $\varepsilon>0$. Suppose that $n$ is chosen so that

$\int_{\text{supp}(\mu)}\left(\hat{d}^{p}(x,\tau_{n}(x)\right)d\mu\leq
W_{p}^{p}(\mu,\hat{\mu}_{n})<\varepsilon^{p}.$

For each $y_{j,n}$, we select a lifting $y_{j,n}^{\omega}\in
st_{\hat{d}}^{-1}(y_{j,n})$; this gives us a $\hat{W}_{p}$-lifting of the
measure $\mu_{n}$, namely

$\mu_{n}^{\omega}:=\sum_{j=1}^{n}\mu(\tau_{n}^{-1}(y_{j,n}))\delta_{y_{j,n}^{\omega}}.$

At the same time, let $E_{j,n}^{\omega}$ be an internal measurable set such
that

$E_{j,n}^{\omega}\subseteq st_{\hat{d}}^{-1}(\tau_{n}^{-1}(y_{j,n}))\text{ and
}\mu_{L}(st_{\hat{d}}^{-1}(\tau_{n}^{-1}(y_{j,n}))\backslash
E_{j,n}^{\omega})<\frac{\varepsilon}{n}.$

(Such an $E_{j,n}^{\omega}$ is guaranteed to exist by Fact 3.) In other words,
$\\{E_{j,n}^{\omega}\\}_{j=1}^{n}$ is a finite family of internal measurable
sets which approximates the $\sigma(\mathcal{B}^{\omega})$-measurable
partition $\\{st_{\hat{d}}^{-1}(\tau_{n}^{-1}(y_{j,n}))\\}_{j=1}^{n}$ (which
partitions $st_{\hat{d}}^{-1}(\text{supp}(\mu))$). Likewise, we may define the
“partial transport map”

$\tilde{\tau}_{n}^{\omega}:\bigcup_{j=1}^{n}E_{j,n}^{\omega}\rightarrow
X^{\omega}$
$\tilde{\tau}_{n}^{\omega}(x^{\omega})=\begin{cases}y_{1,n}^{\omega}&x^{\omega}\in
E_{1,n}^{\omega}\\\ \vdots&\vdots\\\ y_{j,n}^{\omega}&x^{\omega}\in
E_{j,n}^{\omega}\\\ \vdots\\\ y_{n,n}^{\omega}&x^{\omega}\in
E_{j,n}^{\omega}\end{cases}.$

Since $\tilde{\tau}_{n}^{\omega}$ has a range with finitely many values and
every $E_{j,n}^{\omega}$ is internal, it is clear that
$\tilde{\tau}_{n}^{\omega}$ is internally measurable. We then extend
$\tilde{\tau}_{n}^{\omega}$ to a transport map in a slightly arbitrary
fashion; for simplicity, we set

$\tau_{n}^{\omega}(x^{\omega})=\begin{cases}\tilde{\tau}_{n}^{\omega}(x^{\omega})&x^{\omega}\in\bigcup_{j=1}^{n}E_{j,n}^{\omega}\\\
e^{\omega}&\text{else}.\end{cases}$

By the same reasoning, $\tau_{n}^{\omega}$ is an internally measurable map. We
may therefore consider the pushforward measure $\mu^{\omega}$ with respect to
$\tau_{n}^{\omega}$, that is, $\mu^{\omega}\circ(\tau_{n}^{\omega})^{-1}$.

By unpacking definitions, we observe that

$W_{p}^{\omega}(\mu^{\omega},\mu^{\omega}\circ(\tau_{n}^{\omega})^{-1})\leq\left(\int_{X^{\omega}}\left(d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega}))\right)^{p}d\mu^{\omega}\right)^{1/p}.$

