Then, there exists a subsequence $(W_{n_l})_{l \in \mathbb{N}}$ of the sequence $(W_n)_{n \in \mathbb{N}}$ and a measure-valued kernel $W \in W_1$ (resp. $W \in W_+$ or $W \in W_\pm$) such that $(W_{n_l})_{l \in \mathbb{N}}$ converges to the same measure-valued kernel $W$ simultaneously both for $\delta_\square$ and for $\delta'_\square$, the cut distance associated with $d'$.

*Proof.* — We adapt here the general scheme from the proof of Theorem 9.23 in [46], but the argument for the convergence of the $U_k$, defined below, takes into account that measure-valued kernels are infinite-dimensional valued. We set (recall from (3.1) the definition of $\| \cdot \|_\infty$):

$$C = \sup_{n \in \mathbb{N}} \|W_n\|_\infty < +\infty.$$

The proof is divided into four steps.

**Step 1: Without loss of generality, the partitions $P_{n,k}$ are made of intervals.** For every $n \in \mathbb{N}$, we can rearrange the points of $[0, 1]$ by a measure-preserving map so that the partitions $P_{n,k}$ are made of intervals, and we replace $W_n$ by its rearranged version.

An argument similar to the next lemma is used in the proof in [46, Proof of Theorem 9.23] without any reference. So, we provide a proof and stress that diameters of the partitions shrinking to zero is an important assumption (see Remark 8.3 below). We say a measurable map $\varphi$ from $[0, 1]$ to $[0, 1]$ is a.e. invertible if there exists a measurable function $\psi$ from $[0, 1]$ to $[0, 1]$ such that $\varphi \circ \psi(x) = \psi \circ \varphi(x) = x$ for a.e. $x \in [0, 1]$; and that it is bi-measurable if $\varphi$ is measurable and for all Borel set $B \subset [0, 1]$, $\varphi(B)$ is also a Borel set.

**LEMMA 8.2 (Kernel rearrangement with interval partitions).** — Let $(P_k)_{k \in \mathbb{N}}$ be a refining sequence of finite partitions of $[0, 1]$ whose diameter converges to zero. Then, there exist a measure-preserving map $\varphi \in \bar{S}_{[0,1]}$ which is bi-mesurable and a.e. invertible, and a refining sequence of partitions made of intervals $(Q_k)_{k \in \mathbb{N}}$ such that for all $k \in \mathbb{N}$, and all set $S \in P_k$ there exists a set $R \in Q_k$ such that a.e. $1_R = 1_{\varphi^{-1}(S)}$.

In particular, if $W$ is a signed measure-valued kernel, then for $U = W^\varphi$, we have that a.e. $U_{Q_k} = (W^\varphi)_{Q_k} = (W_{P_k})^\varphi$ for all $k \in \mathbb{N}$.

Notice that, according to Remark 4.4, the sequence of refining partition $(P_k)_{k \in \mathbb{N}}$, with a partition diameter converging to 0, separates points and thus generates the Borel $\sigma$-field of $[0, 1]$.

*Proof.* — Consider the infinite Ulam-Harris tree $\mathcal{T}^\infty = \{u_1 \cdots u_k : k \in \mathbb{N}, u_1, \dots, u_k \in \mathbb{N}^*\}$, where for $k = 0$ the empty word $u = \partial$ is called the root node of the tree; for a node $u = u_1 \cdots u_k \in \mathcal{T}^\infty$, we define its height as $h(u) = k$, and if $k > 0$ we define its parent node as $p(u) = u_1 \cdots u_{k-1}$.
and we say that $u$ is a child node of $p(u)$. We order vertices on the tree $\mathcal{T}^\infty$ with the lexicographical (total) order $<_{lex}$. As a first step, we construct a subtree $\mathcal{T} \subset \mathcal{T}^\infty$ that indexes the sets in the partitions $(\mathcal{P}_k)_{k \in \mathbb{N}}$, such that for every $k \in \mathbb{N}$, $\mathcal{P}_k = \{S_u : u \in \mathcal{T}, h(u) = k\}$, and such that if $S_v \subset S_u$ with $S_v \in \mathcal{P}_k$ and $S_u \in \mathcal{P}_{k-1}$, then $p(v) = u$.

Without loss of generality, we may assume that $\mathcal{P}_0 = \{[0, 1]\}$, and we label its only set by the empty word $\partial$, and we set $S_\partial = [0, 1]$. Then, suppose we have already labeled the sets from $\mathcal{P}_0, \dots, \mathcal{P}_k$, and we proceed to label the sets from $\mathcal{P}_{k+1}$. Because the partition $\mathcal{P}_{k+1}$ is a refinement of $\mathcal{P}_k$, we can group the sets of $\mathcal{P}_{k+1}$ by their unique parent set from $\mathcal{P}_k$, i.e. for every $S_u \in \mathcal{P}_k$, let $\mathcal{O}_u = \{S \in \mathcal{P}_{k+1} : S \subset S_u\}$, then $S_u = \cup_{S \in \mathcal{O}_u} S$. For $S_u \in \mathcal{P}_k$, we fix an arbitrary enumeration of $\mathcal{O}_u = \{S^1, \dots, S^l\}$ with $l = |\mathcal{O}_u|$, then label the set $S^j$ by $uj$, and set $S^j = S_{uj}$; remark that the parent node of $w = uj$ is $p(w) = u$, and the height of node $w$ is $h(w) = h(u) + 1 = k + 1$. Hence, we have labeled every set from $\mathcal{P}_{k+1}$. To finish the construction, we set $\mathcal{T} = \{u : \exists k \in \mathbb{N}, \exists S \in \mathcal{P}_k, S \text{ has label } u\}$.

We now proceed to construct a measure-preserving map $\psi$ such that the image of every set $S_u$ is a.e. equal to an interval, and such that those intervals are ordered w.r.t. to the order of their labels in $\mathcal{T}$.

Define the map $\sigma : [0, 1] \to \mathcal{T}^\mathbb{N}$ by $\sigma(x) = (u^k(x))_{k \in \mathbb{N}} \in \mathcal{T}^\mathbb{N}$ where $u^k(x)$ is the only node of $\mathcal{T}$ with height $k$ such that $x \in S_{u^k(x)}$ (and thus $u^{k+1}(x)$ is a child node of $u^k(x)$). Remark that if $u^{k_0}(x) <_{lex} u^{k_0}(y)$ for some $k_0 \in \mathbb{N}$, then $u^k(x) <_{lex} u^k(y)$ for every $k \ge k_0$. We extend naturally the total order $<_{lex}$ from $\mathcal{T}$ to a the total order on $\mathcal{T}^\mathbb{N}$: for $(u^k)_{k \in \mathbb{N}}, (v^k)_{k \in \mathbb{N}} \in \mathcal{T}^\mathbb{N}$, $(v^k)_{k \in \mathbb{N}} <_{lex} (v^k)_{k \in \mathbb{N}}$ if $u^{k_0} <_{lex} v^{k_0}$ where $k_0$ is the smallest $k$ such that $u^k \ne v^k$.

For every $u \in \mathcal{T}$, define:

$$A^{-}(u) = \bigcup_{v <_{\text{lex}} u : h(v) = h(u)} S_{v} \quad \text{and} \quad A^{+}(u) = A^{-}(u) \cup S_{u},$$

and then define $C^{-}(u) = \lambda(A^{-}(u))$ and $C^{+}(u) = \lambda(A^{+}(u))$. Now, define $\psi$ as, for $x \in [0, 1]$:

$$\psi(x) = \lambda(A^{-}(x)) \\ \text{where } A^{-}(x) = \{y \in [0, 1] : \sigma(y) <_{\text{lex}} \sigma(x)\} = \cup_{k \in \mathbb{N}} A^{-}(u^{k}(x)).$$

Moreover, as the sequence of partitions $(\mathcal{P}_k)_{k \in \mathbb{N}}$ has a diameter that converges to zero, and thus separates points, the map $\sigma$ is injective. Thus, we
also have:

$$ \psi(x) = \lambda(A^{+}(x)) \\ \text{where } A^{+}(x) = \{y \in [0, 1] : \sigma(y) \leq_{\text{lex}} \sigma(x)\} = A^{-}(x) \cup \{x\}. $$

Remark that both $A^{-}(x)$ and $A^{+}(x)$ are Borel measurable.

Remark that for every $k \in \mathbb{N}$, we have $A^{-}(u^{k}(x)) \subset A^{-}(x) \subset A^{+}(x) \subset A^{+}(u^{k}(x))$. In particular, for every $u \in \mathcal{T}$, we have $\psi(S_{u}) \subset [C^{-}(u), C^{+}(u)];$ however $\psi(S_{u})$ is not necessarily an interval, but we shall see that $\lambda(\psi(S_{u})) = C^{+}(u) - C^{-}(u)$, i.e. $\psi(S_{u})$ is a.e. equal to $[C^{-}(u), C^{+}(u)]$. Remark that, as the sequence of partitions $(\mathcal{P}_{k})_{k \in \mathbb{N}}$ is refining, we get that $[C^{-}(u), C^{+}(u)] = \cup_{v: p(v)=u} [C^{-}(v), C^{+}(v)]$ for every $u \in \mathcal{T} \setminus \{\partial\}$.

As the diameter of the partitions $(\mathcal{P}_{k})_{k \in \mathbb{N}}$ converges to zero, we have the following alternative formula for $\psi$:

$$ \psi(x) = \lim_{k \to \infty} C^{-}(u^{k}(x)) = \lim_{k \to \infty} C^{+}(u^{k}(x)). $$

For every $k \in \mathbb{N}$, the map $x \mapsto C^{-}(u^{k}(x))$ is a simple function (constant on each $S \in \mathcal{P}_{k}$ and takes finitely-many values), and thus $\psi$ is measurable as a limit of measurable maps.

