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Metrization of the Gromov-Hausdorff
(-Prokhorov) Topology for Boundedly-Compact
Metric Spaces
Ali Khezeli∗
January 19, 2019
Abstract
In this work, a metric is presented on the set of boundedly-compact
pointed metric spaces that generates the Gromov-Hausdorff topology.
A similar metric is defined for measured metric spaces that generates
the Gromov-Hausdorff-Prokhorov topology. This extends previous works
which consider only length spaces or discrete metric spaces. Completeness
and separability are also proved for these metrics. Hence, they provide the
measure theoretic requirements to study random (measured) boundedly-
compact pointed metric spaces, which is the main motivation of this work.
In addition, we present a generalization of the classical theorem of Strassen
which is of independent interest. This generalization proves an equivalent
formulation of the Prokhorov distance of two finite measures, having pos-
sibly different total masses, in term of approximate coupling. A Strassen-
type result is also proved for the Gromov-Hausdorff-Prokhorov metric for
compact spaces.
Contents
1 Introduction 2
1.1 Introduction to the Gromov-Hausdorff Topology . . . . . . . . . 2
1.2 Introduction to the Contributions of the Present Paper . . . . . . 3
2 The Hausdorff and Prokhorov Metrics 5
2.1 Notations ............................... 5
2.2 TheHausdorffMetric ........................ 6
2.3 The Prokhorov Metric . . . . . . . . . . . . . . . . . . . . . . . . 6
∗Tarbiat Modares University, khezeli@modares.ac.ir
1
3 The Gromov-Hausdorff-Prokhorov Metric 9
3.1 Pointed Measured Metric (PMM) Spaces . . . . . . . . . . . . . . 9
3.2 The Metric in the Compact Case . . . . . . . . . . . . . . . . . . 11
3.3 The Metric in the Boundedly-Compact Case . . . . . . . . . . . . 17
3.4 The Topology of the GHP Metric . . . . . . . . . . . . . . . . . . 21
3.5 Completeness, Separability and Pre-Compactness . . . . . . . . . 25
3.6 Random PMM Spaces and Weak Convergence . . . . . . . . . . . 27
4 Special Cases and Connections to Other Notions 29
4.1 A Metrization of the Gromov-Hausdorff Convergence . . . . . . . 29
4.2 LengthSpaces............................. 30
4.3 RandomMeasures .......................... 31
4.4 Benjamini-Schramm Metric For Graphs . . . . . . . . . . . . . . 31
4.5 DiscreteSpaces............................ 32
4.6 The Gromov-Hausdorff-Vague Topology . . . . . . . . . . . . . . 32
4.7 The Skorokhod Space of C`adl`ag Functions . . . . . . . . . . . . . 33
1 Introduction
Subsection 1.1 below provides an introduction to the notion of Gromov-Hausdorff
convergence on the set of all boundedly-compact pointed metric spaces. The con-
tributions of the present paper are introduced in Subsection 1.2.
1.1 Introduction to the Gromov-Hausdorff Topology
The Gromov-Hausdorff Metric. The Hausdorff metric, denoted by dH,
defines the distance of two compact subsets of a given metric space. Gromov
defined a metric on the set Ncof all compact metric spaces which are not
necessarily contained in a given space (isometric metric spaces are regarded
equivalent). This metric is called the Gromov-Hausdorff metric in the literature.
The distance of two compact metric spaces Xand Yis defined by
dc
GH (X, Y ) := inf dH(f(X), g(Y)),(1.1)
where the infimum is over all metric spaces Zand all pairs of isometric em-
beddings f:X→Zand g:Y→Z(an isometric embedding is a distance-
preserving map which is not necessarily surjective).
The Gromov-Hausdorff metric has been defined for group-theoretic purposes.
However, it has found important applications in probability theory as well, since
it enables one to study random compact metric spaces. Specially, this is used in
the study of scaling limits of random graphs and other random objects. This
goes back to the novel work of Aldous [3] who proved that a random tree with n
vertices, chosen uniformly at random and scaled properly, converges to a random
object called the Brownian continuum random tree in a suitable sense as ntends
to infinity. Using Gromov’s definition, Aldous’s result can be restated in terms
of weak convergence of probability measures on Nc(see [17] and [11]). Since
2
then, scaling limits of various random discrete models have also been studied.
An important topological property needed for probability-theoretic applications
is that the set Nc(or other relevant sets) is complete and separable, and hence,
can be used as a standard probability space.
The Gromov-Hausdorff-Prokhorov Metric. The Prokhorov metric, de-
noted by dP, defines the distance of two finite measures on a common met-
ric space. By using this metric, the Gromov-Hausdorff metric is generalized
to define the distance of two compact measured metric spaces ([12], [18], [22]
and [1]), where a compact measured metric space is a compact metric space
Xtogether with a finite measure µon X. The metric is usually called the
Gromov-Hausdorff-Prokhorov metric and is defined by
dc
GHP ((X, µ),(Y, ν )) := inf{dH(f(X), g(Y)) ∨dP(f∗µ, g∗ν)},
where the infimum is over all metric spaces Zand isometric embeddings f:
X→Zand g:Y→Z(f∗µdenotes the push-forward of the measure µby f).
This metric plays an important role in defining and studying random mea-
sured metric spaces (see e.g., [1] and the papers citing it). In particular, since
every discrete set can be naturally equipped with the counting measure, this
metric can be used to prove stronger convergence results in scaling limits of
discrete objects. Also, it has applications in mass-transportation problems (see
e.g., [22] and the references mentioned therein).
The Non-Compact Case. To relax the assumption of compactness, it
is convenient to consider boundedly-compact metric spaces; i.e., metric spaces
in which every closed ball of finite radius is compact. Also, it is important in
many aspects to consider pointed metric spaces; i.e., metric spaces with a dis-
tinguished point, which is called the origin here. Then, the notion of Gromov-
Hausdorff convergence is defined for sequences of boundedly-compact pointed
metric spaces (see e.g., [9]), which goes back to Gromov [14]. Heuristically, the
idea is to consider large balls centered at the origins and compare them using
the Gromov-Hausdorff metric in the compact case (the precise definition takes
into account the discontinuity issues caused by the points which are close to
the boundaries of the balls). This gives a topology on the set N∗of boundedly-
compact pointed metric spaces, called the Gromov-Hausdorff topology. The no-
tion of Gromov-Hausdorff-Prokhorov convergence and topology [22] (also called
measured Gromov-Hausdorff convergence) is defined similarly on the set M∗of
boundedly-compact pointed measured metric spaces (in which the measures are
boundedly-finite). The next subsection provides more discussion on the matter.
1.2 Introduction to the Contributions of the Present Pa-
per
The main focus of this work is on boundedly-compact pointed metric spaces
and measured metric spaces. In the boundedly-compact case, under some re-
strictions on the metric spaces under study, similar metrics are defined in the
literature that generate the Gromov-Hausdorff (-Prokhorov) topology restricted
3
to the corresponding subsets of N∗or M∗. For instance, [1] considers only length
spaces (i.e., metric spaces in which the distance of any two points is the infi-
mum length of the curves connecting them) and [6] considers discrete metric
spaces. Also, in the case of graphs (where every graph is equipped with the
graph-distance metric), the Benjamini-Schramm metric [7] does the job. These
papers use the corresponding metrics to study random real trees, random dis-
crete metric spaces and random graphs respectively, in the non-compact case.
The main contribution of the present paper is the definition of a metric on
the set N∗of all boundedly-compact pointed metric spaces (which are not nec-
essarily length spaces or discrete spaces) that generates the Gromov-Hausdorff
topology. The same is done for measured metric spaces as well (connections
with the metric defined in [5] will be discussed in the last section). This enables
one to define and study random (measured) boundedly-compact pointed metric
spaces, which is the main motivation of this paper.
To define the distance of two boundedly-compact pointed metric spaces
(X, o) and (X0, o0), the idea is, as in the Gromov-Hausdorff convergence, to
compare large balls centered at oand o0(this idea is sometimes called the local-
ization method, which is commonly used in various situations in the literature
some of which are discussed in Section 4). There are some pitfalls caused by
boundary-effects of the balls; e.g., the value dc
GH (Br(o), Br(o0)), where Br(o) is
the closed ball of radius rcentered at o, is not monotone in r. The definition
of this paper is based on the following value, which has a useful monotonicity
property: inf dc
GH (Br(o), Y ), where the infimum is over all compact subsets
Y⊆X0such that Y⊇Br−1/r(o0) (the last condition can also be removed and
most of the results remain valid). Here, a version of the metric dc
GH for pointed
metric spaces should be used. For measured metric spaces, a similar metric
is also provided which gives the Gromov-Hausdorff-Prokhorov topology. The
definition of this metric is based on a similar idea.
It is also proved that the set N∗(resp. M∗) of boundedly-compact pointed
(measured) metric spaces is complete and separable, and hence, can be used as
a standard probability space. This is important if one wants to consider random
(measured) metric spaces in the boundedly-compact case.
Meanwhile, as a tool in the proofs, a generalization of K¨onig’s’s infinity
lemma is proved for compact sets, which is of independent interest. The argu-
ments based on this lemma are significantly simpler in comparison with similar
arguments in the literature.
Other variants of the metric are also available, for instance
Z∞
0
e−r1∧dc
GH (Br(o), Br(o0))dr.
By the results of this paper, one can show that this formula defines a metric on
N∗as well and has similar properties (formulas like this are common in various
settings in the literature; e.g., [1]), but the definition of the present paper enables
one to have more quantitative bounds in the arguments.
In addition, a generalization of Strassen’s theorem [20] is presented, which
is of independent interest and is useful in the arguments. The result provides
4
an equivalent formulation of the Prokhorov distance between two given finite
measures on a common metric space. The original theorem of Strassen does this
in the case of probability measures. A Strassen-type result is also presented for
the Gromov-Hausdorff-Prokhorov metric in the compact case.
Finally, the connections to other notions in the literature are discussed.
This includes random measures, Benjamini-Schramm metric for graphs, the
Skorokhod space of c`adl`ag functions, the work of [5] for metric measure spaces,
and more.
The structure of the paper is as follows. Section 2 recalls the Hausdorff and
Prokhorov metrics and also provides the generalization of Strassen’s theorem.
In Section 3, the Gromov-Hausdorff-Prokhorov metric is recalled in the compact
case and a Strassen-type theorem is proved for it. The metric is also extended to
the general boundedly-compact pointed case (it contains the Gromov-Hausdorff
metric as a special case). The properties of this metric are also studied therein.
Finally, Section 4 discusses special cases of the metric which already exist in the
literature and also discusses the connections to other notions.
2 The Hausdorff and Prokhorov Metrics
In this section, the definitions and basic properties of the Hausdorff and Prokhorov
metrics are recalled. Also, a generalization of Strassen’s theorem [20] is provided
(Theorem 2.1) which gives an equivalent formulation of the Prokhorov metric.
It will be used in the next section.
2.1 Notations
The set of nonnegative real numbers is denoted by R≥0. The minimum and
maximum binary operators are denoted by ∧and ∨respectively.
For all metric spaces Xin this paper, the metric on Xis always denoted by d
if there is no ambiguity. For a closed subset A⊆X, the (closed) r-neighborhood
of Ain Xis the set Nr(A) := {x∈X:∃y∈A:d(x, y)≤r}. The complement
of Ais denoted by Acor X\A. The two projections from X×Yonto Xand
Yare denoted by π1and π2respectively. Also, all measures on Xare assume
to be Borel measures. The Dirac measure at a∈Xis denoted by δa. If µis a
measure on X, the total mass of µis defined by
||µ|| := µ(X).