Indeed, since $\tau_{n}^{\omega}$ is internal, it can be written as
$[(\tau_{n,i})]$ for maps $\tau_{n,i}:X_{i}\rightarrow X_{i}$. At the same
time, $\mu^{\omega}=[(\mu_{i})]$ for some sequence
$(\mu_{i})_{i\in\mathbb{N}}$ of measures in $\mathcal{P}(X_{i})$. For each of
these maps, consider the induces transport plan
$\gamma_{i}:=(Id\times\tau_{n,i})_{\\#}\mu_{i}$, which is a transport plan
whose first marginal is $\mu_{i}$ and whose second marginal is
$\mu_{i}\circ\tau_{n,i}^{-1}$. Then $\gamma^{\omega}:=[(\gamma_{i})]$ is a
transport plan between $[(\mu_{i})]:=\mu^{\omega}$ and
$[(\mu_{i}\circ\tau_{n,i}^{-1})]:=[(\mu_{i})]\circ[(\tau_{n,i}^{-1})]:=\mu^{\omega}\circ(\tau_{n}^{\omega})^{-1}$.
It remains only to note that

$\displaystyle
W_{p}^{\omega}(\mu^{\omega},\mu^{\omega}\circ(\tau_{n}^{\omega})^{-1})$
$\displaystyle\leq\left(\int_{X^{\omega}\times
X^{\omega}}(d^{\omega}(x^{\omega},y^{\omega}))^{p}d\gamma^{\omega}\right)^{1/p}$
$\displaystyle:=\left[\left(\left(\int_{X_{i}\times
X_{i}}(d_{i}(x_{i},y_{i}))^{p}d\gamma_{i}(x_{i},y_{i})\right)^{1/p}\right)\right]$
$\displaystyle=\left[\left(\left(\int_{X_{i}}(d_{i}(x_{i},\tau_{n,i}(x_{i})))^{p}d\mu_{i}(x_{i})\right)^{1/p}\right)\right]$
$\displaystyle:=\left(\int_{X^{\omega}}(d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega})))^{p}d\mu^{\omega}\right)^{1/p}.$

Now,

$\displaystyle\int_{X^{\omega}}\left(d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega}))\right)^{p}d\mu^{\omega}$
$\displaystyle=\sum_{j=1}^{n}\int_{E_{j,n}^{\omega}}\left(d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega}))\right)^{p}d\mu^{\omega}+\int_{X^{\omega}\backslash\bigcup_{j=1}^{n}E_{j,n}^{\omega}}\left(d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega}))\right)^{p}d\mu^{\omega}$
$\displaystyle<\sum_{j=1}^{n}\int_{E_{j,n}^{\omega}}\left(d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega}))\right)^{p}d\mu^{\omega}+D^{p}\varepsilon^{p}$
$\displaystyle\approx\sum_{j=1}^{n}\int_{E_{j,n}^{\omega}}st\left(d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega}))\right)^{p}d\mu_{L}+D^{p}\varepsilon^{p}$
$\displaystyle(*)$
$\displaystyle\approx\sum_{j=1}^{n}\int_{E_{j,n}^{\omega}}\left(st(d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega})))\right)^{p}d\mu_{L}+D^{p}\varepsilon^{p}$

In particular, in line $(*)$ we have used the fact that $(\cdot)^{p}$ is
continuous, and $d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega}))$ is
nearstandard — this nearstandardness, in turn, follows from the fact that each
$E_{j,n}^{\omega}$ has finite diameter.

By definition,
$st((d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega})))=\hat{d}(st_{\hat{d}}(x^{\omega}),st_{\hat{d}}(\tau_{n}^{\omega}(x^{\omega})))$;
by construction,

$\displaystyle st_{\hat{d}}(\tau_{n}^{\omega}(x^{\omega}))$
$\displaystyle=y_{j,n}$ $\displaystyle x^{\omega}\in E_{j,n}^{\omega}$
$\displaystyle=\tau_{n}(st_{\hat{d}}(x^{\omega}))$

since $st_{\hat{d}}(E_{j,n}^{\omega})\subseteq\tau_{n}^{-1}(y_{j,n})$.
Therefore,

