* It must induce a decomposition of $\mathcal{C}(\mu, \nu)$, see Equation (34);

* For all $i \in \mathcal{I} \setminus \mathcal{I}^=$, the measures $\mu_i - \eta_i$ and $\nu_i - \eta_i$, $i \in \mathcal{I} \setminus \mathcal{I}^=$$^+$ must be supported on unions of intervals that pairwise, possibly overlapping only a (see Equation (37)).

**Definition C.2** (Admissible decompositions). We say that $\mathcal{D} = ((\mu_i, \nu_i))_{i \in \mathcal{I}} \subset \mathcal{M}_+^2$ is an admissible decomposition of $(\mu, \nu)$ if the following two conditions hold:

* $\forall \pi \in \mathcal{C}(\mu, \nu), \exists(\pi_i)_{i \in \mathcal{I}} \in \prod_{i \in \mathcal{I}} \mathcal{C}(\mu_i, \nu_i), \pi = \sum_{i \in \mathcal{I}} \pi_i; \quad (34)$

* There exist index sets $(J_i)_{i \in \mathcal{I} \setminus \mathcal{I}^=}$ and corresponding families $((a_i^j, b_i^j))_{j \in J_i} \in (\mathbb{R} \times \mathbb{R})^{J_i}$ with $a_i^j < b_i^j$ such that, for all $i \in \mathcal{I} \setminus \mathcal{I}^=$, $\mu_i - \eta_i$ and $\nu_i - \eta_i$ are concentrated on $\bigcup_{j \in J_i} [a_i^j, b_i^j]$, and

$$ \forall (i_1, i_2) \in (\mathcal{I} \setminus \mathcal{I}^=)^2, \forall (j_1, j_2) \in J_{i_1} \times J_{i_2} : (i_1, j_1) \neq (i_2, j_2) \implies (b_{i_2}^{j_2} \le a_{i_1}^{j_1} \text{ or } b_{i_1}^{j_1} \le a_{i_2}^{j_2}). \quad (35) $$

We denote by $\mathcal{A}^*$ the set of admissible decompositions.

**Notation C.3.** When considering multiple decompositions $\mathcal{D}_1, \mathcal{D}_2$, we write $\mathcal{D}_k = \{(\mu_i^k, \nu_i^k)\}_{i \in \mathcal{I}_k}$ instead of $\mathcal{D} = \{(\mu_i, \nu_i)\}_{i \in \mathcal{I}}$ and $\eta_i^k$ instead of $\eta_i$.

**Remark C.4.** 1. Unlike in Definition 2.43, an admissible decomposition may include several pairs of the form $(\eta, \eta)$, and a pair $(\eta, \eta)$ can appear multiple times in the decomposition. Observe that Requirement (34) states that $\mathcal{C}(\mu, \nu) \subset \sum_{i \in \mathcal{I}} \mathcal{C}(\mu_i, \nu_i)$ (see Notation 1.1): this is weaker than the first Requirement of Definition 2.43. In fact, we will prove that both requirements are equivalent provided that Equation (35) holds (see Remark C.6, Point 1). Also note that if the family $([\min(s_{\mu_i-\eta_i}, s_{\nu_i-\eta_i}), \max(S_{\mu_i-\eta_i}, S_{\nu_i-\eta_i})])_{i \in \mathcal{I} \setminus \mathcal{I}^=}$ consists of disjoint sets, Requirement (35) is satisfied: take $J_i = \{1\}$, $a_i^1 = \min(s_{\mu_i-\eta_i}, s_{\nu_i-\eta_i})$ and $b_i = \max(S_{\mu_i-\eta_i}, S_{\nu_i-\eta_i})$. In particular, this shows that the second requirement in Definition 2.43 implies Requirement (37). Thus, we have $\mathcal{A}^* \subset \mathcal{A}$.

