Proof of Theorem 1 The Gromov-Prohorov metric

180 Electronic Communications in Probability

3.1 Proof of Theorem 1

We adapt the proof of Gromov’s reconstruction theorem for metric measure spaces, given by A. Ver- shik – see Chapter 3 1 2 .5 and 3 1 2 .7 in [ 16 ] – to the marked case. Let x = X , r X , µ X , y = Y, r Y , µ Y ∈ ▼ I . It is clear that ν x = ν y if x = y. Thus, it remains to show that the converse is also true, i.e. we need to show that ν x = ν y implies that x and y are measure-preserving isometric see Definition 2.1. If ν x = ν y , then there exists ν ∈ M 1 ❘ ◆ 2 + × I ◆ × ❘ ◆ 2 + × I ◆ putting mass 1 on the diagonal and having ν x and ν y as projections on the first resp. second coordinate. We define a probability measure µ ∈ M 1 X × I ◆ × Y × I ◆ by µA × B := ν R X ,r X A × R Y,r Y B , A ∈ B X × I ◆ , B ∈ B Y × I ◆ . Here B denotes the Borel- σ algebra. Then we have recall 7 that R X ,r X ◦ π X ×I ◆ ∗ µ = ν x , R Y,r Y ◦ π Y ×I ◆ ∗ µ = ν y , and R X ,r X ◦ π X ×I ◆ x, u, y, v = R Y,r Y ◦ π Y ×I ◆ x, u, y, v, 15 for µ-almost all x, u, y, v = x 1 , u 1 , x 2 , u 2 , . . . , y 1 , v 1 , y 2 , v 2 , . . . . Then in particular, by the Glivenko-Cantelli theorem, for µ-almost all x, u, y, v, 1 n n X k=1 δ x k ,u k n→∞ ==⇒ µ X and 1 n n X k=1 δ y k ,v k n→∞ ==⇒ µ Y . 16 Now, take any x, u, y, v such that 15 and 16 hold as well as x n , u n ∈ suppµ X , y n , v n ∈ supp µ Y , n ∈ ◆. By 15 we find that u = v. Define ϕ : suppπ X ∗ µ X → suppπ Y ∗ µ Y as the only continuous map satisfying ϕx n = y n , n ∈ ◆ and recall the definition of e ϕ in 4. By 15, we obtain that r X x m , x n = r Y y m , y n = r Y ϕx m , ϕx n , m, n ∈ N, which extends to supp π X ∗ µ X by continuity. In addition, by 16 and continuity, e ϕ ∗ µ X = µ Y and so X , r X , µ X and Y, r Y , µ Y are measure-preserving isometric, i.e. x = y.