If in addition, ρ:X→Yis measurable, ρ∗µdenotes the push-forward of µ
under ρ; i.e., ρ∗µ(·) = µ(ρ−1(·)). If µand νare measures on X, the total
variation distance of µand νis defined by
||µ−ν|| := sup{|µ(A)−ν(A)|:A⊆X}.
5
2.2 The Hausdorff Metric
The following definitions and results are borrowed from [9]. Let Zbe a metric
space. For two closed subsets A, B ⊆Z, the Hausdorff distance of Aand B
is defined by
dH(A, B) := inf {≥0 : A⊆N(B) and B⊆N(A)}.(2.1)
Let F(Z) be the set of closed subsets of Z. It is well known that dHis a metric
on F(Z). Also, if Zis complete and separable, then F(Z) is also complete and
separable. In addition, if Zis compact, then F(Z) is also compact. See e.g.,
Proposition 7.3.7 and Theorem 7.3.8 of [9].
2.3 The Prokhorov Metric
Fix a complete separable metric space Z. For two finite Borel measures µand
νon Z, the Prokhorov distance of µand ν(see e.g., [15]) is defined by
dP(µ, ν) := inf { > 0 : ∀A:µ(A)≤ν(N(A)) + , ν (A)≤µ(N(A)) + },(2.2)
where Aranges over all closed subsets of Z.
It is well known that dPis a metric on the set of finite Borel measures on
Zand makes it a complete and separable metric space. Moreover, the topology
generated by this metric coincides with that of weak convergence (see e.g., [15]).
The following theorem is the main result of this subsection. It provides
another formulation of the Prokhorov distance using the notion of approximate
couplings [2] and will be useful afterwards. Let αbe a finite Borel measure on
X×X. The discrepancy of αw.r.t. µand ν[2] is defined by
D(α;µ, ν) := ||π1∗α−µ|| +||π2∗α−ν||.
One has D(α;µ, ν) = 0 if and only if αis a coupling of µand ν; i.e., π1∗α=µ
and π2∗α=ν.
Theorem 2.1 (Generalized Strassen’s Theorem).Let µand νbe finite Borel
measures on a complete separable metric space Z.
(i) dP(µ, ν)≤if and only if there is a Borel measure αon Z×Zsuch that
D(α;µ, ν) + α({(x, y ) : d(x, y)> })≤. (2.3)
(ii) Equivalently,
dP(µ, ν) = min{≥0 : ∃α:D(α;µ, ν ) + α({(x, y) : d(x, y)> })≤}
(2.4)
and the minimum is attained.
(iii) In addition, if µ(Z)≤ν(Z), then the infimum in (2.4) is attained for
:= dP(µ, ν)and some αsuch that π1∗α=µand π2∗α≤ν. Moreover, α
can be chosen to be supported on supp(µ)×supp(ν).
6
Proof. Let ≥0 and αbe a measure satisfying (2.3). We will prove that
dP(µ, ν)≤. Let 1:= ||π1∗α−µ||,2:= ||π2∗α−ν|| and δ:= α({(x, y) :
d(x, y)> }). Let A⊆Zbe a closed subset and B:= {(x, y) : x∈A, d(x, y)≤
}. One has π2(B) = N(A). Therefore,
µ(A)≤π1∗α(A) + 1=α(π−1
1(A)) + 1
≤α(B) + 1+δ≤α(π−1
2(N(A))) + 1+δ
=π2∗α(N(A))) + 1+δ≤ν(N(A)) + 1+2+δ
≤ν(N(A)) + ,
where the last inequality holds by the assumption (2.3). Similarly, one can show
ν(A)≤µ(N(A)) + . Since this holds for all A, one gets dP(µ, ν )≤.
Conversely, assume dP(µ, ν )≤. One can assume ν(Z) = µ(Z) + δand
δ≥0 without loss of generality. Let r > be arbitrary. The former assumption
implies that ν(A)≤µ(Nr(A)) + rfor every closed set A⊆Z. It follows that
ν(Z\A)≥µ(Z\Nr(A)) −r+δ. (2.5)
Let B⊆Zbe an arbitrary closed subset, s>rbe arbitrary and Abe the
closure of Z\Ns(B). Note that Z\A⊆Ns(B) and Z\Nr(A)⊇B. It follows
from (2.5) that ν(Ns(B)) ≥µ(B)−r+δ. By letting rand stend to and by
∩s>Ns(B) = B, one gets that
µ(B)≤ν(N(B)) + −δ,
for all closed sets B⊆Z. Now, add a point ato Z, let Z0=Z∪ {a}and
let ν0:= ν+ (−δ)δa, which is a measure on Z0. Let K:= {(x, y)∈Z×
Z:d(x, y)≤} ∪ (Z× {a}). Then, for any closed subset A⊂Z, one has
µ(A)≤ν0({y∈Z0:∃x∈A: (x, y)∈K}). Therefore, by Lemma 2.4 below,
one finds a measure βon Ksuch that π1∗β=µand π2∗β≤ν0. Let γbe the
restriction of βto Z×Z. One has π1∗γ≤µand π2∗γ≤ν. Let µ1:= µ−π1∗γ
and ν1:= ν−π2∗γ. The assumption µ(Z)≤ν(Z) implies that µ1(Z)≤µ1(Z).
Therefore, if ν1= 0, then µ1= 0 and γhas the desired properties. So, assume
ν16= 0. Also, one can obtain ||µ1|| ≤ ν0(a) = −δ. Define
α:= γ+1
ν1(Z)µ1⊗ν1.(2.6)
We claim that αsatisfies the desired properties. It is straightforward that
π1∗α=µand π2∗α≤ν. This implies that D(α;µ, ν) = ||ν|| − ||π2∗α|| =
||ν|| − ||µ|| =δ. Also, since γis supported on K, (2.6) implies that
α(Kc)≤ || 1
ν1(Z)µ1⊗ν1|| =||µ1|| ≤ −δ.
Therefore, D(α;µ, ν) + α(Kc)≤. So, αsatisfies (2.3). Finally, it can be seen
that αis supported on supp(µ)×supp(ν) and the claim is proved.
7
It is shown below how Theorem 2.1 implies Strassen’s theorem [20].
Corollary 2.2 (Strassen’s Theorem).Let µand νbe finite Borel measures on
Zsuch that µ(Z) = ν(Z). Then, there exists a coupling αof µand νsuch that
α({(x, y) : d(x, y)> })≤, (2.7)
where := dP(µ, ν).
Proof. Let αbe the measure in part (iii) of Theorem 2.1 for := dP(µ, ν).
One has π1∗α=µand π2∗α≤ν. The assumption µ(Z) = ν(Z) implies that
π2∗α=ν. So, αis a coupling of µand νand D(α;µ, ν ) = 0. Since the infimum
in (2.4) is attained at α, one has α({(x, y) : d(x, y)> })≤and the claim is
proved.
Remark 2.3. A variant of the Prokhorov metric is defined in [2] by a formula
similar to (2.4) (by changing the + to ∨in (2.4)). This definition, although not
identical to the classical Prokhorov metric (2.2), only differs by a factor at most
2, and hence, generates the same topology.
The following lemma is used in the proof of Theorem 2.1. It is a continuum
version of Hall’s marriage theorem and also generalizes Theorem 11.6.3 of [10].
Lemma 2.4. Let Xand Ybe separable metric spaces and µand νbe finite
Borel measures on Xand Yrespectively. Assume K⊆X×Yis a closed
subset such that for every closed set A⊆X, one has µ(A)≤ν(K(A)), where
K(A) := {y∈Y:∃x∈A: (x, y)∈K}. Then there is a Borel measure αon K
such that π1∗α=µand π2∗α≤ν.
Proof. If µand νhave finite supports and integer values, then the claim follows
easily from Hall’s marriage theorem (to show this, by splitting the atoms of µ
and νinto finitely many points, one can reduce the problem to the case where
every atom has measure one). By scaling, the same holds if µand νhave finite
supports and rational values. Note that such measures are dense in the set of
finite measures (see e.g., Lemma 4.5 in [15]).
Now, let µand νbe arbitrary measures that satisfy the assumptions of
the lemma. By the above arguments, there exist sequences (µn)nand (νn)n
of finite measures on Xand Yrespectively that converge weakly to µand ν
respectively and every µnor νnhas finite support and rational values. So the
claim holds for µnand µnfor each n. For m∈N, one can find n=n(m)
such that dP(µn, µ)<1
mand dP(νn, ν)<1
m. Add a point ato Yand define
ν0
n:= νn+2
mδaand
Km:= {(x, y) : ∃(x0, y0)∈K:d(x, x0)≤1
m, d(y, y0)≤1
m} ∪ (X× {a}).
Therefore, for any closed set A⊆X, one has
µn(A)≤µ(N1/m(A)) + 1
m≤ν(K(N1/m(A))) + 1
m
≤νn(N1/m(K(N1/m (A)))) + 2
m=ν0
n(Km(A)),
8
where Km(A)⊆Y∪{a}is defined similarly to K(A). Note that µnand ν0
nhave
finite supports and rational values. So the claim of the lemma holds for them.
Therefore, one can find a Borel measure αmon Kmsuch that π1∗αm=µn
and π2∗αm≤ν0
n. By the finiteness of µand ν, it is easy to see that the
set of measures αmis tight. So one finds a convergent subsequence of αm’s,
say converging weakly to α. Since the sets Kmare closed and nested, it can
be seen that αis supported on Kmfor any m, and hence, it is supported on
∩mKm=K∪(X×{a}). Moreover, since αm(X×{a})≤2
m,X×{a}is disjoint
from Kand Kis closed, it follows that αis supported on Konly. Finally, by
π1∗αm=µnand π2∗αm≤νn+2
mδa, one can get π1∗α=µand π2∗α≤ν. So,
the claim is proved.
3 The Gromov-Hausdorff-Prokhorov Metric
This section presents the main contribution of the paper. Roughly speaking,
the Gromov-Hausdorff metric and the Gromov-Hausdorff-Prokhorov metric are
generalized to the non-compact case (Subsection 3.3); and more precisely, to
boundedly-compact pointed (measured) metric spaces. Here, no further restric-
tions on the metric spaces are needed (e.g., being a length space or a discrete
space as in [1] and [6] respectively). As mentioned in the introduction, this
provides a metrization of the Gromov-Hausdorff (-Prokhorov) topology, where
the latter has been defined earlier in the literature. In addition, completeness,
separability, pre-compactness and weak convergence of probability measures
are studied for the Gromov-Hausdorff (-Prokhorov) metric. Moreover, in the
compact case, a Strassen-type theorem is proved for the Gromov-Hausdorff-
Prokhorov metric.
Since the Gromov-Hausdorff metric is a special case of the Gromov-Hausdorff-
Prokhorov metric (by considering metric spaces equipped with the zero mea-
sure), only the latter is discussed in this section. If the reader is interested in the
Gromov-Hausdorff metric only, he or she can assume that all of the measures
in this section are equal to zero (except in Subsection 3.6). Further discussion
is provided in Subsection 4.1.