We outline the rest of the proof. We first prove that $\psi$ is measure-preserving. Secondly, we prove that $\psi$ is a.e. invertible and construct its a.e. inverse map $\varphi$. Thirdly, we prove that $(\varphi^{-1}(\mathcal{P}_{k}))_{k \in \mathbb{N}}$ is a refining sequence of partitions. And lastly, we approximate almost everywhere the sequence of partitions $(\varphi^{-1}(\mathcal{P}_{k}))_{k \in \mathbb{N}}$ by a sequence of refining partitions composed of intervals.

We now prove that $\psi$ is measure preserving. Remark that $\psi(x)$ is a non-decreasing function of $\sigma(x)$ for the total relation order $<_{\text{lex}}$, i.e. $\psi(y) \leq \psi(x)$ if and only if $\sigma(y) \leq_{\text{lex}} \sigma(x)$. Hence, $\psi^{-1}([0, \psi(x)]) = \{y \in [0, 1] : \sigma(y) \leq_{\text{lex}} \sigma(x)\}$, and we have:

$$ \lambda(\psi^{-1}([0, \psi(x)])) = \lambda(\{y \in [0, 1] : \sigma(y) \leq_{\text{lex}} \sigma(x)\}) = \psi(x). $$

Thus, to show that $\psi$ is measure preserving we just need to show that $\psi([0, 1])$ is dense in $[0, 1]$. For every $u \in \mathcal{T}$, as $\psi(S_{u}) \subset [C^{-}(u), C^{+}(u)]$, we know that the interval $[C^{-}(u), C^{+}(u)]$ contains at least one point of the form $\psi(x)$. Remark that for all $k \in \mathbb{N}$, we have:

$$ [0, 1] = \cup_{u \in \mathcal{T}: h(u)=k} [C^{-}(u), C^{+}(u)]. $$

Hence, as $\lambda([C^{-}(u), C^{+}(u)]) = \lambda(S_{u}) \leq \text{diam}(\mathcal{P}_{h(u)})$ for every $u \in \mathcal{T}$, and as the diameter of the partitions $(\mathcal{P}_{k})_{k \in \mathbb{N}}$ converges to zero, we know that
each interval of positive length contains a point of the form $\psi(x)$ for some $x \in [0, 1]$, which implies that $\psi([0, 1])$ is indeed dense in $[0, 1]$.

We now prove that $\psi$ is a.e. invertible and construct its a.e. inverse map $\varphi$. Without loss of generality, assume that there is no set $S_u$ with null measure. Consider two distinct elements $x, y \in [0, 1]$ such that $\sigma(x) <_{lex} \sigma(y)$. Assume that $\psi(x) = \psi(y)$, and let $N \in \mathbb{N}$ be the last index $k$ such that $u^k(x) = u^k(y)$. Then, for every $k > N$, we have $u^k(x) <_{lex} u^k(y)$, which implies that $\psi(x) \le C^+(u^k(x)) \le C^-(u^k(y)) \le \psi(y)$; and thus $\psi(x) = \psi(y) = C^+(u^k(x)) = C^-(u^k(y))$, which in turn implies that there is no node of $\mathcal{T}$ between $u^k(x)$ and $u^k(y)$. Remark that this situation is analogous to the terminating decimal versus repeating decimal situation. Hence, we proved that there is no node between $u^{N+1}(x)$ and $u^{N+1}(y)$ and that for every $k > N$, $u^{k+1}(x)$ is the right-most child of $u^k(x)$, and $u^{k+1}(y)$ is the left-most child of $u^k(y)$ (i.e. $u^{k+1}(x) = u^k(x)|O_{u^k(x)}|$ and $u^{k+1}(y) = u^k(y)1$). Recall that the map $\sigma$ is injective. Putting all of this together, we get that the set $\{(x, y) \in [0, 1] : \psi(x) = \psi(y), x < y\}$ can be indexed by the nodes of $\mathcal{T}$, and is thus at most countable. The set $D = \{x \in [0, 1] : \psi^{-1}(\{\psi(x)\})$ is a singleton$\}$ is a Borel set as $[0, 1] \setminus D$ is at most countable. Hence, the map $\psi$ is injective on the subset $D \subset [0, 1]$ with measure one, and as $\psi$ is measure preserving, we get that $\psi(D)$ has measure one, and thus $\psi$ is bijective from $D$ to $\psi(D)$, that is, $\psi$ is a.e. invertible. We construct the map $\varphi$ as the inverse map of $\psi$ for $x \in \psi(D)$ and $\varphi(x) = 0$ for $x \in [0, 1] \setminus \psi(D)$. Without loss of generality, we assume that $0 \notin D$. Thus, $\varphi$ is the a.e. inverse map of $\psi$.

We are left to prove that $\varphi$ is bi-measurable and measure preserving. As we saw that each point $z \in [0, 1]$ has a pre-image $\psi^{-1}(z) = \{x \in [0, 1] : \psi(x) = z\}$ at most countable (indeed of cardinal at most 2), thus [51] insures that $\psi$ is bi-measurable. Let $B \subset (0, 1]$ be a Borel set. We have that $\varphi^{-1}(B) = \varphi^{-1}(B \cap D) = \psi(B \cap D)$ is a Borel set, where the first equality uses that $\varphi([0, 1]) = D \cup \{0\}$, the second equality uses that $\psi$ is the inverse of $\varphi$ on $D$, and lastly we used that $\psi$ is bi-measurable. We also have that $\varphi^{-1}(B \cup \{0\}) = \varphi^{-1}(B) \cup ([0, 1] \setminus \psi(D))$ is a Borel set. Thus, we deduce that $\varphi$ is measurable. We now prove that $\varphi$ is bi-measurable. Let $B \subset [0, 1]$ be a Borel set. We have that $\varphi(B)$ is equal to $\psi^{-1}(B)$ if $B \subset \psi(D)$ and $\psi^{-1}(B \cap \psi(D)) \cup \{0\}$ otherwise. In both cases, this is a Borel set as $\psi(D)$ is a Borel set since $\psi$ is bi-measurable and $D$ is a Borel set.

Moreover, we have:

$$\lambda(\varphi^{-1}(B)) = \lambda(\psi(B \cap D)) = \lambda(\psi^{-1}(\psi(B \cap D))) = \lambda(B \cap D) = \lambda(B),$$
where we used that $\varphi^{-1}(B) = \psi(B \cap D)$ for the first equality, that $\psi$ is measure preserving for the second equality, that $\psi$ is bijective from $D$ to $\psi(D)$ for the third equality, and that $D$ has measure one for the last equality. We also have:

$$ \lambda(\varphi^{-1}(B \cup \{0\})) = \lambda(\varphi^{-1}(B)) + \lambda([0, 1] \setminus \psi(D)) = \lambda(B) = \lambda(B \cup \{0\}), $$

where we used that $\varphi^{-1}(B) \subset \psi(D)$ and $[0, 1] \setminus \psi(D)$ are disjoint sets for the first equality, that $\lambda(\varphi^{-1}(B)) = \lambda(B)$ and that $\psi(D)$ has measure one for the second equality. Hence, the map $\varphi$ is measurable and measure preserving.

We now prove that $(\varphi^{-1}(\mathcal{P}_k))_{k \in \mathbb{N}}$ is a refining sequence of partitions. For $k \in \mathbb{N}$, as $\mathcal{P}_k$ is a finite partition of $[0, 1]$, we have that $\varphi^{-1}(\mathcal{P}_k) = \{\varphi^{-1}(S_u) : u \in \mathcal{T}, h(u) = k\}$ is also a finite partition of $[0, 1]$. Moreover, as $(\mathcal{P}_k)_{k \in \mathbb{N}}$ is a refining sequence of partitions, we get that the sequence of partitions $(\varphi^{-1}(\mathcal{P}_k))_{k \in \mathbb{N}}$ is also refining. Remark that the sets $\varphi^{-1}(S_u)$ are not necessarily intervals, they are intervals minus some at most countable sets (this is similar to the unit line minus the Cantor set).

To finish the proof, we are left to construct a refining sequence of partitions made of intervals $(Q_k)_{k \in \mathbb{N}}$ that agrees almost everywhere with the refining sequence of partitions $(\varphi^{-1}(\mathcal{P}_k))_{k \in \mathbb{N}}$. For $u \in \mathcal{T}$, define $R_u = [C^{-}(u), C^{+}(u)]$ (and $R_u = [C^{-}(u), C^{+}(u)]$ if $u$ is the unique node such that $v \le \text{lex } u$ for every $v \in \mathcal{T}$ with $h(v) = h(u)$). As $\psi$ is measure preserving, and as $\psi(S_u) \subset [C^{-}(u), C^{+}(u)]$ with $\lambda(S_u) = C^{+}(u) - C^{-}(u)$, we get that $\lambda([C^{-}(u), C^{+}(u)] \setminus \psi(S_u)) = 0$. As $\varphi$ is the a.e. inverse map of $\psi$, we have that a.e. $1_{\varphi^{-1}(S_u)} = 1_{\psi(S_u)} = 1_{[C^{-}(u), C^{+}(u)]} = 1_{R_u}$, i.e. $R_u$ agrees almost everywhere with $\varphi^{-1}(S_u)$. For $k \in \mathbb{N}$, define the finite partition $Q_k = \{R_u : h(u) = k\}$. Then, by definition of the sets $R_u$, the sequence of partitions $(Q_k)_{k \in \mathbb{N}}$ is refining. This concludes the proof. $\square$