$\displaystyle\int_{E_{j,n}^{\omega}}\left(st(d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega})))\right)^{p}d\mu_{L}$
$\displaystyle=\int_{E_{j,n}^{\omega}}\left(\hat{d}(st_{\hat{d}}(x^{\omega}),\tau_{n}(st_{\hat{d}}(x^{\omega})))\right)^{p}d\mu_{L}$
$\displaystyle\leq\int_{st_{\hat{d}}^{-1}(\tau_{n}^{-1}(y_{j,n}))}\left(\hat{d}(st_{\hat{d}}(x^{\omega}),\tau_{n}(st_{\hat{d}}(x^{\omega})))\right)^{p}d\mu_{L}$
$\displaystyle=\int_{\tau_{n}^{-1}(y_{j,n})}\left(\hat{d}(x,\tau_{n}(x))\right)^{p}d(\mu_{L}\circ
st_{\hat{d}}^{-1}).$

Since $\mu=\mu_{L}\circ st_{\hat{d}}^{-1}$, and
$\text{supp}(\mu)=\bigsqcup_{j=1}^{n}\tau_{n}^{-1}(y_{j,n})$, we deduce that

$\displaystyle\int_{X^{\omega}}\left(d^{\omega}(x^{\omega},\tau_{n}^{\omega}(x^{\omega}))\right)^{p}d\mu^{\omega}$
$\displaystyle<\int_{\text{supp}(\mu)}\left(\hat{d}(x,\tau_{n}(x))\right)^{p}d\mu+D^{p}\varepsilon^{p}$
$\displaystyle\leq W_{p}^{p}(\mu,\mu_{n})+D^{p}\varepsilon^{p}$
$\displaystyle\leq(D^{p}+1)\varepsilon^{p}.$

Therefore
$W_{p}^{\omega}(\mu^{\omega},\mu^{\omega}\circ(\tau_{n}^{\omega})^{-1})<(D^{p}+1)^{1/p}\varepsilon$.

At the same time, we claim that

$W_{p}^{\omega}(\mu_{n}^{\omega},\mu^{\omega}\circ(\tau_{n}^{\omega})^{-1})<D\varepsilon.$

Indeed,

$\mu_{n}^{\omega}=\sum_{j=1}^{n}\mu(\tau_{n}^{-1}(y_{j,n}))\delta_{y_{j,n}^{\omega}};\qquad\mu^{\omega}\circ(\tau_{n}^{\omega})^{-1}=\mu^{\omega}\left(X^{\omega}\backslash\bigcup_{j=1}^{n}E_{j,n}^{\omega}\right)\delta_{e^{\omega}}+\sum_{j=1}^{n}\mu^{\omega}(E_{j,n}^{\omega})\delta_{y_{j,n}^{\omega}}.$

Since for all $j=1,\ldots,n$,

$\mu_{L}\left(st_{\hat{d}}^{-1}(\tau_{n}^{-1}(y_{j,n}))\backslash
E_{j,n}^{\omega}\right)<\frac{\varepsilon}{n}$

and

$\mu(\tau_{n}^{-1}(y_{j,n}))=\mu_{L}(st_{\hat{d}}^{-1}(\tau_{n}^{-1}(y_{j,n})))$

we see that

$\mu(\tau_{n}^{-1}(y_{j,n}))-\frac{\varepsilon^{p}}{n}<\mu^{\omega}(E_{j,n}^{\omega})\lessapprox\mu(\tau_{n}^{-1}(y_{j,n}))$

and

$\mu^{\omega}\left(X^{\omega}\backslash\bigcup_{j=1}^{n}E_{j,n}^{\omega}\right)<\varepsilon.$

In particular, this implies that

$\displaystyle
TV^{\omega}(\mu_{n}^{\omega},\mu^{\omega}\circ(\tau_{n}^{\omega})^{-1})$
$\displaystyle:=\frac{1}{2}\left(\sum_{j=1}^{n}|\mu^{\omega}(E_{j,n}^{\omega})-\mu(\tau_{n}^{-1}(y_{j,n}))|+\mu^{\omega}\left(X^{\omega}\backslash\bigcup_{j=1}^{n}E_{j,n}^{\omega}\right)\right)$
$\displaystyle\leq\frac{1}{2}\left(\sum_{j=1}^{n}\frac{\varepsilon}{n}+\varepsilon\right)=\varepsilon.$