3.1 Pointed Measured Metric (PMM) Spaces
This subsection provides the basic definitions and properties regarding (mea-
sured) metric spaces. Given metric spaces Xand Z, a function f:X→Z
is an isometric embedding if it preserves the metric; i.e., d(f(x1), f (x2)) =
d(x1, x2) for all x1, x2∈X. It is an isometry if it is a surjective isometric
embedding. For a metric space X,x∈Xand r≥0, let
Br(x) := Br(X, x) := {y∈X:d(x, y)< r},
Br(x) := Br(X, x) := {y∈X:d(x, y)≤r}.
The set Br(x) (resp. Br(x)) is called the open ball (resp. closed ball) of
radius rcentered at x. Note that Br(x) is closed, but is not necessarily the
9
closure of Br(x) in X. The metric space Xis boundedly compact if every
closed ball in Xis compact.
The rest of the paper is focused on pointed metric spaces, abbreviated
by PM spaces (Remark 3.5 explains the non-pointed case). Such a space is a
pair (X, o), where Xis a metric space and ois a distinguished point of Xcalled
the root (or the origin). A pointed measured metric space, abbreviated
by a PMM space, is a tuple X= (X, o, µ) where Xis a metric space, µis a
non-negative Borel measure on Xand ois a distinguished point of X. The balls
centered at oin Xform other PMM spaces as follows:
X(r):= Br(o), o, µ
Br(o),
X(r):= Br(o), o, µ
Br(o).
Convention 3.1. All measures in this paper are Borel measures. A PMM space
X= (X, o, µ) is called compact if Xis compact and µis a finite measure.
Also, Xis called boundedly compact if Xis boundedly compact and µis
boundedly finite; i.e., every ball in Xhas finite measure under µ.
Apointed isometry ρ: (X, o)→(X0, o0) between two PM spaces (X, o)
and (X0, o0) is an isometry ρ:X→X0such that ρ(o) = o0. A GHP-isometry
between two PMM spaces (X, o, µ) and (X0, o0, µ0) is a pointed isometry ρ:
(X, o)→(X0, o0) such that ρ∗µ=µ0. If there exists a GHP-isometry between
(X, o, µ) and (X0, o0, µ0), then they are called GHP-isometric.
Let N∗be the set of equivalence classes of boundedly compact PM spaces
under pointed isometries1. Define Nc
∗similarly by considering only compact
spaces. Also, let M∗be the set of equivalence classes of boundedly compact
PMM spaces under GHP-isometries and define Mc
∗similarly by considering
only compact PMM spaces. It can be seen that they are indeed sets.
Lemma 3.2. Let X= (X, o, µ)be a boundedly-compact PMM space.
(i) The curve t7→ Bt(o)is c`adl`ag under the Hausdorff metric and its left
limit at t=ris the closure of Br(o).
(ii) The curve t7→ µ
Bt(o)is c`adl`ag under the Prokhorov metric and its left
limit at t=ris µ
Br(o).
In fact, it will be seen that the curve t7→ X (t)is c`adl`ag under the Gromov-
Hausdorff-Prokhorov metric (see Lemma 4.2).
Proof. Let r≥0 and > 0. By compactness of the balls, it is straightforward
to show that there exists δ > 0 such that
N(Br(o)) ⊇Br+δ(o), µ(Br(o)) + ≥µ(Br+δ(o)).
1The ∗sign stands for ‘pointed’ and is included in the symbol mainly for compatibility
with the literature.
10
This implies that
dH(Br(o), Br+δ(o)) ≤, dP(µ
Br(o), µ
Br+δ(o))≤.
It follows that the curves t7→ Bt(o) and t7→ µ
Bt(o)are right-continuous.
Similarly, one can see that δcan be chosen such that
N(Br−δ(o)) ⊇Br(o), µ(Br−δ(o)) + ≥µ(Br(o)).
Since Br−δ(o)⊆Br(o), it follows that
dH(Br(o), Br−δ(o)) ≤, dP(µ
Br(o), µ
Br−δ(o))≤.
This shows that the left limits of the curves are as desired and the claim is
proved.
Definition 3.3. Let X= (X, o, µ) be a boundedly-compact PMM space. A
real number r > 0 is called a continuity radius for Xif Br(o) is the closure of
Br(o) in Xand µBr(o)\Br(o)= 0. Otherwise, it is called a discontinuity
radius for X. Equivalently, ris a continuity radius for Xif and only if the curves
t7→ Bt(o) and t7→ µ
Bt(o)(equivalently, the curve t7→ X (t)) are continuous at
t=r.
Lemma 3.4. Every boundedly-compact PMM space has at most countably many
discontinuity radii.
Proof. Every c`adl`ag function in a metric space has at most countably many
discontinuity points. So the claim is implied by Lemma 3.2.
3.2 The Metric in the Compact Case
In this subsection, the compact case of the Gromov-Hausdorff-Prokhorov metric
is recalled from [1]. A Strassen-type result is also presented for the Gromov-
Hausdorff-Prokhorov metric (Theorem 3.6). In addition, the notion of PMM-
subspace (Definition 3.11) is introduced and its properties are studied. The
latter will be used in the next subsection.
Recall that Mc
∗is the set of (equivalence classes of) compact PMM spaces.
For compact PMM spaces X= (X, oX, µX) and Y= (Y, oY, µY), define the
(compact) Gromov-Hausdorff-Prokhorov distance of Xand Y, abbrevi-
ated here by the cGHP distance, by
dc
GHP (X,Y) := inf {d(f(oX), g(oY)) ∨dH(f(X), g (X)) ∨dP(f∗µX, g∗µY)},
(3.1)
where the infimum is over all metric spaces Zand all isometric embeddings
f:X→Zand g:Y→Z.
The Gromov-Hausdorff-Prokhorov distance is define in [22] and [18] for non-
pointed metric spaces and in the case where µXand µYare probability measures.
The general case of the metric is defined in [1] by a similar formula in which +
11
is used instead of ∨, but is equivalent to (3.1) up to a factor of 3. It is proved
in [1] that dc
GHP is a metric on Mc
∗and makes it a complete separable metric
space. The same proofs work by considering the slight modification mentioned
above. The reason to consider ∨instead of + is to ensure a Strassen-type result
(Theorem 3.6 below) that provides a useful formulation of the cGHP metric in
terms of approximate couplings and correspondences.
Remark 3.5 (Non-Pointed Spaces).In the compact case, a similar metric
is defined between non-pointed spaces. It is obtained by removing the term
d(f(oX), g(oY)) from (3.1). Equivalently, by letting the distance of (X, µX) and
(Y, µY) be
min ndc
GHP (X, x, µX),(Y, y, µY):x∈X , y ∈Yo.
The results of this subsection have analogues for non-pointed spaces as well.
However, considering pointed spaces is essential in the non-compact case dis-
cussed in the next subsection.
Acorrespondence R(see e.g., [9]) between Xand Yis a relation between
points of Xand Ysuch that it is a Borel subset of X×Yand every point in X
corresponds to at least one point in Yand vice versa. The distortion of Ris
dis(R) := sup{|d(x, x0)−d(y, y0)|: (x, y )∈R, (x0, y0)∈R}.
The following is the main result of this subsection. It is a Strassen-type result
for the metric dc
GHP and is based on Theorem 2.1.
Theorem 3.6. Let X= (X, oX, µX)and Y= (Y , oY, µY)be compact PMM
spaces and ≥0.
(i) dc
GHP (X,Y)≤if and only if there exists a correspondence Rbetween
Xand Yand a Borel measure αon X×Ysuch that (oX, oY)∈R,
dis(R)≤2and D(α;µX, µY) + α(Rc)≤.
(ii) In other words,
dc
GHP (X,Y) = inf
R,α 1
2dis(R)∨D(α;µX, µY) + α(Rc)(3.2)
and the infimum is attained.
(iii) In addition, if ||µX|| ≤ ||µY||, then the infimum is attained for some R
and αsuch that π1∗α=µand π2∗α≤ν.
Remark 3.7. The formula (3.2) resembles the definition of a metric in [2] which
uses ∨instead of +. The definition in [2], although is not equal to the classical
Gromov-Hausdorff-Prokhorov metric, but is equivalent to it.
12
Remark 3.8. Theorem 3.6 generalizes Theorem 7.3.25 of [9] and Proposition 6
of [18]. The former is a result for the Gromov-Hausdorff distance; i.e., the case
where µXand µYare the zero measures. The latter is the case where µXand
µYare probability measures, where αcan be chosen to be a coupling of µXand
µYand the term D(α;µX, µY) disappears.
Proof of Theorem 3.6. Assume Ris a correspondence such that (oX, oY)∈R
and dis(R)≤2. By Theorem 7.3.25 in [9], without loss of generality, one can
assume X, Y ⊆Z,dH(X, Y )≤and if (x, y)∈R, then d(x, y)≤. Assume α
is a measure such that D(α;µX, µY) + α(Rc)≤. One has α({(x, y) : d(x, y)>
})≤α(Rc). So, Theorem 2.1 implies that dP(µX, µY)≤. This implies that
dc
GHP (X,Y)≤.
Conversely, assume dc
GHP (X,Y)≤. Let δ > . By (3.1), one can find two
isometric embeddings f:X→Zand g:Y→Zfor some Zsuch that
d(f(oX), g(oY)) ≤δ,
dH(f(X), g(Y)) ≤δ,
dP(f∗µX, g∗µY)≤δ,
(3.3)
where dHand dPare defined using this metric on Z. Let Rδ:= {(x, y)∈X×Y:
d(f(x), g(y)) ≤δ}. The first condition in (3.3) implies that (oX, oY)∈Rδ. The
second condition in (3.3) implies that Rδis a correspondence. One also has
dis(Rδ)≤2δ. The third condition in (3.3) and Theorem 2.1 imply that there
exists a measure βon Z×Zsuch that D(β;f∗µX, g∗µY) + β({(x, y)∈Z×Z:
d(x, y)> δ})≤δ. The third part of Theorem 2.1 shows that βcan be chosen
to be supported on f(X)×g(Y). Therefore, βinduces a measure αδon X×Y
by the inverses of the isometries fand g. Thus,
D(αδ;µX, µY) + αδ(Rc
δ)≤δ. (3.4)
Now, we will consider the limits of Rδand αδas δ↓. Since X×Yis
compact, Blaschke’s theorem (see e.g., Theorem 7.3.8 in [9]) implies that there
exists a subsequence of the sets Rδthat is convergent in the Hausdorff metric
to some closed subset of X×Y. Let R⊆X×Ybe the limit of this sequence.
Since each Rδis a correspondence, it can be seen that Ris also a correspondence
and (oX, oY)∈R. Also, it can be seen that the fact dis(Rδ)≤2δimplies that
dis(R)≤2. Prokhorov’s theorem on tightness [19] (see also [8] or [15]) implies
that there is a further subsequence such that the measures αδconverge weakly.
So assume αδ→αalong this subsequence. From now on, we assume δis always
in the subsequence without mentioning it explicitly.
Let hbe any continuous function on X×Ywhose support is disjoint from R
and h≤1. This implies that supp(h)∩Rδ=∅for sufficiently small δ. Therefore,
Rhdαδ≤αδ(Rc
δ). The weak convergence αδ→αgives Rhdα ≤lim inf αδ(Rc
δ).
By considering this for all h, one gets
α(Rc)≤lim inf αδ(Rc
δ).(3.5)
For considering the discrepancy D(α;µX, µY) of α, assume βis chosen in the
above argument such that the condition in part (iii) of Theorem 2.1 is satisfied,
13
hence π1∗αδ=µXand π2∗αδ≤µY. One can easily obtain π1∗α=µXand
π2∗α≤µY. Therefore, one gets
D(αδ;µX, µY) = µY(Y)−αδ(X×Y),
D(α;µX, µY) = µY(Y)−α(X×Y).