**Remark 8.3 (The shrinking diameter assumption is important).** — Even if it is not stressed in [46, Proof of Theorem 9.23], the measure preserving map $\varphi$ in Lemma 8.2, which is bi-measurable and a.e. invertible, cannot be obtained without any assumption on the refining sequence of partitions $(\mathcal{P}_k)_{k \in \mathbb{N}}$ (in our case, we assumed that their diameter converges to zero). Indeed consider the sequence of partitions where for every $k \in \mathbb{N}$, $\mathcal{P}_k$ is composed of the sets:

$$ S_{k,j} = [j2^{-k-1}, (j+1)2^{-k-1}) \cup [1/2 + j2^{-k-1}, 1/2 + (j+1)2^{-k-1}), \\ 0 \le j < 2^k, $$
i.e. $S_{k,j}$ is the union of two dyadic interval translated by $1/2$, (also add 1 to the set $S_{k,0}$ to get a complete partition). Then, for every $x \in [0, 1/2)$, $x$ and $x+1/2$ belong to the same set of $\mathcal{P}_k$ for every $k \in \mathbb{N}$; in particular the diameter of the partitions $(\mathcal{P}_k)_{k \in \mathbb{N}}$ does not converge to zero. By contradiction, assume there exist a measure preserving map $\varphi \in \bar{S}_{[0,1]}$ and a sequence of interval partitions $(\mathcal{Q}_k)_{k \in \mathbb{N}}$ such that for all $k \in \mathbb{N}$ and all set $S_{k,j} \in \mathcal{P}_k$ with $0 \le j < 2^k$, there exists an interval set $I_{k,j} \in \mathcal{Q}_k$ such that a.e. $1_{I_{k,j}} = 1_{\varphi^{-1}(S_{k,j})}$. In particular, the set $I_{k,j}$ must be an interval of length $2^{-k}$. Hence, $\mathcal{Q}_k$ is a dyadic partition with stepsize $2^{-k}$, and thus the diameter of the partitions $(\mathcal{Q}_k)_{k \in \mathbb{N}}$ converges to zero. For every $x \in [0, 1/2)$, we get that $\text{diam}(\varphi^{-1}(\{x, x+1/2\})) \le \text{diam}(\mathcal{Q}_k) = 2^{-k}$ for all $k \in \mathbb{N}$; this implies that $\varphi^{-1}(\{x, x+1/2\})$ is a singleton, i.e. either $x \notin \varphi([0, 1])$ or $x+1/2 \notin \varphi([0, 1])$. Hence, as $\varphi$ is bi-measurable, considering the measurable set $A = \varphi([0, 1])$, we have $\lambda([0, 1/2) \cap A) = \lambda([1/2, 1) \setminus A)$ and $\lambda([0, 1/2) \setminus A) = \lambda([1/2, 1) \cap A)$. As $\lambda([0, 1/2)) = \lambda([0, 1/2) \cap A) + \lambda([0, 1/2) \setminus A) = 1/2$ because $\varphi$ is measure preserving, we get that $\lambda(A) = \lambda([0, 1/2) \cap A) + \lambda([1/2, 1) \cap A) = 1/2$, which contradicts the fact that $\varphi$ is measure preserving.

Now, for every $n \in \mathbb{N}$, applying Lemma 8.2 to $(\mathcal{P}_{n,k})_{k \in \mathbb{N}}$ and $W_n$, we get a measure-preserving map $\varphi_n$ and a refining sequence of partitions $(\mathcal{P}'_{n,k})_{k \in \mathbb{N}}$ made of intervals such that for all $k \in \mathbb{N}$, and all set $R \in \mathcal{P}'_{n,k}$ there exists a set $S \in \mathcal{P}_{n,k}$ such that a.e. $1_R = 1_{\varphi_n^{-1}(S)}$. In particular, for all $k \in \mathbb{N}$, the sequence of partitions $(\mathcal{P}_{n,k})_{k \in \mathbb{N}}$ still satisfy (i)-(ii). Set $W'_n = W_n^{\varphi_n}$ and $W'_{n,k} = W_{n,k}^{\varphi_n}$ so that almost everywhere:

$$W'_{n,k} = ((W_n)_{\mathcal{P}_{n,k}})^{\varphi_n} = (W_n^{\varphi_n})_{\mathcal{P}'_{n,k}} = (W')_{\mathcal{P}'_{n,k}}.$$

As $d$ and $d'$ are invariant, we have for every $n, k \in \mathbb{N}$ that $d(W_n, W_{n,k}) = d(W'_n, W'_{n,k})$, and similarly for $d'$. This insures that the signed measure-valued kernels $(W'_n)_{n \in \mathbb{N}}$ and $(W'_{n,k})_{n \in \mathbb{N}}$, $k \in \mathbb{N}$, still satisfy (iii)-(iv). Recall that for a measure-valued kernel $W$ and a measure-preserving map $\varphi$, $\delta_{\square,m}(W, W^{\varphi}) = 0$. Hence, we can replace the signed measure-valued kernels $(W_n)_{n \in \mathbb{N}}$ and $(W_{n,k})_{n \in \mathbb{N}}$, $k \in \mathbb{N}$, by $(W'_n)_{n \in \mathbb{N}}$ and $(W'_{n,k})_{n \in \mathbb{N}}$, $k \in \mathbb{N}$, and assume that the partitions $\mathcal{P}_{n,k}$ are made of intervals.

**Step 2: There exists a subsequence $(W_{n_\ell})_{\ell \in \mathbb{N}}$ such that for every $k \in \mathbb{N}$ and $\epsilon \in \{+, -\}$, the subsequence $(W_{n_\ell,k}^\epsilon)_{\ell \in \mathbb{N}}$ weakly converges, as $\ell \to \infty$, almost everywhere to a limit, say $U_k^\epsilon$ which is a step-function adapted to a partition with $m_k$ elements (some elements might be empty sets).

Fix some $k \in \mathbb{N}$. The stepfunctions $(W_{n,k} = (W_n)_{\mathcal{P}_{n,k}})_{n \in \mathbb{N}}$ all have the same number of steps $m_k$. For $n \in \mathbb{N}$, denote by $\mathcal{P}_{n,k} = \{S_{n,k,i} : 1 \le i \le k\}$
$m_k$} the interval partition adapted to $W_{n,k}$ where the intervals are order according to the natural order on $[0, 1]$ (note that some intervals might be empty, simply put them at the end). For $n \in \mathbb{N}$ and $1 \le i \le m_k$, let $\lambda(S_{n,k,i})$ denote the length of the interval $S_{n,k,i} \in \mathcal{P}_{n,k}$. As the lengths of steps take values in the compact set $[0, 1]$, there exists a subsequence of indices $(n_\ell)_{\ell \in \mathbb{N}}$ such that for every $1 \le i \le m_k$, there exists $s_{k,i} \in [0, 1]$ such that $\lim_{\ell \to \infty} \lambda(S_{n_\ell,k,i}) = s_{k,i}$. Denote by $\mathcal{P}_k = \{S_{k,i} : 1 \le i \le m_k\}$ the interval partition composed of $m_k$ intervals where the i-th interval $S_{k,i}$ has length $s_{k,i}$ (note that some intervals might be empty). Up to a diagonal extraction, we can assume that the convergence holds for every $k \in \mathbb{N}$ simultaneously. Remark that for all $n, k \in \mathbb{N}$, the fact that $\mathcal{P}_{n,k+1}$ is a refinement of $\mathcal{P}_{n,k}$ can be simply restated as linear relations on the interval lengths $(\lambda(S_{n,k,i}))_{1 \le i \le m_k}$ and $(\lambda(S_{n,k+1,i}))_{1 \le i \le m_{k+1}}$. As linear relations are preserved when taking the limit, we get that the partition $\mathcal{P}_{k+1}$ is a refinement of $\mathcal{P}_k$ for all $k \in \mathbb{N}$. We assume from now on that $(W_n)_{n \in \mathbb{N}}$ and $(W_{n,k})_{n \in \mathbb{N}}$, $k \in \mathbb{N}$, are the corresponding subsequences.

For every $n \in \mathbb{N}$, we decompose $W_n = W_n^+ - W_n^-$ into its positive and negative kernel parts, see Lemma 3.3. For $n, k \in \mathbb{N}$ and $\epsilon \in \{+, -\}$, we define $W_{n,k}^{\epsilon} = (W_n^{\epsilon})_{\mathcal{P}_{n,k}}$. In particular, remark that $W_{n,k} = W_{n,k}^+ - W_{n,k}^-$ and for all $\ell \ge k$, that $W_{n,k}^{\epsilon} = (W_{n,\ell}^{\epsilon})_{\mathcal{P}_{n,k}}$. Let $\epsilon \in \{+, -\}$ and $1 \le i, j \le m_k$ such that $s_{k,i}s_{k,j} > 0$ be fixed. For every $n \in \mathbb{N}$, we have on $S_{n,k,i} \times S_{n,k,j}$ that $W_{n,k}^{\epsilon} = \mu_{n,k}^{i,j,\epsilon} \in \mathcal{M}_+(\mathbf{Z})$ with:

$$ \mu_{n,k}^{i,j,\epsilon}(\cdot) = \frac{1}{\lambda(S_{n,k,i})\lambda(S_{n,k,j})} W_n^{\epsilon}(S_{n,k,i} \times S_{n,k,j}; \cdot). $$

We have that:

$$ \|\mu_{n,k}^{i,j,\epsilon}\|_{\infty} \le \|W_n\|_{\infty} \le C. $$

This gives that the sequence $(\mu_{n,k}^{i,j,\epsilon})_{n \in \mathbb{N}}$ in $\mathcal{M}_{\pm}(\mathbf{Z})$ is bounded. We now prove it is tight. Let $\eta > 0$. As $\lim_{n \to \infty} \lambda(S_{n,k,\ell}) = s_{k,\ell} > 0$ for $\ell = i, j$, we deduce that there exists $c > 0$ such that for every $n \in \mathbb{N}$ large enough and $\ell = i, j$, we have $\lambda(S_{n,k,\ell}) > c$. Set $\eta' = c^2\eta$. As the sequence $(W_n)_{n \in \mathbb{N}}$ in $\tilde{\mathcal{W}}_{\pm}$ is tight, there exists a compact set $K \subset \mathbf{Z}$ such that for every $n \in \mathbb{N}$, $M_{W_n}(K^c) \le \eta'$. Hence, for every $n \in \mathbb{N}$ large enough, we have:

$$ \mu_{n,k}^{i,j,\epsilon}(K^c) \le \frac{1}{\lambda(S_{n,k,i})\lambda(S_{n,k,j})} M_{W_n}(K^c) \le \eta. $$