At the same time, all the points $e^{\omega}$ and $y_{j,n}^{\omega}$ (for
$j=1,\ldots,n$) reside in a domain with diameter at most $D$. Hence, by Lemma
26, we know that

$W_{p}^{\omega}(\mu_{n}^{\omega},\mu^{\omega}\circ(\tau_{n}^{\omega})^{-1})\leq
D\cdot
TV^{\omega}(\mu_{n}^{\omega},\mu^{\omega}\circ(\tau_{n}^{\omega})^{-1})<D\varepsilon.$

So from the triangle inequality, we have that

$W_{p}^{\omega}(\mu^{\omega},\mu_{n}^{\omega})<\left((D^{p}+1)^{1/p}+D\right)\varepsilon.$

Lastly, let $\tilde{\mu}^{\omega}$ be any $\hat{W}_{p}$-lifting of $\mu$; from
the fact that $W_{p}(\mu,\mu_{n})<\varepsilon$, we know that
$W_{p}^{\omega}(\tilde{\mu}^{\omega},\mu_{n}^{\omega})<\varepsilon$.
Therefore,

$W_{p}^{\omega}(\mu^{\omega},\tilde{\mu}^{\omega})<\left((D^{p}+1)^{1/p}+D+1\right)\varepsilon.$

But $\varepsilon>0$ was arbitrary, so we conclude that
$W_{p}^{\omega}(\mu^{\omega},\tilde{\mu}^{\omega})\approx 0$. Hence
$\mu^{\omega}$ is also a lifting of $\mu$, as desired. ∎

##  3\. Ultralimits of $CD(K,\infty)$ spaces

The aim of this section is to prove the following result.

######  Theorem 28.

Fix a non-principal ultrafilter $\omega$ on $\mathbb{N}$. Consider a sequence
$(X_{i},d_{i},e_{i},\mu_{i})$ of pointed metric measure spaces with
$\mu_{i}\in\mathcal{P}(X_{i})$, where each $(X_{i},d_{i},\mu_{i})$ satisfies
the synthetic Ricci curvature bound “$CD(K_{i},\infty)$”, for $\omega$-almost
all $i\in\mathbb{N}$. Let $(X,d,e,\mu)$ be the metric measure ultralimit of
$(X_{i},d_{i},e_{i},\mu_{i})_{i\in\mathbb{N}}$ in the sense of Definition 9,
so that $\mu\in\mathcal{P}(X)$. Then $(X,d,\mu)$ satisfies the synthetic Ricci
curvature lower bound “$CD(K,\infty)$” with $K=st(K^{\omega})$.

######  Remark.

We draw the reader’s attention to a number of complications which are present.
We also mention why it is not possible to directly prove the stability of the
“strong $CD(K,\infty)$” property using a similar strategy to ours.

First, one concern is the following: we might like to witness the
$CD(K,\infty)$ property in the limiting space $\mathcal{P}_{2}(\hat{X})$ by
taking an ultraproduct of geodesics in $\mathcal{P}_{2,ac(\mu_{i})}(X_{i})$
along which the relative entropy functional is $K_{i}$-geodesically convex,
and then pushing down the resulting curve to $\mathcal{P}_{2}(\hat{X})$. But
what can conceivably happen is that the ultraproduct of the geodesics only
gives us a limiting geodesic in the bigger space
$\widehat{\mathcal{P}_{2}(X)}$, whereas the geodesics inside
$\mathcal{P}_{2}(\hat{X})$ are instead produced by ultraproducts of sequences
of geodesics that do _not_ witness the (weak) $K_{i}$-geodesic convexity of
the relative entropy. It turns out that a careful argument is able to sidestep
this problem, but this is one reason why our proof is not a single paragraph
argument (as with a proof of the stability of Alexandrov-type synthetic
sectional curvature bounds w.r.t. ultralimits).