These equations enable us to obtain that D(α;µX, µY) = lim D(αδ;µX, µY).
Finally, (3.4) and (3.5) imply that D(α;µX, µY) + α(Rc)≤. Therefore, Rand
αsatisfy the claim. This proves parts (i) and (ii) of the theorem.
As mentioned above, if βis chosen such that π1∗β=f∗µXand π2∗β≤g∗µY,
then the claim of part (iii) is obtained. So the proof is completed.
Theorem 3.6 readily implies the following.
Corollary 3.9. The infimum in the definition of the cGHP metric (3.1) is
attained.
The following are further properties of dc
GHP which are needed later.
Lemma 3.10. For compact PMM spaces X= (X, oX, µX)and Y= (Y, oY, µY),
max{d(oY, y) : y∈Y} ≤ max{d(oX, x) : x∈X}+ 2dc
GHP (X,Y).
Proof. Let := dc
GHP (X,Y). By Theorem 3.6, there is a correspondence R
between Xand Ysuch that (oX, oY)∈Rand dis(R)≤2. Let y∈Ybe
arbitrary. There exists x∈Xthat R-corresponds to y. Since dis(R)≤2, one
gets d(oY, y)≤d(oX, x)+2. This implies the claim.
The following definition and results are needed for the next subsection.
Definition 3.11. Let X= (X, o, µ) and X0= (X0, o0, µ0) be PMM spaces. X0
is called a PMM-subspace of Xif X0⊆X,o0=oand µ0≤µ. The following
symbol is used to express that X0is a PMM-subspace of X:
X0 X .
For two PMM-subspaces Xi= (Xi, o, µi) of X(i= 1,2), their Hausdorff-
Prokhorov distance is defined by
dHP (X1,X2) := dH(X1, X2)∨dP(µ1, µ2).(3.6)
This equation immediately gives
dc
GHP (X1,X2)≤dH P (X1,X2).(3.7)
Lemma 3.12. Let Xand Ybe compact PMM spaces.
(i) If X0is a compact PMM-subspace of X, then there exists a compact PMM-
subspace Y0of Ysuch that
dc
GHP (X0,Y0)≤dc
GHP (X,Y).
14
Figure 1: A schematic picture of the sets in the proof of Lemma 3.12. The sets
X, Y, X0, Y 0, Y0and Br(oX) are depicted as intervals one of whose end points is
the lower left corner of the figure and the other end is shown by a label.
(ii) Let := dc
GHP (X,Y)and r≥2be arbitrary. If in addition to (i), one
has X(r) X 0 X , then Y0can be chosen such that Y(r−2) Y0 Y.
Proof. Let X=: (X, oX, µX), Y=: (Y, oY, µY) and := dc
GHP (X,Y). By
Theorem 3.6, there exists a correspondence Rbetween Xand Yand a measure
αon X×Ysuch that (oX, oY)∈R, dis(R)≤2and D(α;µX, µY) + α(Rc)≤.
By part (iii) of the theorem, we may assume π1∗α≤µXand π2∗α≤µY. Also,
by replacing Rwith its closure in X×Yif necessary, we might assume Ris
closed without loss of generality. Let X0=: (X0, oX, µ0
X).
(i). Let Y0be the set of points in Ythat R-correspond to some point in
X0. Let α1:= α
X0×Y0. By Lemma 1 of [21], there exists a measures α0≤α1
on X0×Y0such that π1∗α0=π1∗α1∧µ0
X. Consider the measure µ0
Y:= π2∗α0
on Y0. We claim that Y0:= (Y0, oY, µ0
Y) satisfies the desired property. Note
that Y0is a closed subset of Y,oY∈Y0and µ0
Y≤π2∗α1≤π2∗α≤µY. So
Y0 Y. Let R0:= R∩(X0×Y0). The definition of Y0gives that R0is a
correspondence between X0and Y0and (oX, oY)∈R0. Also, it is clear that
dis(R0)≤dis(R)≤2. By Theorem 3.6, it remains to prove that
D(α0;µ0
X, µ0
Y) + α0((R0)c)≤. (3.8)
Let C1:= X0×(Y\Y0) and C2:= (X0×Y0)\R0(see Figure 1). One has
α0((R0)c)≤α(C2). Since π1∗α0≤µ0
X, one gets
||π1∗α0−µ0
X|| =||µ0
X|| − ||π1∗α0|| =||µ0
X|| − ||π1∗α1∧µ0
X||.
15
Since µ0
Xand π1∗α1are bounded by µX
X0, one can easily deduce that
||π1∗α0−µ0
X|| ≤ ||µX
X0|| − ||π1∗α1||
=µX(X0)−α1(X0×Y0)
=µX(X0)−α(X0×Y) + α(C1)
≤ ||π1∗α−µX|| +α(C1).(3.9)
Since ||π2∗α0−µ0
Y|| = 0, one gets that
D(α0;µ0
X, µ0
Y)≤ ||π1∗α−µX|| +α(C1)≤D(α;µX, µY) + α(C1).
Therefore,
D(α0;µX
X0, µ0
Y) + α0((R0)c)≤D(α;µX, µY) + α(C1∪C2)
≤D(α;µX, µY) + α(Rc)
≤,
where the first inequality is because C1∩C2=∅and the second inequality is
because C1and C2are disjoint from R, which is easy to see. So, (3.8) is proved
and the proof is completed.
(ii). Let Y0:= Br−2(oY). Define Y0, R0, α1and α0as in part (i) and
replace µ0
Yby µ00
Y:= π2∗α0∨µY
Y0. Let y∈Y0be arbitrary. Since Ris a
correspondence, there exists x∈Xsuch that (x, y)∈R. Since dis(R)≤2, one
gets that d(x, oX)≤d(y, oY) + 2≤r. This implies that x∈Br(oX)⊆X0. The
definition of Y0implies that y∈Y0. Hence, Y0⊇Y0and so µ00
Yis supported
on Y0. We will show that Y00 := (Y0, oY, µ00
Y) satisfies the claim. Note that
(µY
Y0)≤µ00
Y≤(µY
Y0). This gives that Y(r−2) Y00 Y.
Define C1and C2as in part (i). The proof of part (i) shows that (oX, oY)∈
R0, dis(R0)≤dis(R)≤2,α0((R0)c) = α0(C2) and (3.9) holds. To bound
||π2∗α0−µ00
Y||, note that π2∗α0≤µY
Y0on Y0. So the definition of µ00
Ygives that
||π2∗α0−µ00
Y|| =µY(Y0)−π2∗α0(Y0)
=µY(Y0)−α0(X0×Y0)
=µY(Y0)−α0(Br(oX)×Y0)−α0(C3),
where C3:= (X0\Br(oX)) ×Y0. Since µ0
Xagrees with µXon Br(oX), one gets
that π1∗α1≤µ0
Xon Br(oX). So the definition of α0implies that π1∗α0=π1∗α1
on Br(oX). The condition α0≤α1gives that α0=α1=αon Br(oX)×Y0. So,
by letting C4:= (X\X0)×Y0, the above equation gives
||π2∗α0−µ00
Y|| =µY(Y0)−α(Br(oX)×Y0)−α0(C3)
=µY(Y0)−α(X×Y0) + α(C3∪C4)−α0(C3)
≤ ||µY−π2∗α|| +α(C3∪C4)−α0(C3).
16
The above discussions show that C3∩R=∅, which implies that C3⊆C2. Also,
note that the four sets C1, C2, C4, R are pairwise disjoint. So, by summing up,
we get
D(α0;µ0
X, µ00
Y) + α0((R0)c)≤ ||µX−π1∗α|| +α(C1) +
||µY−π2∗α|| +α(C3∪C4)−α0(C3) + α0(C2)
≤D(α;µX, µY) + α(C1∪C3∪C4) + α0(C2\C3)
≤D(α;µX, µY) + α(C1∪C2∪C4)
≤D(α;µX, µY) + α(Rc)
≤.
Finally, Theorem 3.6 implies that dc
GHP (X0,Y00 )≤and the claim is proved.
Lemma 3.13. If Xis a compact PMM space, then the set of compact PMM-
subspaces of Xis compact under the topology of the metric dc
GHP .
Proof. By (3.7), it is enough to show that the set of compact PMM-subspaces
of Xis compact under the metric dHP . Let X=: (X, o, µ) and consider a
sequence Xn= (Xn, o, µn) of PMM-subspaces of X. Blaschke’s theorem (see
e.g., Theorem 7.3.8 in [9]) implies that the set of compact subsets of Xis
compact under dH. Also, the set of measures on Xwhich are bounded by µ
is tight and closed (under weak convergence). So Prokhorov’s theorem implies
that the latter is compact. So by passing to a subsequence, one may assume that
dH(Xn, Y )→0 and dP(µn, ν )→0 for some compact subset Y⊆Xand some
measure ν≤µ. It is left to the reader to show that o∈Yand νis supported
on Y. This implies that dH P (Xn,(Y, o, ν)) →0 and the claim is proved.
3.3 The Metric in the Boundedly-Compact Case
This subsection presents the definition of the Gromov-Hausdorff-Prokhorov met-
ric in the boundedly-compact case and proves that it is indeed a metric. Mean-
while, K¨onig’s infinity lemma is generalized to compact sets (Lemma 3.16) and
is used in the proofs. The Gromov-Hausdorff metric is a special case and will
be discussed in Subsection 4.1.
Let Xand Ybe boundedly-compact PMM spaces. According to the heuristic
mentioned in the introduction, the idea is that Xand Yare close if two large
compact portions of the two spaces are close under the metric dc
GHP . In the
definition, for a fixed large r, the ball X(r)is not needed to be close to Y(r)
due to the points that are close to the boundaries of the balls. Instead, the
former should be close to a perturbation of the latter. This is made precise in
the following (see Remark 3.20 for another definition and also Theorem 3.24).
For r≥≥0, define
a(, r;X,Y) := inf{dc
GHP (X(r),Y0)},(3.10)
where the infimum is over all compact PMM-subspaces Y0of Y(Definition 3.11)
such that Y(r−) Y0 Y (by removing the condition Y(r−) Y0, all of the
17
results will remain valid except maybe those in Subsection 3.4). Lemma 3.17
below proves that the infimum is attained. The case r= 1/ is mostly used in
the following. So, for 0 < ≤1, define
a(X,Y) := a(, 1/;X,Y).
Of course, this is not a symmetric function of Xand Y.
Definition 3.14. Let Xand Ybe boundedly-compact PMM spaces. The
Gromov-Hausdorff-Prokhorov (GHP) distance of Xand Yis defined
by
dGHP (X,Y) := inf {∈(0,1] : a(X,Y)∨a(Y,X)<
2},(3.11)
with the convention that inf ∅:= 1.
In fact, Lemma 3.19 below implies that the infimum in (3.11) is not attained.
Note that we always have
0≤dGHP (X,Y)≤1.(3.12)
The following theorem is the main result of this subsection. Further prop-
erties of the function dGHP are discussed in the next subsections.
Theorem 3.15. The GHP distance (3.11) induces a metric on M∗.
To prove this theorem, the following lemmas are needed.
Lemma 3.16 (K¨onig’s Infinity Lemma For Compact Sets).Let Cnbe a compact
set for each n∈Nand fn:Cn→Cn−1be a continuous function for n > 1.
Then, there exists a sequence x1∈C1, x2∈C2, . . . such that fn(xn) = xn−1for
each n > 1.