3.2 The Gromov-Prohorov metric

In this section, we define the marked Gromov-Prohorov metric on ▼ I , which generates a topol- ogy which is at least as strong as the marked Gromov-weak topology, see Lemma 3.5. However, since we establish in Proposition 3.6 that both topologies have the same compact sets, we see in Proposition 3.7 that the topologies are the same, and hence, the marked Gromov-Prohorov metric metrizes the marked Gromov-weak topology. We use the same notation for ϕ and e ϕ as in Defi- nition 2.1. Recall that the topology of weak convergence of probability measures on a separable space is metrized by the Prohorov metric see [ 9 , Theorem 3.3.1]. Definition 3.1 The marked Gromov-Prohorov topology. For x i = X i , r i , µ i ∈ ▼ I , i = 1, 2, set d MGP x 1 , x 2 := inf Z,ϕ 1 , ϕ 2 d Pr e ϕ 1 ∗ µ 1 , e ϕ 2 ∗ µ 2 , 17 where the infimum is taken over all complete and separable metric spaces Z, r Z , isometric em- beddings ϕ 1 : X 1 → Z, ϕ 2 : X 2 → Z and d Pr denotes the Prohorov metric on M 1 Z × I, based Marked metric measure spaces 181 on the metric er Z = r Z + r I on Z × I, metrizing the product topology. Here, d MGP denotes the marked Gromov-Prohorov metric MGP metric. The topology induced by d MGP is called the marked Gromov-Prohorov topology MGP topology. Remark 3.2 Equivalent definition of the MGP metric. For x i = X i , r i , µ i ∈ ▼ I , i = 1, 2, denote by X 1 ⊔ X 2 the disjoint union of X 1 and X 2 . Then, d MGP x 1 , x 2 := inf r X1⊔X2 d Pr e ϕ 1 ∗ µ 1 , e ϕ 2 ∗ µ 2 , 18 where the infimum is over all metrics r X 1 ⊔X 2 on X 1 ⊔ X 2 extending the metrics on X 1 and X 2 and ϕ i : X i → X 1 ⊔ X 2 , i = 1, 2 denote the canonical embeddings. Remark 3.3 d MGP is a metric . The fact that d MGP indeed defines a metric follows from an easy extension of Lemma 5.4 in [ 13 ]. Symmetry and non-negativity are clear from the definition, and positive definiteness is a consequence of Theorem 1. Furthermore the triangle inequality holds by the following argument: For three mmm-spaces x i = X i , r i , µ i ∈ ▼ I , i = 1, 2, 3 and any ǫ 0, by the same construction as in Remark 3.2, we can choose a metric r X 1 ⊔X 2 ⊔X 3 on X 1 ⊔ X 2 ⊔ X 3 , extending the metrics r X 1 , r X 2 , r X 3 , such that d Pr e ϕ 1 ∗ µ 1 , e ϕ 2 ∗ µ 2 − d MGP x 1 , x 2 ǫ, d Pr e ϕ 2 ∗ µ 2 , e ϕ 3 ∗ µ 3 − d MGP x 2 , x 3 ǫ. 19 Then, we can use the triangle inequality for the Prohorov metric on M 1 X 1 ⊔ X 2 ⊔ X 3 × I and let ǫ → 0 to obtain the triangle inequality for d MGP . Lemma 3.4 Equivalent description of the MGP topology. Let x = X , r X , µ X , x 1 = X 1 , r 1 , µ 1 , x 2 = X 2 , r 2 , µ 2 , . . . ∈ ▼ I . Then, d MGP x n , x n→∞ −−→ 0 if and only if there is a complete and separable metric space Z, r Z and isometric embeddings ϕ X : X → Z, ϕ 1 : X 1 → Z, ϕ 2 : X 2 → Z, . . . with d Pr e ϕ n ∗ µ n , e ϕ X ∗ µ X n→∞ −−→ 0. 20 Proof. The assertion is an extension of Lemma 5.8 in [ 13 ] to the marked case. The proof of the present lemma follows the same lines, which we sketch briefly. First, the “if”-direction is clear. For the “only if” direction, fix a sequence ǫ 1 , ǫ 2 , · · · 0 with ǫ n → 0 as n → ∞. By the same construction as in Remark 3.3, we can construct a metric r Z on Z, defined as the completion of X ⊔ X 1 ⊔ X 2 ⊔ · · · , with the property that d Pr e ϕ n ∗ µ n , e ϕ X ∗ µ X − d MGP x n , x ǫ n , 21 where ϕ X : X → Z and ϕ n : X n → Z, n ∈ ◆ are canonical embeddings. The assertion follows. Lemma 3.5 MGP convergence implies MGW convergence. Let x , x 1 , x 2 , · · · ∈ ▼ I be such that d MGP x n , x n→∞ −−→ 0. Then, x n n→∞ −−→ x in the MGW topology. Proof. Let x = X , r, µ, x 1 = X 1 , r 1 , µ 1 , x 2 = X 2 , r 2 , µ 2 , . . . . Take