This gives that the sequence $(\mu_{n,k}^{i,j,\epsilon})_{n \in \mathbb{N}}$ in $\mathcal{M}_+(\mathbf{Z})$ is bounded and tight, and thus by Lemma 2.8, it has a convergent subsequence. By diagonal extraction, we can assume there is a subsequence $(W_{n_\ell})_{\ell \in \mathbb{N}}$ such that for all $k \in \mathbb{N}$, all $1 \le i, j \le m_k$ such that $s_{k,i}s_{k,j} > 0$, and all $\epsilon \in \{+, -\}$, the
subsequence $(\mu_{n_{\ell},k}^{i,j,\epsilon})_{\ell \in \mathbb{N}}$ weakly converges to a limit say $\mu_k^{i,j,\epsilon}$. Define the stepfunction $U_k^\epsilon \in W_+$ adapted to the partition $\mathcal{P}_k$ which is equal to $\mu_k^{i,j,\epsilon}$ on $S_{k,i} \times S_{k,j}$ (if $s_{k,i}s_{k,j} = 0$, set $\mu_k^{i,j,\epsilon} = 0$). We have in particular obtained that, for every $k \in \mathbb{N}$, the subsequence $(W_{n_{\ell},k})_{\ell \in \mathbb{N}}$ weakly converges a.e. to $U_k^\epsilon$ which is a stepfunction adapted to a partition with $m_k$ elements; this implies that the subsequence $(W_{n_{\ell},k})_{\ell \in \mathbb{N}}$ also weakly converges a.e. to $U_k = U_k^+ - U_k^-$. We now assume that $(W_n)_{n \in \mathbb{N}}$ is such a subsequence. With this convention, notice that for all $k, n \in \mathbb{N}$ and $\epsilon \in \{+, -\}$:

$$ (8.1) \qquad \|U_k^\epsilon\|_\infty \le \sup_{n \in \mathbb{N}} \|W_{n,k}^\epsilon\|_\infty \le \sup_{n \in \mathbb{N}} \|W_n\|_\infty = C < +\infty. $$

**Step 3:** There exists a subsequence of $(U_k)_{k \in \mathbb{N}}$ which weakly converges to a limit $U \in W_\pm$ almost everywhere on $[0, 1]^2$. The proof of this step is postponed to the end. Without loss of generality we still write $(U_k)_{k \in \mathbb{N}}$ for this subsequence.

**Step 4:** We have $\lim_{n \to \infty} \delta_{\square}(U, W_n) = \lim_{n \to \infty} \delta'_{\square}(U, W_n) = 0$. Let $\eta > 0$. As the cut distances $d$ is smooth, we deduce from Step 3, that for $k$ large enough $d(U, U_k) \le \eta$. By hypothesis (iii) on the sequence $(W_{n,k})_{n \in \mathbb{N}}$, we also have that for $k$ large enough $d(W_n, W_{n,k}) \le \eta$. For such large $k$, as by step 2 the sequence $(W_{n,k})_{n \in \mathbb{N}}$ weakly converges almost everywhere to $U_k$, and again as the cut distances $d$ is smooth, there is a $n_0$ such that for every $n \ge n_0$, $d(U_k, W_{n,k}) \le \eta$. Then for all $n \ge n_0$, we have:

$$ \begin{aligned} \delta_{\square}(U, W_n) &\le \delta_{\square}(U, U_k) + \delta_{\square}(U_k, W_{n,k}) + \delta_{\square}(W_{n,k}, W_n) \\ &\le d(U, U_k) + d(U_k, W_{n,k}) + d(W_{n,k}, W_n) \\ &\le 3\eta. \end{aligned} $$

This gives that $\lim_{n \to \infty} \delta_{\square}(W_n, U) = 0$.

If we consider a second distance $d'$ as in the lemma, then similarly $\lim_{n \to \infty} \delta'_{\square}(W_n, U) = 0$.

**Proof of Step 3.** Assume that the claim is true for measure-valued kernels. Then, if $(U_k)_{k \in \mathbb{N}}$ is a sequence of signed-measure valued kernels, applying the claim to $(U_k^\epsilon)_{k \in \mathbb{N}}$, for $\epsilon \in \{+, -\}$, we get a measure-valued $U^\epsilon \in W_+$ such that the sequence $(U_k^\epsilon)_{k \in \mathbb{N}}$ weakly conveges a.e. to $U^\epsilon$. Thus, the sequence $(U_k)_{k \in \mathbb{N}}$ weakly conveges a.e. to $U = U^+ - U^-$.

Hence, we are left to prove the claim for measure-valued kernels. The proof is divided in four steps. The first three steps also work for signed-measure valued kernels, but the last argument of step 3.d. only works for measures.
**Step 3.a: The sequence $(U_k)_{k \in \mathbb{N}}$ inherit the tightness property from the sequence $(W_n)_{n \in \mathbb{N}}$.** Let $\eta > 0$. Since the sequence $(W_n)_{n \in \mathbb{N}}$ is tight, there exists a compact set $K \subset \mathbb{Z}$ such that for every $n \in \mathbb{N}$, we have $M_{W_n}(K^c) \le \eta$. Remark that:

$$M_{W_{n,k}} = \sum_{1 \le i,j \le m_k} \lambda(S_{n,k,i})\lambda(S_{n,k,j})\mu_{n,k}^{i,j} = M_{W_n}$$

$$\text{and} \quad M_{U_k} = \sum_{1 \le i,j \le m_k} s_{k,i}s_{k,j}\mu_k^{i,j}.$$

For all $k \in \mathbb{N}$ and $1 \le i,j \le m_k$, as the sequence $(\mu_{n,k}^{i,j})_{n \in \mathbb{N}}$ weakly converges to $\mu_k^{i,j}$, using [14, Theorem 2.7.4] with the open subset $K^c \subset \mathbb{Z}$, we get that $\mu_k^{i,j}(K^c) \le \liminf_{n \to \infty} \mu_{n,k}^{i,j}(K^c)$. As $\lim_{n \to \infty} \lambda(S_{n,k,i}) = s_{k,i}$ for all $1 \le i \le m_k$, and summing those bounds, we get:

$$M_{U_k}(K^c) \le \liminf_{n \to \infty} M_{W_{n,k}}(K^c) = \liminf_{n \to \infty} M_{W_n}(K^c) \le \eta.$$

Consequently, the sequence $(U_k)_{k \in \mathbb{N}}$ is tight.

**Step 3.b: Convergence of the measures $\hat{U}_k$ in $M_+([0, 1]^2 \times \mathbb{Z})$ defined for $k \in \mathbb{N}$ as:**

$$(8.2) \qquad \hat{U}_k(dx, dy, dz) = U_k(x, y; dz) \lambda_2(dx, dy).$$

Since the sequence $(M_{U_k})_{k \in \mathbb{N}}$ in $M_+(\mathbb{Z})$ is tight, for all $\eta > 0$, there exists a compact set $K \subset \mathbb{Z}$ such that for every $k \in \mathbb{N}$, $M_{U_k}(K^c) \le \eta$; and thus $\hat{U}_k(\hat{K}^c) = M_{U_k}(K^c) \le \eta$ where $\hat{K} = [0, 1]^2 \times K$ is a compact subset of $[0, 1]^2 \times \mathbb{Z}$, that is, the sequence $(\hat{U}_k)_{k \in \mathbb{N}}$ in $M_+([0, 1]^2 \times \mathbb{Z})$ is tight. The sequence $(\hat{U}_k)_{k \in \mathbb{N}}$ is also bounded as $\|\hat{U}_k\|_{\infty} \le \|U_k\|_{\infty} \le C$ thanks to (8.1). Hence, using Lemma 2.8, there exists a subsequence $(\hat{U}_{k_l})_{l \in \mathbb{N}}$ of the sequence $(\hat{U}_k)_{k \in \mathbb{N}}$ that converges to some measure, say $\hat{U}$, in $M_+([0, 1]^2 \times \mathbb{Z})$. Remark that, when considering the subsequence of indices $(k_l)_{l \in \mathbb{N}}$, the sub-sequences $(W_{n,k_l})_{l \in \mathbb{N}}$, $n \in \mathbb{N}$, still satisfy properties (i)-(iv) of Lemma 8.1, and for all $l \in \mathbb{N}$, the sequence $(W_{n,k_l})_{n \in \mathbb{N}}$ still weakly converges to $U_{k_l}$. Without loss of generality, we now work with this subsequence and thus write $k$ instead of $k_l$.

**Step 3.c: The measure $\hat{U}(dx, dy, dz)$ can be disintegrated w.r.t. $\lambda_2(dx, dy)$ giving us an element of $W_+$.** To prove this, we need the following disintegration theorem for measures, see [38, Theorem 1.23] (stated in more the general framework of Borel spaces) which generalizes the disintegration theorem for probability measures [37, Theorem 6.3]. The notation $\mu \sim \nu$ for two measures $\mu$ and $\nu$ means that $\mu \ll \nu$ and $\nu \ll \mu$, where $\mu \ll \nu$ means that $\mu$ is absolutely continuous w.r.t. $\nu$.
**LEMMA 8.4** (Disintegration theorem for measures, [38, Theorem 1.23]).
Let $\rho$ be a measure on $S \times T$, where $S$ is a measurable space and $T$ a Polish space. Then there exist a measure $\nu \equiv \rho(\cdot \times T)$ on $S$ and a probability kernel $\mu: S \to \mathcal{M}_1(T)$ such that $\rho = \nu \otimes \mu$ (i.e. $\rho(ds, dt) = \nu(ds)\mu(s; dt)$). Moreover, the measures $\mu_s = \mu(s; \cdot)$ are unique for $\nu$-a.e. $s \in S$.

Using Lemma 8.4 with $S = [0, 1]^2$ and $T = \mathbb{Z}$, we get that there exists a probability kernel $U'$ in $\mathcal{W}_1$ such that:

$$ \hat{U}(dx, dy, dz) = U'(x, y; dz) \pi(dx, dy), $$

where $\pi = \hat{U}(\cdot \times \mathbb{Z})$ is a measure on $[0, 1]^2$.