This lemma is a generalization of K¨onig’s infinity lemma, which is the special
case where each Cnis a finite set.
Proof. Let C0be a single point and f1:C1→C0be the unique function. For
m>n, let fm,n := fn+1 ◦ · · · ◦ fm. Note that for every n, the sets fm,n(Cm) for
m=n+1, n +2, . . . are nested. We will define the sequence xn∈Cninductively
such that xnis in the image of fm,n for every m > n. Let x0:= 0 which has
that property. Assuming xn−1is defined, let xnbe an arbitrary point in the
intersection of f−1
n(xn−1) and T∞
m=n+1 fm,n(Cm) (note that the intersection is
nonempty by compactness and the induction hypothesis). It can be seen that
xnsatisfies the induction claim and the lemma is proved.
Lemma 3.17. The infimum in (3.10) is attained.
Proof. The claim is implied by Lemma 3.13 and the fact that dc
GHP is a metric
on Mc
∗.
18
Lemma 3.18. The number a(, r;X,Y)is non-increasing w.r.t. . Moreover,
if a(, r0;X,Y)≤
2, then a(, r;X,Y)is non-decreasing w.r.t. rin the interval
r∈[, r0].
Proof. The first claim is easy to check. For the second claim, it is enough to
prove that for r∈[, r0), one has a(, r;X,Y)≤a(, r0;X,Y).
Let a(, r0;X,Y) =: δ≤
2. By Lemma 3.17, there is a compact PMM-
subspace Y0of Ysuch that Y(r0−) Y0and dc
GHP (X(r0),Y0)≤δ. By
Lemma 3.12, there is a further compact PMM-subspace Y00 of Y0such that
Y0(r−2δ) Y00 and dc
GHP (X(r),Y00 )≤δ. Since 2δ≤and r < r0by assump-
tion, one gets that Y(r−) Y00. Therefore, a(, r;X,Y)≤δby definition. This
proves the claim.
Lemma 3.19. For δ:= dGHP (X,Y)≤1, one has
aδ(X,Y)∨aδ(Y,X)≥δ
2.
In addition, if dGHP (X,Y)< γ ≤1, then
aγ(X,Y)∨aγ(Y,X)<γ
2.
Proof. For the first claim, assume that for δ:= dGHP (X,Y), one has aδ(X,Y)∨
aδ(Y,X)<δ
2−α, where α > 0. So there exists a compact PMM-subspace
Y(1/δ−δ) Y0 Y such that dc
GHP (X(1/δ),Y0)<δ
2−α. Let Y=: (Y, oY, µY)
and Y0=: (Y0, oY, µ0
Y). Lemma 3.2 implies that there exists < δ such that
dHP (X(1/),X(1/δ))<α
4. Let Y0:= B1/−(oY). By a similar argument to
Lemma 3.2, <δcan be chosen such that dHP (Y0,Y00)<α
4, where Y00 :=
(Y0∪Y0, oY, µ0
Y∨µY
Y0). Note that Y(1/−) Y00 Y. The triangle inequality
implies that dHP (X(1/),Y00 )<δ
2−α
2. Now, if is chosen such that >δ−α,
the definition (3.10) gives that a(X,Y)<
2. Similarly, < δ can be chosen
such that a(Y,X)<
2. This gives that dGHP (X,Y)≤<δ, which is a
contradiction.
For the second claim, since dGHP (X,Y)< γ ≤1, (3.11) implies that there
exists <γsuch that a(X,Y)∨a(Y,X)<
2. The second claim in Lemma 3.18
implies that a(, 1/γ;X,Y)∨a(, 1/γ;Y,X)<
2. Therefore, the first claim in
Lemma 3.18 implies that aγ(X,Y)∨aγ(Y,X)<
2<γ
2.
Proof of Theorem 3.15. It is easy to see that dGH P (X,Y) depends only on the
isometry classes of Xand Y. Therefore, it induces a function on M∗×M∗,
which is denoted by the same symbol dGHP . It is immediate that dGH P is
symmetric and dGHP (X,X) = 0.
Let X,Yand Zbe boundedly compact PMM spaces. Assume dGH P (X,Y)<
and dGHP (Y,Z)< δ. For the triangle inequality, it is enough to show that
19
dGHP (X,Z)≤+δ. If +δ≥1, the claim is clear by (3.12). So assume +δ < 1.
By Lemma 3.19, one gets that a(X,Y)∨a(Y,X)<
2and aδ(Y,Z)∨aδ(Z,Y)<
δ
2. Lemma 3.18 implies that a(, 1/(+δ); X,Y)<
2. Therefore, by (3.10), there
is a compact PMM-subspace Y0= (Y0, oY, µ0
Y) of Ysuch that Y(1/(+δ)−) Y0
and
dc
GHP (X(1/(+δ)) ,Y0)<
2.
So, Lemma 3.10 implies that Y0 Y(1/(+δ)+). It is straightforward to deduce
from 0 < +δ < 1 that +1
+δ<1
δ. Therefore, Y0 Y(1/δ).
On the other hand, by (3.10), there exists a compact PMM-subspace Z0=
(Z0, oZ, µ0
Z) of Zsuch that Z(1/δ−δ) Z0and dc
GHP (Y(1/δ),Z0)<δ
2. By
Lemma 3.12, there is a further compact PMM-subspace Z00 of Z0such that
Z0(1/(+δ)−−δ) Z00 and
dc
GHP (Y0,Z00 )<δ
2.
The triangle inequality for dc
GHP gives
dc
GHP (X(1/(+δ)) ,Z00)≤
2+δ
2.
Since Z(1/(+δ)−−δ) Z00, (3.10) implies that a+δ(X,Z)<(+δ)/2. Similarly,
one obtains a+δ(Z,X)<(+δ)/2. Therefore, dGHP (X,Z)≤+δand the
triangle inequality is proved.
The last step is to prove that dGHP (X,Y) = 0 implies that Xand Yare
GHP-isometric. Fix r≥0 and let 0 <<1 be arbitrary. Lemma 3.19 implies
that a(X,Y)<
2. Therefore, assuming r < 1
, (3.10) and Lemma 3.12 imply
that there exists a PMM-subspace Yof Ysuch that
dc
GHP (X(r),Y)<
2.
By Lemmas 3.10 and 3.13, There is a convergent subsequence of the subspaces
under the metric dc
GHP , say Yn→ Y0 Y , where n→0. It follows that
dc
GHP (X(r),Y0) = 0. Since dc
GHP is a metric on Mc
∗,X(r)is GHP-isometric to
Y0. In particular, Lemma 3.10 implies that Y0 Y (r). On the other hand, one
can similarly find a PMM-subspace X0of Xwhich is GHP-isometric to Y(r)and
X0 X (r). These facts imply that X(r)and Y(r)are themselves GHP-isometric
as follows: If f:X(r)→ Y0and g:Y(r)→ X 0are GHP-isometries, then,
g◦f:X(r)→ X 0is also a GHP-isometry. Compactness of X(r), finiteness of
the measure on X(r)and X0 X (r)imply that g◦fis surjective and X0=X(r).
To prove that Xis GHP-isometric to Y, let Cnbe the set of GHP-isometries
from X(n)to Y(n)for n= 1,2, . . ., which is shown to be non-empty. The
topology of uniform convergence makes Cna compact set. The restriction map
20
f7→ f
X(n−1) induces a continuous function from Cnto Cn−1. Therefore, the
generalization of K¨onig’s infinity lemma (Lemma 3.16) implies that there is a
sequence of GHP-isometries ρn∈Cnsuch that ρn−1is the restriction of ρn
to X(n−1) for each n. Thus, these isometries can be glued together to form a
GHP-isometry between Xand Y, which proves the claim.
Remark 3.20. By Lemma 3.2, it is easy to see that
Z∞
0
e−r1∧dc
GHP (X(r),Y(r))dr (3.13)
is well defined for all X,Y ∈ M∗and defines a semi-metric on M∗(such formulas
are common in various settings in the literature). With similar arguments to
those in the present section, it can be shown that this is indeed a metric and
makes M∗a complete separable metric space as well. However, we preferred
to use the formulation of Definition 3.14 to avoid the issues regarding non-
monotonicity of dc
GHP (X(r),Y(r)) as a function of r. In addition, Lemma 3.12
enables us to have more quantitative bounds in the arguments. Nevertheless,
Theorem 3.24 below implies that the two metrics generate the same topology.
Remark 3.21. Let Zbe a metric space, Fbe the set of boundedly-compact
subsets of Zand Mbe the set of boundedly-finite Borel measures on Z(up to
no equivalence relation). By formulas similar to either (3.11) or (3.13), one can
extend the Hausdorff metric and the Prokhorov metric to Fand Mrespectively.
This can be done by fixing a point o∈Z, letting X(r):= X∩Br(o) for X⊆Z
and letting µ(r):= µ
Br(o)for measures µon Z(let dH(∅, X) := ∞whenever
X6=∅). By similar arguments, one can show that formulas similar to (3.11)
or (3.13) give metrics on Fand Mrespectively. Moreover, if Zis complete and
separable, then Fand Mare also complete and separable (this can be proved
similarly to the results of Subsection 3.5 below). In this case, the metrics on F
and Mare metrizations of the Fell topology and the vague topology respectively.
The details are skipped for brevity. See Subsection 4.3 and [16] for further
discussion.
3.4 The Topology of the GHP Metric
Gromov [13] has defined a topology on the set of boundedly-compact pointed
metric spaces, which is called the Gromov-Hausdorff topology in the literature
(see also [9]). In addition, the Gromov-Hausdorff-Prokhorov topology (see [22])
is defined on the set M∗of boundedly-compact PMM spaces (it is called the
pointed measured Gromov-Hausdorff topology in [22]). In this subsection, it is
shown that the metric dGHP of the present paper is a metrization of the Gromov-
Hausdorff-Prokhorov topology. The main result is Theorem 3.24 which provides
criteria for convergence under the metric dGHP . The Gromov-Hausdorff topol-
ogy will be studied in Subsection 4.1.
Lemma 3.22. Let X,Y ∈ M∗be PMM spaces.
21
(i) For all r≥0,
dGHP (X,Y)≤1
r∨2dc
GHP (X(r),Y(r)).
(ii) If X,Y ∈ Mc
∗are compact, then
dGHP (X,Y)≤2dc
GHP (X,Y).
(iii) The topology on Mc
∗induced by the metric dGHP is coarser than that of
dc
GHP .
Proof. (i). Since dGHP (X,Y)≤1, we can assume r≥1 without loss of general-
ity. Let > 1/r∨2dc
GHP (X(r),Y(r)). It is enough to prove that ≥dGHP (X,Y).
This is trivial if ≥1. So assume < 1. By letting Y0:= Y(r)in (3.10), one gets
that a(, r;X,Y)≤dc
GHP (X(r),Y(r))<
2. So, the fact 1
< r and Lemma 3.18
imply that a(X,Y)<
2. Similarly, one gets a(Y,X)<
2. So (3.11) gives that
dGHP (X,Y)≤and the claim is proved.
(ii). The claim is implied by part (i) by letting rlarge enough such that
X(r)=X,Y(r)=Yand 1
r≤2dc
GHP (X,Y).
(iii). By the previous part, any convergent sequence under dc
GHP is also
convergent under dGHP . This implies the second claim.