Simultaneously, it is quite possible that $\mathcal{P}_{2,ac}(\hat{X})$
contains geodesics which do _not_ arise as pushdowns of ultraproducts of
geodesics; rather, of curves in $\mathcal{P}_{2}(X)_{\lim}^{\omega}$ whose
length is _infinitesimally close_ to the $W_{2}^{\omega}$-distance between the
endpoints — so, for instance, ultraproducts of sequences of a.c. curves of the
form $g_{i}:[0,1]\rightarrow\mathcal{P}_{2,ac}(X_{i})$ where $\omega$-a.s.,

$|\text{length}(g_{i})-W_{2}(g_{i}(0),g_{i}(1))|\rightarrow 0.$

Since the $CD(K,\infty)$ property has nothing to say about approximate
geodesics (rather, one would need to use something like the _lax_
$CD(K,\infty)$ property from [44]), it is non-obvious how the pushdown of
$[(g_{i})]$ interacts with the relative entropy on $\mathcal{P}_{2}(\hat{X})$
even if the pushdown is well-defined and has range in
$\mathcal{P}_{2,ac(\mu)}(\hat{X})$. We note that similar problems in the
setting of Gromov-Hausdorff-type convergence of metric measure spaces have
already been discussed, for instance at the very end of the last chapter of
[49] which addresses the (non)-stability of the curvature-dimension condition
“strong $CD(0,N)$”. The type of example discussed therein by Villani, of a
sequence of spaces where the limiting space has vastly more (and ill-behaved)
geodesics makes it hard to suggest that, without some other side condition
(such as the $RCD(K,\infty)$ property), it is plausible that “ultralimit of
strong $CD(K,\infty)$ spaces is strong $CD(K,\infty)$” fails to hold in
general.

We begin with a number of preparatory lemmas.

######  Lemma 29.

Let $(\mu_{i})$ and $(\nu_{i})$ be sequences of probability measures on
$(X_{i},d_{i})$, with ultraproducts $\mu^{\omega}$ and $\nu^{\omega}$ in
$(\mathcal{P}_{\lim}^{\omega}(X),W_{2}^{\omega})$.

(1) We have $\nu_{i}\ll\mu_{i}$ $\omega$-a.s., with Radon-Nikodym derivative
$\frac{d\nu_{i}}{d\mu_{i}}:=h_{i}$, iff

$\forall
B\in\mathcal{B}^{\omega}\quad\nu^{\omega}(B)=\int_{B}h^{\omega}d\mu^{\omega}.$

(2) Furthermore, we have that $h_{i}\in L^{p}(\mu_{i})$ for some fixed
$p\in(1,\infty)$, with $\|h_{i}\|_{L^{p}(\mu_{i})}\leq C$ $\omega$-a.s. for
some uniform constant $C$, iff

$\int_{X^{\omega}}|h^{\omega}|^{p}d\mu^{\omega}<C^{p}.$

Similarly, if $h_{i}\in L^{\infty}(\mu_{i})$, then
$\|h_{i}\|_{L^{\infty}(\mu_{i})}\leq C$ $\omega$-a.s., iff

$\mu^{\omega}\\{x^{\omega}:h^{\omega}>C\\}=0.$

(3) Suppose that

$\forall
B\in\mathcal{B}^{\omega}\quad\nu^{\omega}(B)=\int_{B}h^{\omega}d\mu^{\omega}.$

Then if $h^{\omega}$ is $S$-integrable with respect to $\mu_{L}$,

$\forall B\in\sigma(\mathcal{B}^{\omega})\quad\nu_{L}(B)=\int_{B}st\circ
h^{\omega}d\mu_{L}.$

###### ?proofname?.

(1, 2) These are immediate consequences of Łoś’s theorem.