We now need to prove that $\pi \ll \lambda_2$. By contradiction, assume this is false, then there exists a measurable set $A \in \mathcal{B}([0, 1]^2)$ such that $\lambda_2(A) = 0$ and $\pi(A) > 0$. As the measure $\int_A U'(x, y; \cdot) \pi(dx, dy)$ is not null, there exists $f \in C_b(\mathbb{Z})$ such that $\int_A U'(x, y; f) \pi(dx, dy) \neq 0$. As the sequence $(\hat{U}_k)_{k \in \mathbb{N}}$ weakly converges to $\hat{U}$ in $\mathcal{M}_+([0, 1]^2 \times \mathbb{Z})$ by step 3.b, we have that the sequence of measures $\hat{U}_k(dx, dy; f) = U_k(x, y; f) \lambda_2(dx, dy)$ weakly converges as $k \to \infty$ to $\hat{U}(dx, dy; f) = U'(x, y; f) \pi(dx, dy)$ in $\mathcal{M}_+([0, 1]^2)$. Moreover, as the maps $x, y \mapsto U_k(x, y; f)$ are uniformly bounded (by $\|f\|_{\infty} \|U_k\|_{\infty} \le C \|f\|_{\infty}$, see (8.1)), they are also uniformly integrable (w.r.t. $\lambda_2$), and applying [12, Corollary 4.7.19] there exist a subsequence $(U_{k_l})_{l \in \mathbb{N}}$ and a bounded function $g_f$ on $[0, 1]^2$ such that for every bounded measurable function $h \in L^\infty([0, 1]^2)$, we have:

$$ \lim_{l \to \infty} \int U_{k_l}(x, y; f) h(x, y) \lambda_2(dx, dy) = \int g_f(x, y) h(x, y) \lambda_2(dx, dy). $$

In particular, the sequence of measures $(U_{k_l}(x, y; f) \lambda_2(dx, dy))_{l \in \mathbb{N}}$ weakly converges to the measure $g_f(x, y) \lambda_2(dx, dy)$, which imposes the equality between measures:

$$ \hat{U}(dx, dy, f) = U'(x, y; f)\pi(dx, dy) = g_f(x, y) \lambda_2(dx, dy). $$

Hence, taking $h = 1_A$, we get:

$$ \hat{U}(A, f) = \int_A U'(x, y; f) \pi(dx, dy) = \int_A g_f(x, y) \lambda_2(dx, dy) = 0, $$

which yields a contradiction. Consequently, the measure $\pi$ is absolutely continuous w.r.t. $\lambda_2$, with density still denoted by $\pi$, and we set $\lambda_2$-a.e. on $[0, 1]^2$:

$$ (8.3) \quad U(x, y; dz) = \pi(x, y) U'(x, y; dz) $$

and thus $\hat{U}(dx, dy, dz) = U(x, y; dz) \lambda_2(dx, dy)$.
**Step 3.d: The sequence $(U_k)_{k \in \mathbb{N}}$ weakly converges to $U$ almost everywhere on $[0, 1]^2$.** Recall that by construction, the stepfunction $U_k$ is adapted to the partition $\mathcal{P}_k$ defined in Step 2, and that $\mathcal{P}_{k+1}$ is a refinement of $\mathcal{P}_k$. A closer look at Step 2 yields that for all $\ell \ge k$, since $W_{n,k} = (W_{n,\ell})_{\mathcal{P}_{n,k}}$, we also get:

$$ (8.4) \qquad U_k = (U_\ell)_{\mathcal{P}_k}. $$

We prove (8.4) for $\ell = k+1$, the other cases follow by induction. As $\mathcal{P}_{n,k+1}$ is a refinement of $\mathcal{P}_{n,k}$, we already know that $U_k$ and $(U_{k+1})_{\mathcal{P}_k}$ are both stepfunctions adapted to the finite partition $\mathcal{P}_k$. Thus, we only need to verify that $U_k$ and $(U_{k+1})_{\mathcal{P}_k}$ take the same value on each step. For every $n \in \mathbb{N}$, the fact that $W_{n,k} = (W_{n,k+1})_{\mathcal{P}_{n,k}}$ implies that for all $1 \le i, j \le m_k$ such that $\lambda(S_{n,k,i})\lambda(S_{n,k,j}) > 0$, we have:

$$ \mu_{i,j}^{n,k} = \sum_{i' \in I_i, j' \in I_{j'}} \frac{\lambda(S_{n,k+1,i'})\lambda(S_{n,k+1,j'})}{\lambda(S_{n,k,i})\lambda(S_{n,k,j})} \mu_{i',j'}^{n,k+1}, $$

and this equation is preserved when taking the limit $n \to \infty$, which gives us:

$$ \mu_{i,j}^k = \sum_{i' \in I_i, j' \in I_j} \frac{s_{k+1,i'} s_{k+1,j'}}{s_{k,i} s_{k,j}} \mu_{i',j'}^{k+1}, $$
for all $1 \le i, j \le m_k$ such that $s_{k,i} s_{k,j} > 0$.

This proves that the stepfunctions $U_k$ and $(U_{k+1})_{\mathcal{P}_k}$ take the same value on each step $S_{k,i} \times S_{k,j}$ with positive size $s_{k,i} s_{k,j} > 0$ (on a step with null size $s_{k,i} s_{k,j} = 0$, $U_k$ and $(U_{k+1})_{\mathcal{P}_k}$ are both equal to the null measure). This gives that $U_k = (U_{k+1})_{\mathcal{P}_k}$.

Let $f \in C_b(\mathbb{Z})$ be a bounded continuous function, and $X, Y$ be independent uniform random variables on $[0, 1]$. Then (8.4) and (8.1) imply that the sequence $N^f = (N_k^f = U_k(X, Y; f))_{k \in \mathbb{N}}$ is a martingale bounded by $C\|f\|_\infty$ for the filtration $(\mathcal{F}_k)_{k \in \mathbb{N}}$, where the $\sigma$-field $\mathcal{F}_k$ is generated by the events $\{X \in S_{k,i}\} \cap \{Y \in S_{k,j}\}$ for $1 \le i, j \le m_k$ and $S_{k,\ell} \in \mathcal{P}_k$. By the martingale convergence theorem, the martingale $N^f$ is almost surely convergent, that is, the sequence $(U_k[f])_{k \in \mathbb{N}}$ converges $\lambda_2$-a.e. to a bounded measurable function $u_f$. Let $g: [0, 1]^2 \to \mathbb{R}$ be a bounded measurable
function. We get:

$$
\begin{align*}
\int_{[0,1]^2} g(x, y) U(x, y; f) \lambda_2(dx dy) &= \int_{[0,1]^2 \times \mathbf{Z}} g(x, y) f(z) \hat{U}(dx, dy, dz) \\
&= \lim_{k \to \infty} \int_{[0,1]^2 \times \mathbf{Z}} g(x, y) f(z) \hat{U}_k(dx, dy, dz) \\
&= \lim_{k \to \infty} \mathbb{E} [g(X, Y) U_k(X, Y; f)] \\
&= \int_{[0,1]^2} g(x, y) u_f(x, y) \lambda_2(dx dy),
\end{align*}
$$

where we used the definition (8.3) of $U$ for the first equality, that $(\hat{U}_k)_{k \in \mathbb{N}}$
weakly converges to $\hat{U}$ for the second, the definition (8.2) of $\hat{U}_k$ for the third,
and the convergence of the martingale $N^f$ for the last. Since $g$ is arbitrary,
we deduce that $\lambda_2$-a.e. $U(\cdot, \cdot; f) = u_f$ and thus that the sequence $(U_k[f])_{k \in \mathbb{N}}$
converges $\lambda_2$-a.e. to $U[f]$. Applying this result for all $f \in \mathcal{F} = (f_m)_{m \in \mathbb{N}}$ a
convergence determining sequence (with the convention $f_0 = 1$), we deduce
that the sequence $(U_k)_{k \in \mathbb{N}}$ weakly converges to $U$ almost everywhere on
$[0, 1]^2$. Recall from Section 2 that convergence determining sequences exist
only for measures and not for signed measures in general, this is why we
worked with measures in Step 3. This ends the proof of Step 3, and thus
ends the proof of the lemma.
$\square$

We are now ready to prove Theorem 5.1.

Proof of Theorem 5.1. — We first prove Point (i) on $W_\pm$ (the proof on $W_1$ is similar). Since the distance $d$ is weakly regular and the sequence $(W_n)_{n\in\mathbb{N}}$ is uniformly bounded and tight in $W_\pm$, we can construct induc- tively for every $n \in \mathbb{N}$ a sequence $(P_{n,k})_{k\in\mathbb{N}}$ of partitions of $[0,1]$ such that hypothesis (i)-(iii) of Lemma 8.1 are satisfied: $P_{n,k+1}$ being obtained by ap- plying the weak regularity property (see Definition 4.10-(i)) with starting partition $Q_{n,k} = P_{n,k} \wedge D_k$, where $D_k$ is the dyadic partition with stepsize $2^{-(k+1)}$. (We may assume that the partitions $P_{n,k}$ for all $n \in \mathbb{N}$ have the same size $m_k$ by adding empty sets.) Then as $d$ is also invariant and smooth on $W_\pm$, the first part of Lemma 8.1 directly gives Point (i).

Before proving Point (ii), we first need to prove the following lemma.