Remark 3.23. In fact, the topology of the metric dGHP on Mc
∗is strictly
coarser than that of dc
GHP since having dGHP (Xn,X)→0 does not imply
dc
GHP (Xn,X)→0; e.g., when Xn:= {0, n}and X:= {0}endowed with the
Euclidean metric and the counting measure (in general, adding the assumption
sup diam(Xn)<∞is sufficient for convergence under dc
GHP ). This is similar to
the fact that the vague topology on the set of measures on a given non-compact
metric space is strictly coarser than the weak topology (see e.g., [15]). A similar
property holds for the set of compact subsets of a given non-compact metric
space.
Theorem 3.24 (Convergence).Let Xand (Xn)n≥0be boundedly compact PMM
spaces. Then the following are equivalent:
(i) Xn→ X in the metric dGHP .
(ii) For every r > 0and > 0, for large enough n, there exists a compact
PMM-subspace X0
nof Xsuch that X(r−) X 0
nXand dc
GHP (X(r)
n,X0
n)<
.
(iii) For every r > 0and > 0, for large enough n, there exist compact PMM-
subspaces of Xand Xnwith dc
GHP -distance less than such that they
contain (as PMM-subspaces) the balls of radii rcentered at the roots of X
and Xnrespectively.
22
(iv) For every continuity radius rof X(Definition 3.3), one has X(r)
n→ X (r)
in the metric dc
GHP as n→ ∞.
(v) There exists an unbounded set I⊆R≥0such that for each r∈I, one has
X(r)
n→ X (r)in the metric dc
GHP as n→ ∞.
(vi) limn→∞ R∞
0e−r1∧dc
GHP (X(r)
n,X(r))dr = 0.
Proof. (i)⇒(ii). Assume Xn→ X . Let r > 0 and > 0 be given. One
may assume < 1
rwithout loss of generality. For large enough n, one has
dGHP (Xn,X)< . If so, Lemma 3.19 imply that a(Xn,X)<
2. So Lemma 3.18
gives a(, r;Xn,X)<
2. Now, the claim is implied by (3.10).
(ii)⇒(iii). The claim of part (iii) is directly implied from part (ii) by replac-
ing rwith r+.
(iii)⇒(i). Let > 0 be arbitrary and r= 1/(2). Assume nis large enough
such that there exist compact PMM-subspaces X(r) X 0 X and X(r)
n
X0
n Xnsuch that dc
GHP (X0,X0
n)< . By Lemma 3.12, there exists a compact
PMM-subspace X(r−2) X 00 X such that dGHP (X(r)
n,X00)< . This implies
that a(2, r;Xn,X)< , hence, a2(Xn,X)< . Similarly, a2(X,Xn)< , which
implies that dGHP (Xn,X)≤2. This proves that Xn→ X .
(ii)⇒(iv). Let Xn=: (Xn, on, µn), X=: (X, o, µ) and rbe a continuity
radius for X. Let > 0 be arbitrary. The assumption on rimplies that there
exists δ > 0 such that
dH(Br+δ(o), Br−δ(o)) ≤,
dP(µ
Br+δ(o), µ
Br−δ(o))≤.
Part (ii), which is assumed, implies that for large enough n, there exists a com-
pact PMM-subspace Ynof Xsuch that X(r−δ) YnXand dc
GHP (X(r)
n,Yn)<
δ/2. The latter and Lemma 3.10 imply that Yn X (r+δ). Now, Ynand X(r)
both contain (ass PMM-subspaces) X(r−δ)and are contained in X(r+δ). By
using the definitions (2.1) and (2.2) of the Hausdorff and the Prokhorov met-
rics directly, one can deduce that dHP (Yn,X(r))≤dH P (X(r+δ),X(r−δ))≤.
So (3.7) implies that dc
GHP (Yn,X(r))≤. Finally, the triangle inequality gives
dc
GHP (X(r)
n,X(r))< +δ
2. Since , δ are arbitrarily small, this implies that
X(r)
n→ X (r)and the claim is proved.
(iv)⇒(v). The claim is implied by Lemma 3.4.
(v)⇒(i). The claim is easily implied by part (i) of Lemma 3.22 and is left
to the reader.
(iv)⇒(vi). By Lemma 4.2, the integrand is a c`adl`ag function of r, and hence,
measurable. Since Xhas countably many discontinuity radii (Lemma 3.4), the
claim follows by Lebesgue’s dominated convergence theorem.
(vi)⇒(iv). To prove this part, some care is needed since the converse of the
dominated convergence theorem does not hold in general, and hence, the above
23
arguments do not work. Let rbe a continuity radius of Xand 0 <<1. By
Definition 3.3, there exists δ > 0 such that δ < /2 and
dHP (X(r+2δ),X(r−2δ))<
2.
Let γn(s) := dc
GHP (X(s)
n,X(s)). By (vi), there exists Nsuch that for all n≥N,
Z∞
0
e−s(1 ∧γn(s)) ds < δe−r.(3.14)
To prove the claim, it is enough to show that for all n≥N, one has γn(r)≤.
Let n≥Nbe arbitrary. First, assume that there exists s>rsuch that
γn(s)≤δ. By Lemmas 3.12 and 3.10, there exists a compact PMM-subspace
X0 X such that dc
GHP (X(r)
n,X0)≤δand X(r−2δ) X 0 X (r+2δ). It can be
seen that the latter implies that
dHP (X(r),X0)≤dH P (X(r+2δ),X(r−2δ))<
2.
The triangle inequality for dc
GHP gives that
γn(r) = dc
GHP (X(r)
n,X(r))≤dc
GHP (X(r)
n,X0) + dHP (X0,X(r))< .
So the claim is proved in this case. Second, assume that for all s > r, one has
γ(s)> δ. This gives that R∞
0e−s(1 ∧γ(s))ds ≥δe−r. This contradicts (3.14).
So the claim is proved.
It is known that convergence under the metric dc
GHP can be expressed using
approximate GHP-isometries (see e.g., page 767 of [22] and Corollary 7.3.28
of [9]). This is expressed in the following lemma, whose proof is skipped.
An -isometry (see e.g., [9]) between metric spaces Xand Yis a function
f:X→Ysuch that sup{|d(x1, x2)−d(f(x1), f(x2))|:x1, x2∈X} ≤ and for
every y∈Y, there exists x∈Xsuch that d(y, f(x)) ≤.
Lemma 3.25. Let X= (X, o, µ)and Xn= (Xn, on, µn)be compact PMM-
spaces (n= 1,2, . . .). Then Xn→ X in the metric dc
GHP if and only if for every
> 0, for large enough n, there exists a measurable -isometry f:Xn→X
such that f(on) = oand dP(f∗µn, µ)< .
In fact, one can prove a quantitative form of this lemma that relates the
existence of such fto the value of dc
GHP (Xn,X) (similarly to Equation (27.3)
of [22]).
The notion of approximate GHP-isometries is also used in [9] and Defini-
tion 27.30 of [22] to define convergence of boundedly-compact PM spaces and
PMM spaces as follows: (Xn, on, µn) tends to (X, o, µ) when there exist se-
quences rk→ ∞ and k→0 and measurable k-isometries fk:Brk(ok)→
Brk(o) such that fk∗µktends to µin the weak-∗topology (convergence against
compactly supported continuous functions). By part (v) of Theorem 3.24, the
reader can verify the following.
24
Theorem 3.26. The metric dGHP is a metrization of the Gromov-Hausdorff-
Prokhorov topology (Definition 27.30 of [22]).
See also Theorem 4.1 for a version of this result for the Gromov-Hausdorff
topology.
3.5 Completeness, Separability and Pre-Compactness
The following two theorems are the main results of this subsection. Recall
that a Polish space is a topological space which is homeomorphic to a complete
separable metric space.
Theorem 3.27. Under the GHP metric, M∗is a complete separable metric
space.
The proof of Theorem 3.27 is postponed to after proving Theorem 3.28.
Recall that a subset Sof a metric space Xis relatively compact (or pre-
compact) when every sequence in Shas a subsequence which is convergent in
X; i.e., the closure of Sin Xis compact. The following gives a pre-compactness
criteria for the GHP metric.
Theorem 3.28 (Pre-compactness).A subset C ⊆ M∗is relatively compact
under the GHP metric if and only if for each r≥0, the set of (equivalence
classes of the) balls Cr:= {X (r):X ∈ C} is relatively compact under the metric
dc
GHP .
For a pre-compactness criteria for the metric dc
GHP , see Theorem 2.6 of [1].
Proof of Theorem 3.28. (⇒). First, assume Cis pre-compact, r≥0 and (Xn)n
is a sequence in C. We will prove that the sequence (X(r)
n)nhas a convergent
subsequence, which proves that Cris pre-compact. By pre-compactness of C,
one finds a convergent subsequence of Xn. So, one may assume Xn→ Y under
the metric dGHP from the beginning without loss of generality. Choose n>
dGHP (Xn,Y) such that n→0. We can assume n<1 for all nwithout loss
of generality. Lemma 3.19 implies that an(Xn,Y)<1
2n. So, Lemma 3.18
implies that δn:= a(1/r, r;Xn,Y)→0. By the definition of ain (3.10) and
Lemma 3.17, one finds a PMM-subspace Ynof Ysuch that
dc
GHP (X(r)
n,Yn)≤δn.(3.15)
Lemma 3.10 gives Yn Y(r+2δn). So, by Lemma 3.13, one can find a convergent
subsequence of the subspaces Ynunder the metric dc
GHP , say tending to Y0 Y.
By passing to this subsequence, one may assume Yn→ Y0from the beginning.
Now, (3.15) implies that dc
GHP (X(r)
n,Y0)→0, which proves the claim (it should
be noted that the limit Y0satisfies Y(r) Y0 Y(r), but is not necessarily
equal to Y(r)).
25
(⇐). Conversely assume Cris pre-compact for every r≥0. Let (Xn)nbe
a sequence in C. The claim is that it has a convergent subsequence under the
metric dGHP . For each given m∈N, by pre-compactness of Cm, one finds a
subsequence of (X(m)
n)nthat is convergent in the dc
GHP metric. By a diagonal
argument, one finds a subsequence n1< n2< . . . such that for every m∈N,
the sequence X(m)
niis convergent as i→ ∞. By passing to this subsequence, we
may assume from the beginning that X(m)
nis convergent as n→ ∞, say, to Ym,
for each m∈N(in the metric dc
GHP ); i.e.,
∀m∈N:X(m)
n→ Ym.
The next step is to show that these limiting spaces Ymcan be glued together to
form a PMM space. Let 1 < m ∈Nbe given. For each n, Lemma 3.12 implies
that there is a PMM-subspace Zm,n of Ymsuch that dc
GHP (X(m−1)
n,Zm,n)≤
dc
GHP (X(m)
n,Ym). This implies that dc
GHP (X(m−1)
n,Zm,n)→0 as n→ ∞. By
Lemma 3.13, the sequence (Zm,n)nhas a convergent subsequence in the metric
dc
GHP , say, tending to Zm Ym. Therefore, dc
GHP (X(m−1)
n,Zm) tends to zero
along the subsequence. On the other hand, the definition of Ym−1implies that
X(m−1)
n→ Ym−1as n→ ∞. Thus, dc
GHP (Ym−1,Zm) = 0; i.e., Ym−1is GHP-
isometric to Zmwhich is a PMM-subspace of Ym. This shows that Ym’s can
be paste together to form a PMM space which is denoted by Y. So, from the
beginning, we may assume Ymis a PMM-subspace of Yfor each m.
In the next step, it will be shown that Yis boundedly-compact. The above
application of Lemma 3.12 also implies that Zm,n contains a large ball in Ym.