(3) Suppose that $\nu^{\omega}=\int h^{\omega}d\mu^{\omega}$ and $h^{\omega}$
is $S$-integrable. From the definition of $S$-integrable function, we have
that

$\int st\circ h^{\omega}d\mu_{L}=st\left(\int h^{\omega}d\mu^{\omega}\right).$

Now, if $B^{\omega}\in\mathcal{B}^{\omega}$ is any internal measurable set,
and $h^{\omega}$ is any internal function, it holds that

$st\left(1_{B^{\omega}}(x^{\omega})\cdot
h^{\omega}(x^{\omega})\right)=\begin{cases}st(h^{\omega}(x^{\omega}))&x^{\omega}\in
B^{\omega}\\\ 0&\text{else}\end{cases}$

in other words, $st\left(1_{B^{\omega}}\cdot
h^{\omega}\right)=1_{B^{\omega}}\cdot st(h^{\omega})$. Therefore,

$\int_{B^{\omega}}st\circ
h^{\omega}d\mu_{L}=st\left(\int_{B^{\omega}}h^{\omega}d\mu^{\omega}\right).$

On the other hand, it holds, directly from the definition of the Loeb measure,
that for any internal measurable set $B^{\omega}\in\mathcal{B}^{\omega}$,

$\nu_{L}(B^{\omega})=st\left(\nu^{\omega}(B^{\omega})\right).$

Hence,

$(\forall
B^{\omega}\in\mathcal{B}^{\omega})\qquad\nu_{L}(B^{\omega})=st\left(\nu^{\omega}(B^{\omega})\right)=st\left(\int_{B^{\omega}}h^{\omega}d\mu^{\omega}\right)=\int_{B^{\omega}}st\circ
h^{\omega}d\mu_{L}.$

This equality then extends to all $B\in\sigma(\mathcal{B}^{\omega})$ by an
application of Fact 3. Indeed, given any $B\in\sigma(\mathcal{B}^{\omega})$,
take some $B^{\omega}\in\mathcal{B}^{\omega}$ with $\mu_{L}(B^{\omega}\Delta
B)=0$. Then,

$\int_{B^{\omega}}st\circ h^{\omega}d\mu_{L}=\int_{B}st\circ
h^{\omega}d\mu_{L}.$

On the other hand, suppose that $\nu_{L}(B^{\omega}\Delta B)=\delta>0$. Then
(again by Fact 3), we can find some internal $D^{\omega}$ with
$D^{\omega}\subset B^{\omega}\Delta B$ and
$\nu_{L}(D^{\omega})>\frac{\delta}{2}$, and hence
$\nu^{\omega}(D^{\omega})>\frac{\delta}{2}$ also. Now, compute that

$\displaystyle\frac{\delta}{2}<\nu^{\omega}(D^{\omega})$
$\displaystyle=\int_{D^{\omega}}h^{\omega}d\mu^{\omega}$
$\displaystyle\approx\int_{D^{\omega}}st\circ h^{\omega}d\mu_{L}$
$\displaystyle\leq\int_{B^{\omega}\Delta B}st\circ h^{\omega}d\mu_{L}$
$\displaystyle=0.$

Thus we have a contradiction; it follows that $\nu_{L}(B^{\omega}\Delta B)=0$.

Consequently,

$\nu_{L}(B)=\nu_{L}(B^{\omega})=\int_{B^{\omega}}st\circ
h^{\omega}d\mu_{L}=\int_{B}st\circ h^{\omega}d\mu_{L}$

as desired. ∎

######  Definition 30.

We write $\nu^{\omega}\ll^{\omega}\mu^{\omega}$ to denote the fact that
$\nu_{i}\ll\mu_{i}$ $\omega$-almost surely.

######  Lemma 31.

(Cf. [8, Section 5]) Let $\mu^{\omega}\in\mathcal{P}_{2}(X)^{\omega}$ be an
internal probability measure whose Loeb measure $\mu_{L}$ pushes forward to a
Radon probability measure $\mu$ on $(\hat{X},d)$. Let $f^{\omega}$ be an
internal measurable function such that
$\int_{X^{\omega}}f^{\omega}d\mu^{\omega}=1$, and further assume that
$f^{\omega}$ is $S$-integrable with respect to $\mu$. Let