LEMMA 8.5 (Compactness theorem for $W_M$). — Let $d$ be an invariant, smooth and weakly regular distance on $W_1$ (resp. $W_+$ or $W_\pm$). Let $M$ be a convex and weakly closed subset of $M_1(\mathbf{Z})$ (resp. $M_+(\mathbf{Z})$ or $M_+(\mathbf{Z}))$. Let $(W_n)_{n\in\mathbb{N}}$ be a sequence of $M$-valued kernels which is tight and uniformly bounded. Then, $(W_n)_{n\in\mathbb{N}}$ has a subsequence that converges for $\delta\square$ to some $M$-valued kernel.
*Proof.* — First remark that, as $\mathcal{M}$ is convex, the image of $W_{\mathcal{M}}$ by the stepping operator $W \mapsto W_{\mathcal{P}}$, where $\mathcal{P}$ is a finite partition of $[0, 1]$, is a subset of $W_{\mathcal{M}}$. Hence, a close look at the proof of Lemma 8.1 (the partitions are constructed as in the proof of Point (i) from Theorem 5.1), and using the notation therein, shows that, up to taking subsequences, one can take the stepping kernels $W_{n,k}$ and $U_k$ in $W_{\mathcal{M}}$, such that $(U_k)_{k \in \mathbb{N}}$ weakly converges to $U$ a.e. and the subsequence $(W_{n_l})_{l \in \mathbb{N}}$ converges to $U$ w.r.t. $\delta_{\square}$. Since $U_k(x, y; \cdot) \in \mathcal{M}$ weakly converges to $U(x, y; \cdot)$ for almost every $x, y \in [0, 1]$ and since $\mathcal{M}$ is weakly closed (and thus sequentially weakly closed), we deduce that $U(x, y; \cdot)$ belongs to $\mathcal{M}$ for almost every $x, y \in [0, 1]$. This means that $U \in W_{\mathcal{M}}$. $\square$

We prove Point (ii) for $\mathcal{M} \subset \mathcal{M}_{\pm}(\mathbf{Z})$ (the proof for $\mathcal{M} \subset \mathcal{M}_1(\mathbf{Z})$ is identical). The fact that $W_{\mathcal{M}}$ and $\tilde{W}_{\mathcal{M}}$ are convex is clear as $\mathcal{M}$ is convex. Let $(W_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\tilde{W}_{\mathcal{M}}$. Since $\mathcal{M}$ is convex, we deduce that $(M_{W_n})_{n \in \mathbb{N}}$ is a sequence in $\mathcal{M}$. As $\mathcal{M}$ is sequentially compact for the weak topology, $\mathcal{M}$ is tight and bounded by Lemma 2.8, and thus the sequence $(W_n)_{n \in \mathbb{N}}$ is tight and uniformly bounded (recall Definition 4.7). Hence, using Lemma 8.5, we get that from any sequence in $\tilde{W}_{\mathcal{M}}$, we can extract a subsequence which converges for $\delta_{\square}$ to an element in $\tilde{W}_{\mathcal{M}}$. This implies that $(\tilde{W}_{\mathcal{M}}, \delta_{\square})$ is compact.

Point (iii) is a direct consequence of Point (ii) as if $\mathbf{Z}$ is compact, so is $\mathcal{M}_1(\mathbf{Z})$. $\square$

*Proof of Point (iii) from Proposition 5.2.* — We prove Point (iii). The fact that $W_{\mathcal{M}}$ and $\tilde{W}_{\mathcal{M}}$ are convex is clear as $\mathcal{M}$ is convex. To prove that $\tilde{W}_{\mathcal{M}}$ is closed, we consider a sequence $(W_n)_{n \in \mathbb{N}}$ in $\tilde{W}_{\mathcal{M}}$ that converges for $\delta_{\square,m}$ to some $W \in \tilde{W}_{\pm}$. As $(W_n)_{n \in \mathbb{N}}$ is a Cauchy sequence for $\delta_{\square,m}$, by Lemma 4.9, $(M_{W_n})_{n \in \mathbb{N}}$ is a Cauchy sequence for $d_m$ and thus is tight. Hence, $(W_n)_{n \in \mathbb{N}}$ is uniformly bounded and tight. Applying Lemma 8.5, there exists a subsequence $(W_{n_k})_{k \in \mathbb{N}}$ of the sequence $(W_n)_{n \in \mathbb{N}}$ which converges for $\delta_{\square,m}$ to some $\mathcal{M}$-valued kernel $U \in \tilde{W}_{\mathcal{M}}$. But as a subsequence, $(W_{n_k})_{k \in \mathbb{N}}$ must also converge for $\delta_{\square,m}$ to $W$. This implies that $W = U$ is a $\mathcal{M}$-valued kernel. $\square$

In order to prove Theorem 5.5, we first prove a lemma that allows to construct the partitions needed to use Lemma 8.1.

LEMMA 8.6 (Construction of partitions for two distances). — Let $d$ and $d'$ be two distances on $W_1$ (resp. $W_+$ or $W_\pm$) which are invariant, smooth, weakly regular and regular w.r.t. the stepping operator (see Definitions 3.10 and 4.10). Let $(W_n)_{n \in \mathbb{N}}$ be a sequence in $W_1$ (resp. $W_+$ or $W_\pm$) which is
tight (resp. uniformly bounded and tight). Then, there exists sequences $(\mathcal{P}_{n,k})_{k \in \mathbb{N}}$, $n \in \mathbb{N}$, of partitions of $[0, 1]$ such that hypothesis (i)-(iv) of Lemma 8.1 are satisfied.

*Proof.* — We prove the result on $W_\pm$ (the proof on $W_1$ and $W_+$ is similar). To simplify notations, write $d^1 = d$ and $d^2 = d'$. We proceed by induction on $k \in \mathbb{N} \cup \{-1\}$. For every $n \in \mathbb{N}$, set $\mathcal{P}_{n,-1} = \{[0, 1]\}$ the trivial partition with size 1. Let $k \in \mathbb{N}$ and assume that we have already constructed partitions $(\mathcal{P}_{n,k-1})_{n \in \mathbb{N}}$ that have the same size $m_{k-1}$. Now we proceed to construct partitions $(\mathcal{P}_{n,k})_{n \in \mathbb{N}}$ that satisfy hypothesis (i)-(iv).

Set $C = \sup_{n \in \mathbb{N}} \|W_n\|_\infty$, which is finite as the sequence $(W_n)_{n \in \mathbb{N}}$ is uniformly bounded. As $d^i$, with $i = 1, 2$, are regular w.r.t. the stepping operator, there exists a finite constant $C_0 > 0$ such that for every $W, U \in W_\pm$, with $\|W\|_\infty \le C$ and $\|U\|_\infty \le C$, and $U$ a stepfunction adapted to a finite partition $Q$:

$$ (8.5) \qquad d^i(W, W_Q) \le C_0 d^i(W, U). $$

Set $\varepsilon = 1/C_0(k+1)$. Since $d^i$, with $i = 1, 2$, are weakly regular and the sequence $(W_n)_{n \in \mathbb{N}}$ is tight and uniformly bounded, there exists $r_k \in \mathbb{N}^*$, such that for every $n \in \mathbb{N}$, there exists a partition $\mathcal{R}_{n,k}^i$ of $[0, 1]$ that refines $Q_{n,k} = \mathcal{P}_{n,k-1} \wedge \mathcal{D}_k$, where $\mathcal{D}_k$ is the dyadic partition with stepsize $2^{-k}$, such that:

$$ (8.6) \qquad |\mathcal{R}_{n,k}^i| \le r_k |Q_{n,k}| \le 2^k r_k |\mathcal{P}_{n,k-1}| \quad \text{and} \quad d^i(W_n, (W_n)_{\mathcal{R}_{n,k}^i}) \le \varepsilon. $$

(Indeed, a close look at the proof shows that $\mathcal{P}_{n,k-1}$ refines $\mathcal{D}_{k-1}$ by construction, thus $Q_{n,k}$ cuts each set of $\mathcal{P}_{n,k-1}$ in at most 2 sets, and we get $|Q_{n,k}| \le 2|\mathcal{P}_{n,k-1}|$.) Now, let $\mathcal{P}_{n,k}$ be the common refinement of $\mathcal{R}_{n,k}^1$ and $\mathcal{R}_{n,k}^2$; it is a refinement of $\mathcal{P}_{n,k-1}$, has diameter at most $2^{-k}$ and size:

$$ |\mathcal{P}_{n,k}| \le 2^{2k} r_k^2 |\mathcal{P}_{n,k-1}|^2 = 2^{2k} r_k^2 m_{k-1}^2. $$

If necessary, by completing $\mathcal{P}_{n,k}$ with null sets, we may assume that $|\mathcal{P}_{n,k}| = m_k$, where $m_k = 2^{2k} r_k^2 m_{k-1}^2$. As $(W_n)_{\mathcal{R}_{n,k}^i}$ is a stepfunction adapted to the partition $\mathcal{P}_{n,k}$, we deduce from (8.5) and (8.6) that for $i = 1, 2$ and $n \in \mathbb{N}$:

$$ d^i(W_n, (W_n)_{\mathcal{P}_{n,k}}) \le C_0 d^i(W_n, (W_n)_{\mathcal{R}_{n,k}^i}) \le C_0 \varepsilon = \frac{1}{k+1}. $$

Hence, for every $n \in \mathbb{N}$, the partition $\mathcal{P}_{n,k}$ satisfies the hypothesis (i)-(iv) of Lemma 8.1. Thus, the induction is complete. $\square$

*Proof of Theorem 5.5.* — Let $d_m$ and $d_{m'}$ be as in Theorem 5.5.
Let $(W_n)_{n \in \mathbb{N}}$ be a sequence of probability-graphons that converges to some $W \in \tilde{W}_1$ for $\delta_{\square,m}$. By Lemma 4.9, the sequence of probability measure $(M_{W_n})_{n \in \mathbb{N}}$ converges to $M_W$ for the distance $d_m$. As $d_m$ induces the weak topology on $M_{\le 1}(Z)$, we have that the sequence $(M_{W_n})_{n \in \mathbb{N}}$ is tight, and thus the sequence $(W_n)_{n \in \mathbb{N}}$ is also tight (recall Definition 4.7). The sequence $(W_n)_{n \in \mathbb{N}}$ is also uniformly bounded as a sequence in $\tilde{W}_1$. Applying Lemma 8.6 with the distances $d = d_{\square,m}$ and $d' = d_{\square,m'}$, which are invariant, smooth, weakly regular and regular w.r.t. the stepping operator, we get sequences of partitions $(P_{n,k})_{k \in \mathbb{N}}$, $n \in \mathbb{N}$, that satisfy hypothesis (i)-(iv) of Lemma 8.1. We then deduce from the last part of Lemma 8.1 that any subsequence of $(W_n)_{n \in \mathbb{N}}$ has a further subsequence which converges to the same limit for both $\delta_{\square,m}$ and $\delta_{\square,m'}$, this limit must then be $W$. This implies that the sequence $(W_n)_{n \in \mathbb{N}}$ converges to $W$ for $\delta_{\square,m'}$.