More precisely, Y(m−1−δn)
m Zm,n for some δn>0 that tends to zero. By
letting ntend to infinity, we get Y(m−1)
m Ym−1(assuming Ym−1is a PMM-
subspace of Ymas above). By an induction, one obtains that Y(m−1)
m0 Ym−1
for every m0≥m. Now, the definition of Yimplies that Y(m) Ym(note also
that Ym Y(m)). This implies that Yis boundedly-compact.
The final step is to show that Xn→ Y in the metric dGHP . Fix > 0
and let m > 1/ be arbitrary. Equation (3.10) and Y(m) Ymimply that
a(, m;Xn,Y)≤dc
GHP (X(m)
n,Ym). By using Lemma 3.18 and the fact that
dc
GHP (X(m)
n,Ym) tends to zero as n→0 one can show that a(Xn,Y)→0. On
the other hand, since Y(m−1) Y(m) Ym, one can use Lemma 3.12 and show
that a(, m −1; Y,Xn)→0 as n→ ∞. By similar arguments, one can show
that a(Y,Xn)→0. This implies that dGH P (Xn,Y)< for large enough n
(see Definition 3.14). Since is arbitrary, one gets Xn→ Y and the claim is
proved.
Proof of Theorem 3.27. The definition of the GHP metric directly implies that
dGHP (X,X(r))≤1
r
26
for every X ∈ M∗and r > 0. Hence, X(r)→ X as r→ ∞. So, the subset
Mc
∗⊆M∗formed by compact spaces is dense. As noted in Subsection 3.2, Mc
∗
is separable under the metric dc
GHP . Lemma 3.22 implies that Mc
∗is separable
under dGHP as well. One obtains that M∗is separable.
For proving completeness, assume (Xn)nis a Cauchy sequence in M∗under
the metric dGHP . Below, we will show that this sequence is pre-compact. This
proves that there exists a convergent subsequence. Being Cauchy implies con-
vergence of the whole sequence and the claim is proved. By Theorem 3.28, to
show pre-compactness of the sequence, it is enough to prove that for a given
r≥0, the sequence of balls (X(r)
n)nis pre-compact under the metric dc
GHP .
Let 0 << 1
r. There exists msuch that for all n>m,dGHP (Xn,Xm)<
. By Lemmas 3.19 and 3.18, one gets a(, r;Xn,Xm)<
2. Therefore, there
exists a compact PMM-subspace Zm,n of Xmsuch that dc
GHP (X(r)
n,Zm,n)≤
2.
Lemma 3.10 gives that Zm,n X (r+)
m. So, by Lemma 3.13, the sequence
(Zm,n)nhas a convergent subsequence under the metric dc
GHP , say, tending to
Zm Xm. Therefore, one finds a subsequence of the balls (X(r)
n)n>m such
that dc
GHP (X(r)
n,Zm)< on the subsequence. Hence, any two elements of the
subsequence have distance less than 2. By doing this for different values of
iteratively; e.g., for =1
2r,1
3r, . . ., and by a diagonal argument, one finds a
sequence n1, n2, . . . such that (X(r)
ni)iis a Cauchy sequence under the metric
dc
GHP . Therefore, by completeness of the metric dc
GHP (see Subsection 3.2),
this sequence is convergent. So, by the arguments of the previous paragraph,
the claim is proved.
3.6 Random PMM Spaces and Weak Convergence
Theorem 3.27 shows that the space M∗, equipped with the GHP metric dGHP ,
is a Polish space. This enables one to define a random PMM space Xas a
random element in M∗and the probability space will be standard. The distri-
bution of Xis the probability measure µon M∗defined by µ(A) := P[X∈A].
In this subsection, weak convergence of random PMM spaces are studied.
Let X1,X2,· · · and Xbe random PMM spaces. Let µn(resp. µ) be the
distribution of Xn(resp. X). Prokhorov’s theorem [19] implies that Xncon-
verges weakly to Xif and only if dP(µn, µ)→0, where dPis the Prokhorov
metric corresponding to the metric dGH P .
In the following, let dc
Pbe the Prokhorov metric corresponding to the metric
dc
GHP on Mc
∗. For given r≥0, it can be seen that the projection X 7→ X (r)
from M∗to Mc
∗is measurable. So the ball X(r)is well defined as a random
element of Mc
∗. Let µ(r)be the distribution of X(r).
Lemma 3.29. Let Xand Ybe random PMM spaces with distributions µand
νrespectively.
27
(i) For every r≥0,
dP(µ, ν)≤1
r∨2dc
P(µ(r), ν(r)).
(ii) If X,Yare compact a.s. (i.e., are random elements of Mc
∗), then
dP(µ, ν)≤2dc
P(µ, ν).
Proof. (i). Let > 1
r∨2dc
P(µ(r), ν(r)). The goal is to prove that ≥dP(µ, ν).
One can assume < 1< r without loss of generality. By Strassen’s theorem
(Corollary 2.2), there exists a coupling of X,Ysuch that
Phdc
GHP (X(r),Y(r))>
2i≤
2.
So part (i) of Lemma 3.22 and the assumption > 1
rgive
P[dGHP (X,Y)> ]≤
2≤.
So the converse of Strassen’s theorem (see Theorem 2.1) implies that dP(µ, ν)≤
and the claim is proved.
(ii). Let > 0 be arbitrary. One can choose r > 1
large enough such
that P[r > diam(X)] < . This implies that PhX(r)6=Xi< . Choose rsuch
that the same holds for Y. So the converse of Strassen’s theorem implies that
dc
P(µ, µ(r))∨dc
P(ν, ν(r))≤. Now, part (i) and the triangle inequality give
dP(µ, ν)≤1
r∨2(dc
P(µ, ν)+2)≤2dc
P(µ, ν)+5.
Since is arbitrary, the claim is proved.
The following result relates weak convergence in M∗to that in Mc
∗. Below,
a number r > 0 is called a continuity radius of µif it is a continuity radius
(Definition 3.3) of Xalmost surely.
Theorem 3.30 (Weak Convergence).Let X1,X2,· · · and Xbe random PMM
spaces with distributions µ1, µ2, . . . and µrespectively. Then the following are
equivalent.
(i) Xn⇒Xweakly; i.e., dP(µn, µ)→0.
(ii) For every continuity radius rof µ,X(r)
n⇒X(r)weakly as random ele-
ments of Mc
∗; i.e., dc
P(µ(r)
n, µ(r))→0.
(iii) There exists an unbounded set I⊆R≥0such that X(r)
n⇒X(r)weakly for
every r∈I.
28
Proof. (i)⇒(ii). Let rbe a continuity radius of µ. Therefore, as δ→0,
dHP (X(r+δ),X(r−δ))→0 a.s. (see (3.6)). So, by fixing > 0 arbitrarily,
the following holds for small enough δ.
PhdHP (X(r+δ),X(r−δ))> i< .
Assume that 0 < δ < r ∧1
r. The assumption of (i) implies that for large enough
n,dP(µn, µ)<δ
2. Fix such n. By Strassen’s theorem (Corollary 2.2), there
exists a coupling of Xnand Xsuch that PdGHP (Xn,X)>δ
2≤δ
2. Similarly
to the proof of (ii)⇒(iv) of Theorem 3.24, by using Lemma 3.12 and the above
inequality, one can deduce that
Pdc
GHP (X(r)
n,X(r))> +δ
2< +δ
2.
Now, the converse of Strassen’s theorem shows that dc
P(µ(r)
n, µ(r))≤+δ
2. Since
the RHS is arbitrarily small, the claim is proved.
(ii)⇒(iii). By Lemma 3.4 and Fubini’s theorem, one can show that the set
of discontinuity radii of µhas zero Lebesgue measure. This implies the claim.
(iii)⇒(i). The claim is implied by part (i) of Lemma 3.29 and is left to the
reader.
Remark 3.31. Part (ii) of Theorem 3.30 is similar to the convergence of finite
dimensional distributions in stochastic processes (note that one can identify a
random PMM space Xwith the stochastic process t7→ X(t)in Mc
∗), but a
stronger result holds: Convergence of one-dimensional marginal distributions,
only for the set of continuity radii, is enough for the convergence of the whole
process in this case. This is due to the monotonicity in Lemma 3.18. See also
Subsection 4.7 below.
4 Special Cases and Connections to Other No-
tions
This section discusses some notions in the literature which are special cases of,
or connected to, the Gromov-Hausdorff-Prokhorov metric defined in this paper.
4.1 A Metrization of the Gromov-Hausdorff Convergence
Here, it is shown that the setting of Section 3 can be used to extend the
Gromov-Hausdorff metric to the boundedly-compact case. Also, it is shown
that this gives a metrization of the Gromov-Hausdorff topology on the set N∗of
boundedly-compact pointed metric spaces. In addition, it is shown that N∗is
a Polish space, which enables one to define random boundedly-compact pointed
metric spaces (see Subsections 4.2 and 4.5 below for metrics on specific subsets
of N∗).
29
First, the Gromov-Hausdorff metric is recalled in the compact case (see [13]
or [9]). The original definition (1.1) is for non-pointed spaces, but we recall the
pointed version since it will be used later. Let X= (X, oX) and Y= (Y , oY) be
compact pointed metric spaces. The Gromov-Hausdorff distance dc
GH (X,Y)
of Xand Yis defined similar to the metric dc
GHP of Subsection 3.2 by deleting
the last term in (3.1); or equivalently, by letting µXand µYbe the zero measures
in (3.1). It is known that dc
GH is a metric on Nc
∗and makes it a complete
separable metric space (see e.g., [9]).
In the boundedly-compact case, the notion of Gromov-Hausdorff con-
vergence is also defined (see [13] or [9]), which can be stated using (3.10) as
follows. Let Xn= (Xn, on) be boundedly-compact PM spaces (n= 1,2, . . .).
The sequence (Xn)nis said to converge to X= (X, o) in the Gromov-Hausdorff
sense (Definition 8.1.1 of [9]) if for every r > 0 and 0 < ≤r, on has
limna(, r;Xn, X ) = 0 (consider the zero measures in (3.10)). This defines
a topology on N∗.
The metric dc
GH is identical to the restriction of the Gromov-Hausdorff-
Prokhorov metric dc
GHP to Nc
∗(by identifying Nc
∗with the subset {(X, o, µ)∈
M∗:µ= 0}of M∗). Now, define the Gromov-Hausdorff metric dGH on N∗
to be the restriction of the metric dGHP (3.11) to N∗. It can also be defined
directly by (3.10) and (3.11) by letting the measures be the zero measures.
Similarly to Theorem 3.26, we have
Theorem 4.1. The metric dGH on N∗, defined above, is a metrization of the
Gromov-Hausdorff topology. Moreover, it makes N∗a complete separable metric
space.
Proof. The first claim is implied by Theorem 3.24. It can be seen that N∗is
a closed subset of M∗. Therefore, Theorem 3.27 implies that N∗is a complete
separable metric space.
In addition, a version of Theorem 3.30 holds for weak convergence of random
boundedly-compact pointed metric spaces.
4.2 Length Spaces
In [1], another version of the Gromov-Hausdorff-Prokhorov distance is defined in
the case of length spaces. It is shown below that it generates the same topology
as (the restriction of) the metric dGHP .
A metric space Xis called a length space if for all pairs x, y ∈X, the
distance of xand yis equal to the infimum length of the curves connecting xto
y. Let Lbe the set of (isometry classes of ) pointed measured complete locally-
compact length spaces (equipped with locally-finite Borel measures). For two
elements X,Y ∈ L, their distance is defined in [1] by the same formula as (3.13).