Z, r Z and isometric embed- dings ϕ X , ϕ 1 , ϕ 2 , . . . such that 20 from Lemma 3.4 holds. It is a consequence of Proposition 3.4.5 in [ 9 ] that S n C n is convergence determining in M 1 ❘ ◆ 2 + × I ◆ ; see also the proof of Proposition 4.1. Let Φ ∈ Π be such that Φ. = 〈 ν . , φ〉 for 182 Electronic Communications in Probability some φ ∈ S ∞ n=0 C n . Since e ϕ n ∗ µ n n→∞ ==⇒ e ϕ X ∗ µ X by 20, we also have that e ϕ n ∗ µ n ⊗ ◆ n→∞ ==⇒ e ϕ X ∗ µ X ⊗ ◆ in M 1 Z × I ◆ . Hence we can conclude that Z φ r Z z k , z l 1≤k l ,u e ϕ n ∗ µ n ⊗ ◆ dz, du n→∞ −−→ Z φ r Z z k , z l 1≤k l , u e ϕ X ∗ µ X ⊗ ◆ dz, du. 22 Since x = Z, r Z , e ϕ X ∗ µ X and x n = Z, r Z , e ϕ n ∗ µ n , n = 1, 2, . . . , this proves that 〈ν x n , φ〉 n→∞ −−→ 〈ν x , φ〉. Because Φ ∈ Π was arbitrary, we have that ν x n n→∞ ==⇒ ν x . Then, by definition, x n n→∞ −−→ x in the MGW topology. Proposition 3.6 Relative compactness in ▼ I . Let Γ ⊆ ▼ I . Then conditions i and ii of Theorem 3 are equivalent to iii The set Γ is relatively compact with respect to the marked Gromov-Prohorov topology. Proof. First, iii⇒i follows from Lemma 3.5. Thus, it remains to show i⇒ii⇒iii. i⇒ii: Note that Π contains functions Φ. = 〈 ν . , φ〉 such that φ does not depend on the variables u ∈ I ◆ , as well as functions φ which only depend on u 1 ∈ I. Denote the former set of functions by Π dist and the latter by Π mark . Assume that the sequence x 1 , x 2 , · · · ∈ Γ converges to x ∈ ▼ I with respect to the MGW topology. Since Φx n n→∞ −−→ Φx for all Φ ∈ Π dist , we find that π 1 x n n→∞ −−→ π 1 x in the Gromov-weak topology. In addition, Φx n n→∞ −−→ Φx for all Φ ∈ Π mark implies π 2 x n n→∞ ==⇒ π 2 x . In particular, ii holds. ii⇒iii: Recall from Theorem 5 of [ 13 ] that the unmarked Gromov-weak and the un- marked Gromov-Prohorov topology coincide. For a sequence in Γ, take a subsequence x 1 = X 1 , r 1 , µ 1 , x 2 = X 2 , r 2 , µ 2 , · · · ∈ Γ and x = X , r X , µ X ∈ ▼ I such that π 1 x n n→∞ −−→ π 1 x ∈ ▼ in the Gromov-Prohorov topology and d Pr π 2 x n , π 2 x n→∞ −−→ 0. 23 Using Lemma 5.7 of [ 13 ], take a complete and separable metric space Z, r Z , isometric embed- dings ϕ X : X → Z, ϕ 1 : X 1 → Z, ϕ 2 : X 2 → Z, . . . such that d Pr π X n ◦ e ϕ n ∗ µ n , π X ◦ e ϕ X ∗ µ X = d Pr π X n ∗ e ϕ n ∗ µ n , π X ∗ e ϕ X ∗ µ X n→∞ −−→ 0. 24 In particular, 23 shows that { π 2 x n = π I ∗ e ϕ n ∗ µ n : n ∈ ◆} is relatively compact in M 1 I and 24 shows that { π X n ∗ e ϕ n ∗ µ n : n ∈ ◆} is relatively compact in M 1 Z. This implies that { e ϕ n ∗ µ n : n ∈ ◆} is relatively compact in M 1 Z × I. Hence, we can find a convergent subsequence, and iii follows by Lemma 3.4. Proposition 3.7 MGW and MGP topologies coincide. The marked Gromov-Prohorov metric generates the marked Gromov-weak topology, i.e. the marked Gromov-weak topology and the marked Gromov-Prohorov topology coincide. Marked metric measure spaces 183 Proof. Let x , x 1 , x 2 , · · · ∈ ▼ I . We have to show that x n n→∞ −−→ x in the MGW topology if and only if x n n→∞ −−→ x in the MGP topology. The ’if’-part was shown in Lemma 3.5. For the ’only if’-direction, assume that x n n→∞ −−→ x in the MGW topology. It suffices to show that for all subsequences of x 1 , x 2 , . . . , there is a further subsequence x n 1 , x n 2 , . . . such that d MGP x n k , x k→∞ −−→ 0. 25 By Proposition 3.6 {x n : n ∈ ◆} is relatively compact in the MGP topology. Therefore, for a subsequence, there exists y ∈ ▼ I and a further subsequence x n 1 , x n 2 , . . . with x n k k→∞ −−→ y in the MGP topology. By the ’if’-direction it follows that x n k k→∞ −−→ y in the MGW topology, which shows that y = x and therefore 25 holds.

3.3 Proofs of Theorems 2 and 3