The role of $d_m$ and $d_{m'}$ being symmetric, we conclude that the distances $\delta_{\square,m}$ and $\delta_{\square,m'}$ induce the same topology on $\tilde{W}_1$. $\square$

## 9. Index of notations

<table><thead><tr><th>Measures</th><th></th></tr></thead><tbody><tr><td>- (Z, O<sub>Z</sub>) a topological Polish space (p. 37)</td><td>- M<sub>≤1</sub>(Z) the set of sub-probability measures on Z, i.e. measures with total mass at most 1 (p. 37)</td></tr><tr><td>- B(Z) the Borel σ-field induced by O<sub>Z</sub> (p. 37)</td><td>- μ<sup>+</sup>, μ<sup>-</sup> the positive and negative parts of μ from its Hahn–Jordan decomposition (p. 37)</td></tr><tr><td>- C<sub>b</sub>(Z) the set of continuous bounded real-valued functions on Z (p. 37)</td><td>- |μ| = μ<sub>+</sub> + μ<sub>-</sub> the total variation measure of μ (p. 37)</td></tr><tr><td>- measure = positive measure (p. 36)</td><td>- ||μ||<sub>∞</sub> = |μ|(Z) the total mass of μ (p. 37)</td></tr><tr><td>- M<sub>±</sub>(Z) the set of signed measures on Z (p. 37)</td><td>- d<sub>m</sub> a distance on either M<sub>≤1</sub>(Z), M<sub>+</sub>(Z) or M<sub>±</sub>(Z) (p. 45)</td></tr><tr><td>- M<sub>+</sub>(Z) the set of measures on Z (p. 37)</td><td>- N<sub>m</sub> a norm on M<sub>±</sub>(Z) (p. 45)</td></tr><tr><td>- M<sub>1</sub>(Z) the set of probability measures on Z (p. 37)</td><td>- d<sub>LP</sub> the Lévy–Prokhorov distance (p. 55)</td></tr><tr><td></td><td>- || · ||<sub>KR</sub> the Kantorovitch–Rubinshtein norm (p. 55)</td></tr></tbody></table>
- $\| \cdot \|_{\text{FM}}$ the Fortet-Mourier norm (p. 55)

- $\| \cdot \|_{\mathcal{F}}$ the norm based on a convergence determining sequence $\mathcal{F}$ (p. 56)

Relabelings and partitions

- $S_{[0,1]}$ the set of bijective measure-preserving maps from $([0,1], \lambda)$ to itself (p. 36)

- $\tilde{S}_{[0,1]}$ the set of measure-preserving maps from $([0,1], \lambda)$ to itself (p. 36)

- $|\mathcal{P}|$ the number of sets in the finite partition $\mathcal{P}$ (p. 44)

Kernels and graphons spaces

- $W_1$ the set of probability-graphons (p. 42)

- $W_+$ the set of measure-valued kernels (p. 42)

- $W_\pm$ the set of signed measure-valued kernels (p. 42)

- $W_M$ the set of $M$-valued kernels with $M \subset M_\pm(Z)$ (p. 43)

- $\tilde{W}_1$ the set of unlabeled probability-graphons (p. 50)

- $\tilde{W}_+$ the set of unlabeled measure-valued kernels (p. 50)

- $\tilde{W}_\pm$ the set of unlabeled signed measure-valued kernels (p. 50)

- $\tilde{W}_M$ the set of unlabeled $M$-valued kernels (p. 50)

Kernels and graphons

- $W^+$ and $W^-$ the positive and negative part of $W \in W_\pm$ (p. 43)

- $|W| = W^+ + W^-$ (p. 43)

- $W(A; \cdot) = \int_A W(x, y; \cdot) dxdy$ for $A \subset [0,1]^2$ (p. 45)

- $W[f](x, y) = W(x, y; f)$ for $f \in C_b(Z)$ (p. 53)

- $W_P$ the stepping of $W$ w.r.t. a partition $\mathcal{P}$ (p. 59)

- $\|W\|_\infty := \sup_{x,y \in [0,1]} \|W(x, y; \cdot)\|_\infty$ (p. 42)

- $M_W(dz) = |W|([0,1]^2; dz)$ (p. 61)

- $W_G$ the probability-graphon associated to a $M_1(Z)$-graph or a weighted graph $G$ (p. 80)

- $\mathbb{H}(k, W)$ the $M_1(Z)$-graph with $k$ vertices sampled from $W \in W_1$ (p. 82)

- $\mathbb{G}(k, W)$ the $M_1(Z)$-graph with $k$ vertices sampled from $W \in W_1$ (p. 82)

- $F^g$ a finite graph whose edges are decorated with functions in $C_b(Z)$ (p. 89)

- $t(F^g, W) = M_W^F(g)$ the homomorphism density of $F^g$ in $W$ (p. 89)

Distances/norms on graphon spaces

- $d_{\square,m}$ the cut distance associated to $d_m$ (p. 45)

- $N_{\square,m}$ the cut norm associated to $N_m$ (p. 45)
- $\delta_{\square}$ the unlabeled distance associated to an arbitrary distance $d$ (p. 49)

- $\delta_{\square,m}$ the unlabeled cut distance associated to $d_{\square,m}$ or $N_{\square,m}$ (p. 49)

- $\|\cdot\|_{\square,\mathbb{R}}$ the cut norm for real-valued kernels (p. 57)

- $\|\cdot\|_{\square,\mathbb{R}}^{+}$ the positive part of the cut norm for real-valued kernels (p. 57)

- *weak isomorphism* of kernels and graphons in Definition 3.16 on page 50

- *tightness* for sets of kernels or graphons in Definition 4.7 on page 62

- *invariant* and *smooth* for a distance $d$ on graphon spaces in Definition 3.10 on page 46

- *weakly regular* and *regular w.r.t.* the stepping operator for a distance $d$ on graphon spaces in Definition 4.10 on page 63

## Definitions

## Acknowledgement

We thank Pierre-André Zitt for some helpful discussion in particular on Lemma 3.20. We also thank the referees for their careful reading and for their helpful suggestions which improved the paper.

## BIBLIOGRAPHY

[1] R. ALBERT, I. ALBERT & G. L. NAKARADO, Structural Vulnerability of the North American Power Grid, Physical Review E 69 (2004), no. 2, article no. 025103, https://doi.org/10.1103/PhysRevE.69.025103.

[2] R. ALBERT & A.-L. BARABÁSI, Statistical Mechanics of Complex Networks, Reviews of Modern Physics 74 (2002), no. 1, p. 47–97, https://doi.org/10.1103/RevModPhys.74.47.

[3] G. ALETTI & G. NALDI, Opinion Dynamics on Graphon: The Piecewise Constant Case, Applied Mathematics Letters 133 (2022), article no. 108227, https://doi.org/10.1016/j.aml.2022.108227.

[4] L. ALONSO, J. A. MÉNDEZ-BERMÚDEZ, A. GONZÁLEZ-MELÉNDREZ & Y. MORENO, Weighted Random-Geometric and Random-Rectangular Graphs: Spectral and Eigenfunction Properties of the Adjacency Matrix, Journal of Complex Networks 6 (2018), no. 5, p. 753–766, https://doi.org/10.1093/comnet/cnx053.

[5] H. AMINI, M. DRAIEF & M. LELARGE, Flooding in Weighted Sparse Random Graphs, SIAM Journal on Discrete Mathematics 27 (2013), no. 1, p. 1–26, https://doi.org/10.1137/120865021.

[6] H. AMINI & M. LELARGE, The Diameter of Weighted Random Graphs, The Annals of Applied Probability 25 (2015), no. 3, p. 1686–1727, https://doi.org/10.1214/14-AAP1034.
[7] E. ARCHER, A. NACHMIAS & M. SHALEV, The GHP Scaling Limit of Uniform Spanning Trees in High Dimensions, Communications in Mathematical Physics 405 (2024), no. 3, article no. 73, https://doi.org/10.1007/s00220-023-04923-2.

[8] S. ATHREYA, S. PAL, R. SOMANI & R. TRIPATHI, Path Convergence of Markov Chains on Large Graphs, 2023, https://arxiv.org/abs/2308.09214.

[9] N. AYI & N. P. DUTEIL, Graph Limit for Interacting Particle Systems on Weighted Random Graphs, 2023, https://arxiv.org/abs/2307.12801.

[10] S. BHAMIDI, R. VAN DER HOFSTAD & G. HOOGHIEMSTRA, First Passage Percolation on Random Graphs with Finite Mean Degrees, The Annals of Applied Probability 20 (2010), no. 5, p. 1907-1965, https://doi.org/10.1214/09-AAP666.

[11] P. BILLINGSLEY, Convergence of Probability Measures, second edition ed., Wiley, New York, 1999.

[12] V. I. BOGACHEV, Measure Theory Vol. 1, vol. 1, Springer, Berlin; New York, 2007.

[13] ——, Measure Theory Vol. 2, vol. 2, Springer, Berlin; New York, 2007.

[14] ——, Weak Convergence of Measures, Mathematical Surveys and Monographs, no. volume 234, American Mathematical Society, Providence, Rhode Island, 2018.

[15] B. BOLLOBÁS, S. JANSON & O. RIORDAN, The Cut Metric, Random Graphs, and Branching Processes, Journal of Statistical Physics 140 (2010), no. 2, p. 289-335, https://doi.org/10.1007/s10955-010-9982-z.

[16] C. BORGS, J. CHAYES, L. LOVÁSZ, V. SÓS & K. VESZTERGOMBI, Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing, Advances in Mathematics 219 (2008), no. 6, p. 1801-1851, https://doi.org/10.1016/j.aim.2008.07.008.

[17] C. BORGS & J. CHAYES, Graphons: A Nonparametric Method to Model, Estimate, and Design Algorithms for Massive Networks, in Proceedings of the 2017 ACM Conference on Economics and Computation (Cambridge Massachusetts USA), ACM, 2017, p. 665-672, https://doi.org/10.1145/3033274.3084097.