It is proved in [1] that this makes La complete separable metric space.
Every element of Lis boundedly-compact by Hopf-Rinow’s theorem (see [1]).
So Lcan be regarded as a subset of M∗. Now, consider the restriction of the
metric dGHP to L. This metric is not equivalent to the metric in (3.13), but
30
generates the same topology (by Theorem 3.24). Moreover, Lis a closed subset
of M∗(see Theorem 8.1.9 of [9]). So Theorem 3.27 implies that Lis also
complete and separable under the restriction of the metric dGHP .
In addition, the pre-compactness result Theorem 3.28 is a generalization of
Theorem 2.11 of [1].
4.3 Random Measures
Let Sbe a boundedly-compact metric space and Mbe the set of boundedly-
finite Borel measures on S. The well known vague topology on M, makes it a
Polish space (see e.g., Lemma 4.6 in [15]). This is the basis for having a standard
probability space in defining random measures on Sas random elements in
M. The metrics defined in Remark 3.21 are metrizations of the vague topology
as well.
One can regard a random measure on Sas a random PMM space by con-
sidering the natural map µ7→ (S, o, µ) from Mto M∗. The cost is considering
measures on Sup to equivalence under automorphisms of (S, o) (see also the
next paragraph). This also allows the base space (S, o) be random, and hence,
a random PMM space can also be called a random measure on a random
environment.
To rule out the issue of the automorphisms in the above discussion, on can
add marks to the points of S, which requires a generalization of the Gromov-
Hausdorff-Prokhorov metric. See [16].
4.4 Benjamini-Schramm Metric For Graphs
Benjamini and Schramm [7] defined a notion of convergence for rooted graphs,
which is particularly interesting for studying the limit of a sequence of sparse
graphs. For simple graphs, convergence under this metric is equivalent to the
Gromov-Hausdorff convergence of the corresponding vertex sets equipped with
the graph-distance metrics. Below, it is shown that, roughly speaking, the
boundedly-compact case of the Gromov-Hausdorff metric defined in this paper
generalizes the Benjamini-Schramm metric for simple graphs. So random rooted
graphs can be regarded as random pointed metric spaces.
For simplicity, we restrict attention to simple graphs. It is also assumed
that the graph is connected and locally-finite; i.e., every vertex has finite de-
gree. For two rooted networks (G1, o1) and (G2, o2), their distance is defined
by 1/(α+ 1), where αis the supremum of those r > 0 such that there is a
graph-isomorphism between Br(o1) and Br(o2) that maps o1to o2. Let G∗be
the set of isomorphism-classes of rooted graphs. It is claimed in [4] that this
distance function makes G∗a complete separable metric space.
Since we assume the graphs are simple, every graph Gcan be modeled as a
metric space, where the metric (which is the graph-distance metric) is integer-
valued. Also, being locally-finite implies that the metric space is boundedly-
compact. So G∗can be identified with a subset of N∗. It can be seen that the
31
restriction of the Gromov-Hausdorff metric on N∗(defined in Subsection 4.1)
to G∗is equivalent to the metric defined in [4] mentioned above.
4.5 Discrete Spaces
Let D∗be the set of all pointed discrete metric spaces (up to pointed isometries)
which are boundedly-finite; i.e., every closed ball contains finitely many points.
To study random pointed discrete spaces, [6] defines a metric on D∗and shows
that D∗is a Borel subset of some complete separable metric space. It is shown
below that random pointed discrete spaces are special cases of random PMM
spaces (or random PM spaces).
First, D∗is clearly a subset of N∗. Therefore, the generalization of the
Gromov-Hausdorff metric on N∗(introduced in Subsection 4.1) induces a metric
on D∗(the topology of this metric is discussed below). It should be noted that
D∗is not a closed subset of N∗, and hence, is not complete (in fact, D∗is dense
in N∗). However, it is a Borel subset of N∗.
Second, by equipping every discrete set Xwith the counting measure on X,
D∗can be regarded as a subset of M∗. It can be seen that it is a Borel subset
which is not closed (e.g., {0,1
n}converges to a single point whose measure is 2).
The closure of D∗in M∗is the set of elements of M∗in which the underlying
metric space is discrete and the measure is integer-valued and has full support.
By Theorem 3.24, it can be seen that the topology on D∗induced from M∗
coincides with the topology defined in [6]. However, it is strictly finer than the
topology induced from N∗. Nevertheless, it can be seen that these topologies
induce the same Borel sigma-field on D∗.
4.6 The Gromov-Hausdorff-Vague Topology
In [5], a variant of the GHP metric is defined on the set M0
∗of boundedly-
compact metric measure spaces and its Polishness is proved. This space is
slightly different from M∗since in the former, the features outside the support
of the underlying measure are discarded (see [22] for more discussion on the
two different viewpoints). More precisely, two pointed metric measure spaces
(X, oX, µX) and (Y, oY, µY) are called equivalent in [5] if there exists a measure
preserving isometry between supp(µX)∪ {oX}and supp(µY)∪ {oY}that maps
oXto oY. The set M0
∗can be mapped naturally into M∗(by replacing Xwith
supp(µX)∪ {oX}). The image of this map is the set of (X, o, µ) in M∗such
that supp(µ)⊇X\ {o}. Since the image of this map is not closed in M∗,
the set M0
∗is not complete under the metric induced by the GHP metric (this
holds even in the compact case). In [5], another metric is defined that makes
M0
∗complete and separable. It can be seen that it generates the same topology
as the restriction of the GHP metric to M0
∗. A second proof for Polishness of
M0
∗can be given by Alexandrov’s theorem by using Polishness of M∗and by
showing that M0
∗corresponds to a Gδsubspace of M∗(given n > 0, it can be
shown that the set of (X, o, µ)∈M∗such that ∀x∈X:µ(B1/n(x)) >0 is
open).
32
The method of [5] is different from the present paper. It defines the met-
ric on M0
∗by modifying (3.13) (since (3.13) does not make M0
∗complete), but
the definition in the present paper is based on the notion of PMM-subspaces,
Lemma 3.12 and (3.11). As mentioned in Remark 3.20, this method gives more
quantitative bounds in the arguments. Despite some similarities in the argu-
ments (which are also similar to those of [1] and other literature that use the
localization method to generalize the Gromov-Hausdorff metric), the results
of [5] do not give a metrization of the Gromov-Hausdorff-Prokhorov topology
on M∗and do not imply its Polishness. Also, the Strassen-type theorems (The-
orems 2.1 and 3.6) and the results based on them are new in the present paper.
The term Gromov-Hausdorff-vague topology is used in [5] to distinguish it
with another notion called the Gromov-Hausdorff-weak topology defined therein.
By considering only probability measures in the above discussion, the two
topologies on the corresponding subset of M0
∗will be identical.
4.7 The Skorokhod Space of C`adl`ag Functions
The Skorokhod space, recalled below, is the space of c`adl`ag functions with values
in a given metric space. By noting that every boundedly-compact PMM space
can be represented as a c`adl`ag curve in Mc
∗(see the following lemma), one can
consider the Skorokhod metric on M∗. This subsection studies the relations of
this metric with the metric dGHP . By similar arguments, one can also study
the connections of the Skorokhod space to the boundedly-compact cases in [1],
[4], [5], [9] and [22], which are introduced earlier in this section.
Lemma 4.2. For every boundedly-compact PMM space X, the curve t7→ X (t)
is a c`adl`ag function with values in Mc
∗. Moreover, the left limit of this curve at
t=ris (Br(o), o, µ
Br(o)), where Br(o)is the closure of Br(o).
Proof. The claim follows from Lemma 3.2 and (3.7).
Let Sbe a complete separable metric space. The Skorokhod space D(S)
is the space of all c`adl`ag functions f: [0,∞)→S. In [8], a metric is defined
on D(S) which is called the Skorokhod metric here. Heuristically, two c`adl`ag
functions x1, x2∈ D(S) are close if by restricting x1to a large interval [0, M ]
and by perturbing the time a little (i.e., by composing x1with a function which
is close to the identity function), the resulting function is close in the sup metric
to the restriction of x2to a large interval. The precise definition is skipped for
brevity (see Section 16 of [8]). Under this metric, D(S) is a complete separable
metric space.
Now let S:= Mc
∗. For every boundedly-compact PMM space X, let ρ(X)
denote the curve t7→ X (t)with values in Mc
∗. By Lemma 4.2, the latter is
c`adl`ag; i.e., is an element of D(S). Now, ρdefines a function from M∗to D(S).
It can be seen that ρis injective and its image is
nx∈ D(S) : ∀r≤s:x(s)(r)=x(r)o.
33
It can also be seen that the latter is a closed subset of D(S). Therefore, the
Skorokhod metric can be pulled back by ρto make M∗a complete separable
metric space.
Proposition 4.3. One has
(i) The topology on M∗induced by the Skorokhod metric (defined above) is
strictly finer than the Gromov-Hausdorff-Prokhorov topology.
(ii) The Borel sigma-field of the Skorokhod metric on M∗is identical with that
of the Gromov-Hausdorff-Prokhorov metric.
Proof. (i). First, assume Xn→ X in the Skorokhod topology. Theorem 16.2
of [8] implies that X(r)
n→ X (r)for every continuity radius rof X. So Theo-
rem 3.24 gives that Xn→ X under the metric dGH P .
Second, let Xn:= {0,1 + 1
n,−1−2
n}and X:= {0,1,−1}equipped with the
Euclidean metric and the zero measure (or the counting measure) and pointed
at 0. Then Xn→ X in the metric dGHP but the convergence does not hold in
the Skorokhod topology (note that for rn= 1 + 1
n, the ball X(rn)
nis close to
{0,1}, but is not close to any ball in Xcentered at 0).
(ii). It can be seen that the set of c`adl`ag step functions with finitely many
jumps is dense in D(S). Also, it can be seen that the the set Iof X ∈ M∗
such that ρ(X) is such a curve (equivalently, the set {X (r):r≥0}is finite)
is dense in M∗under the Skorokhod topology. It can be seen that the sets
AN
(X) for X ∈ Iand > 0 generate the Skorokhod topology on M∗, where
AN
(X) is defined as follows: If r1, r2, . . . , rkare the set of discontinuity points
of X,r0:= 0 and rk< N , consider the set of Y ∈ M∗such that there exists
0 =: t0< t1<· · · < tk< tk+1 := Nsuch that for all i≤k, one has |ti−ri|<
and for all ti≤t < ti+1, one has dc
GHP (Y(t),X(ri))< . It is left to the reader
to show that this is a Borel subset of M∗under the metric dGHP . This proves
the claim.
Remark 4.4. If Xn→ X under the metric dGHP and Xhas no discontinuity
radii, then the convergence holds in the Skorokhod topology as well. This follows
from the fact that the curves ρ(Xn) converge to ρ(X) uniformly on bounded
intervals, which follows from Theorem 3.24 and Lemma 3.12.
Remark 4.5. The above result means that to consider M∗as a standard prob-
ability space, one could consider the Skorokhod metric on M∗from the begging.
This method is identical to considering the GHP metric if one is interested only
in the Borel structure. However, the topology and the notion of weak conver-
gence are different under these metrics. Nevertheless, in most of the examples in
the literature that study scaling limits (e.g., the Brownian continuum random
tree of [3]), both notions of convergence hold since the limiting spaces under
study usually have no discontinuity radii.
34
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