[18] C. BORGS, J. CHAYES, L. LOVÁSZ, V. SÓS & K. VESZTERGOMBI, Convergent Sequences of Dense Graphs II. Multiway Cuts and Statistical Physics, Annals of Mathematics 176 (2012), no. 1, p. 151-219, https://doi.org/10.4007/annals.2012.176.1.2.

[19] S. BORNHOLDT & H. G. SCHUSTER (eds.), Handbook of Graphs and Networks: From the Genome to the Internet, 1 ed., Wiley, 2002.

[20] J.-F. DELMAS, D. DRONNIER & P.-A. ZITT, An Infinite-Dimensional Metapopulation SIS Model, Journal of Differential Equations 313 (2022), p. 1-53, https://doi.org/10.1016/j.jde.2021.12.024.

[21] M. DRAIEF & L. MASSOULIÉ, Epidemics and Rumours in Complex Networks, London Mathematical Society Lecture Note Series, no. 369, Cambridge University Press, Cambridge; New York, 2010.

[22] L. DUBINS & D. FREEDMAN, Measurable Sets of Measures, Pacific Journal of Mathematics 14 (1964), no. 4, p. 1211-1222, https://doi.org/10.2140/pjm.1964.14.1211.

[23] R. ENGELKING, General topology, rev. and completed ed ed., Sigma series in pure mathematics, no. 6, Heldermann, Berlin, 1989.

[24] S. N. ETHIER & T. G. KURTZ, Markov processes: characterization and convergence, vol. 282, John Wiley & Sons, 2009.

[25] V. FALGAS-RAVRY, K. O'CONNELL, J. STRÖMBERG & A. UZZELL, Multicolour Containers and the Entropy of Decorated Graph Limits, 2016, https://arxiv.org/abs/1607.08152.

[26] M. FREEDMAN, L. LOVÁSZ & A. SCHRIJVER, Reflection Positivity, Rank Connectivity, and Homomorphism of Graphs, Journal of the American Mathematical Society 20 (2006), no. 1, p. 37-51, https://doi.org/10.1090/S0894-0347-06-00529-7.
[27] A. FRIEZE & R. KANNAN, Quick Approximation to Matrices and Applications, *Combinatorica* **19** (1999), no. 2, p. 175–220, https://doi.org/10.1007/s004930050052.

[28] F. GARBE, R. HANCOCK, J. HLADKÝ & M. SHARIFZADEH, Limits of Latin Squares, *Discrete analysis* **2023** (2023), article no. 8, https://doi.org/10.19086/da.83253.

[29] D. GARLASCHELLI, The Weighted Random Graph Model, *New Journal of Physics* **11** (2009), no. 7, article no. 073005, https://doi.org/10.1088/1367-2630/11/7/073005.

[30] J. HLADKÝ, A. NACHMIAS & T. TRAN, The Local Limit of the Uniform Spanning Tree on Dense Graphs, *Journal of Statistical Physics* **173** (2018), no. 3-4, p. 502–545, https://doi.org/10.1007/s10955-017-1933-5.

[31] J. HLADKÝ & G. VISWANATHAN, Random Minimum Spanning Tree and Dense Graph Limits, 2023, https://arxiv.org/abs/2310.11705.

[32] Y. HU, X. WEI, J. YAN & H. ZHANG, Graphon Mean-Field Control for Cooperative Multi-Agent Reinforcement Learning, *Journal of the Franklin Institute* (2023), https://doi.org/10.1016/j.jfranklin.2023.09.002.

[33] A. HUMPHREYS & S. SIMPSON, Separable Banach Space Theory Needs Strong Set Existence Axioms, *Transactions of the American Mathematical Society* **348** (1996), no. 10, p. 4231–4255, https://doi.org/10.1090/S0002-9947-96-01725-4.

[34] T. R. HURD & J. P. GLEESON, On Watts’ Cascade Model with Random Link Weights, *Journal of Complex Networks* **1** (2013), no. 1, p. 25–43, https://doi.org/10.1093/commet/cnt003.

[35] S. JANSON, Graphons, Cut Norm and Distance, Couplings and Rearrangements, *New York Journal of Mathematics* **4** (2013), https://arxiv.org/abs/1009.2376.

[36] V. JOG & P.-L. LOH, Recovering Communities in Weighted Stochastic Block Models, in *2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton)* (Monticello, IL), IEEE, 2015, p. 1308–1315, https://doi.org/10.1109/ALLERTON.2015.7447159.

[37] O. KALLENBERG, *Foundations of modern probability*, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002, xx+638 pages.

[38] ——, *Random Measures, Theory and Applications*, Probability Theory and Stochastic Modelling, vol. 77, Springer International Publishing, Cham, 2017.

[39] N. KERIVEN & S. VAITER, What Functions Can Graph Neural Networks Compute on Random Graphs? The Role of Positional Encoding, in *Advances in Neural Information Processing Systems*, vol. 36, Curran Associates, Inc., 2023, p. 11823–11849, https://proceedings.neurips.cc/paper_files/paper/2023/file/271ec4d1a9ff5e6b81a6e21d38b1ba96-Paper-Conference.pdf.

[40] I. Z. KISS, J. C. MILLER & P. L. SIMON, *Mathematics of Epidemics on Networks: From Exact to Approximate Models*, Interdisciplinary Applied Mathematics, vol. 46, Springer International Publishing, Cham, 2017.

[41] I. KOLOSSVÁRY & B. RÁTH, Multigraph Limits and Exchangeability, *Acta Mathematica Hungarica* **130** (2011), no. 1-2, p. 1–34, https://doi.org/10.1007/s10474-010-0037-3.

[42] D. KUNSZENTI-KOVÁCS, L. LOVÁSZ & B. SZEGEDY, Multigraph Limits, Unbounded Kernels, and Banach Space Decorated Graphs, *Journal of Functional Analysis* **282** (2022), no. 2, article no. 109284, https://doi.org/10.1016/j.jfa.2021.109284.

[43] D. LACKER & A. SORET, A Label-State Formulation of Stochastic Graphon Games and Approximate Equilibria on Large Networks, *Mathematics of Operations Research* (2022), article no. moor.2022.1329, https://doi.org/10.1287/moor.2022.1329.

[44] M. LELARGE, L. MASSOULIE & J. XU, Reconstruction in the Labeled Stochastic Block Model, in *2013 IEEE Information Theory Workshop (ITW)* (Sevilla, Spain), IEEE, 2013, p. 1–5, https://doi.org/10.1109/ITW.2013.6691264.
[45] T. G. LEWIS & T. G. LEWIS, *Network Science: Theory and Practice*, Wiley, Hoboken, NJ, 2009.

[46] L. LOVÁSZ, *Large networks and graph limits*, vol. 60, American Mathematical Society, Colloquium Publications, 2012.

[47] L. LOVÁSZ & B. SZEGEDY, *Limits of Dense Graph Sequences*, *Journal of Combinatorial Theory, Series B* **96** (2006), no. 6, p. 933–957, https://doi.org/10.1016/j.jctb.2006.05.002.

[48] ——, *Szemerédi's Lemma for the Analyst*, *GAFA Geometric And Functional Analysis* **17** (2007), no. 1, p. 252–270, https://doi.org/10.1007/s00039-007-0599-6.

[49] ——, *Limits of Compact Decorated Graphs*, 2010, https://arxiv.org/abs/1010.5155.

[50] M. E. J. NEWMAN, *The Structure and Function of Complex Networks*, *SIAM Review* **45** (2003), no. 2, p. 167–256, https://doi.org/10.1137/S003614450342480.

[51] R. PURVES, *Bimeasurable functions*, *Fund. Math.* **58** (1966), p. 149–157, https://doi.org/10.4064/fm-58-2-149-157.

[52] V. S. VARADARAJAN, *Weak Convergence of Measures on Separable Metric Spaces, Sankhyä: The Indian Journal of Statistics (1933-1960)* **19** (1958), no. 1/2, p. 15–22, http://www.jstor.org/stable/25048364.

[53] J. XU, L. MASSOULIÉ & M. LELARGE, *Edge Label Inference in Generalized Stochastic Block Models: from Spectral Theory to Impossibility Results*, in *Proceedings of The 27th Conference on Learning Theory* (Barcelona, Spain) (M. F. Balcan, V. Feldman & C. Szepesvári, eds.), *Proceedings of Machine Learning Research*, vol. 35, PMLR, 13–15 Jun 2014, p. 903–920, https://proceedings.mlr.press/v35/xu14.html.

[54] M. XU, V. JOG & P.-L. LOH, *Optimal Rates for Community Estimation in the Weighted Stochastic Block Model*, *The Annals of Statistics* **48** (2020), no. 1, p. 183–204, https://doi.org/10.1214/18-AOS1797.

[55] S.-Y. YUN & A. PROUTIERE, *Optimal Cluster Recovery in the Labeled Stochastic Block Model*, in *Advances in Neural Information Processing Systems* (D. Lee, M. Sugiyama, U. Luxburg, I. Guyon & R. Garnett, eds.), vol. 29, Curran Associates, Inc., 2016, https://proceedings.neurips.cc/paper_files/paper/2016/file/a8849b052492b5106526b2331e526138-Paper.pdf.

Manuscript received 22nd December 2023, accepted 19th February 2025.

Romain ABRAHAM
Institut Denis Poisson
Université d'Orléans, Université de Tours, CNRS
France

romain.abraham@univ-orleans.fr

Jean-François DELMAS
CERMICS
Ecole des Ponts
France

jean-francois.delmas@enpc.fr

Julien WEIBEL
Institut Denis Poisson
Université d'Orléans, Université de Tours, CNRS
France
and CERMICS
Ecole des Ponts
France

julien.weibel@normalesup.org

© The authors, 2025

<figure><img src="image_6.png" /></figure>

This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0
INTERNATIONAL LICENSE.

http://creativecommons.org/licenses/by/4.0/

<figure><img src="image_7.png" /></figure>

Innovations in Graph Theory is a member of the
Mersenne Center for Open Publishing
ISSN: 3050-743X