2. We now present two equivalent formulations of Requirement (35). The first, offered as an intuitive formulation, states as follows: Equation (35) holds if and only if there exists a family $(J_i)_{i \in \mathcal{I} \setminus \mathcal{I}^=}$ and families $(A_i^j)_{j \in J_i} \in \mathcal{S}Q^{J_i}$ of elements of $\mathcal{S}Q$ such that, for all $i \in \mathcal{I} \setminus \mathcal{I}^=$, elements of $\mathcal{C}(\mu_i - \eta_i, \nu_i - \eta_i)$ are concentrated on $\bigcup_{j \in J_i} A_i^j$, and the following condition holds:

$$ \forall (i_1, i_2) \in (\mathcal{I} \setminus \mathcal{I}^=)^2, \forall (j_1, j_2) \in J_{i_1} \times J_{i_2} : (i_1, j_1) \neq (i_2, j_2) \implies (A_{i_2}^{j_2} \le \mathbb{R}^2 A_{i_1}^{j_1} \text{ or } A_{i_1}^{j_1} \le \mathbb{R}^2 A_{i_2}^{j_2}). \quad (36) $$

Indeed, if Requirement (35) holds just take $A_i^j = [a_i^j, b_i^j]^2$, whereas if Condition (36) holds, just take $a_i^j = \inf(p_1(A_i^j))$ and $b_i^j = \sup(p_1(A_i^j))$. The second formulation will often be used in the remainder of this subsection: Equation (35) is satisfied if and only if $(C_i)_{i \in \mathcal{I} \setminus \mathcal{I}^=}$ forms a family of disjoint classes such that

$$ \bigcup_{i \in \mathcal{I} \setminus \mathcal{I}^=} C_i \in \mathcal{DR}. \quad (37) $$
To see that Condition (37) implies Condition (36), observe that, if $(A_i^j)_{j \in J_i}$ is an injective family satisfying $C_i = \{A_i^j; j \in J_i\}$, then the families $(A_i^j)_{j \in J_i}$ for $i \in \mathcal{I}$ satisfy Condition (36). The establish that Condition (36) implies Condition (37) observe that $(A_i^j)_{i \in J_i} \in \mathfrak{DR}(\mu_i, \nu_i)$ (see Definition B.7). The minimality property of $C_i$ implies that $C_i \preceq \{A_i^j\}_{j \in J_i}$ for every $i \in \mathcal{I} \setminus \mathcal{I}^-$.
As Condition (36) holds, $\bigcup_{j \in J_i} \{A_i^j\}_{j \in J_i} \in \mathfrak{DR}:$ Condition (37) then follows directly. This is a consequence of the fact that, for all $i \in \mathcal{I}$, $C_i \preceq \{A_i^j; j \in J_i\}$. Observe that Equation (37) implies the following:

$$ \forall(i, j) \in (\mathcal{I} \setminus \mathcal{I}^-)^2, \forall(C, C') \in C_i \times C_j : i \neq j \implies C_* \cap C'_* = \emptyset. \quad (38) $$

This follows directly from the fact that, for any pair $(A, B) \in \mathcal{S}Q^2$, $A \leq_{\mathbb{R}^2} B$ implies $A_* \cap B_* = \emptyset$.

3. We already know that $\mathcal{D}_K$ belongs to $\mathcal{A} \subset \mathcal{A}^*$. Similarly $\mathcal{D}_K^* \in \mathcal{A}^*$: Requirement (34) follows from Theorem A.4, while Requirement (37) follows from the equalities $\mathcal{C}(\tilde{\mu}_k^+, \tilde{\nu}_k^+) = \mathcal{C}(\mu_k^+, \nu_k^+) = \{\mathrm{cl}(A_k^+)\}$ and $\mathcal{C}(\tilde{\mu}_k^-, \tilde{\nu}_k^-) = \mathcal{C}(\mu_k^-, \nu_k^-) = \{\mathrm{cl}(A_k^-)\}$. Therefore, we conclude $\mathcal{D}_K^* \in \mathcal{A}^*$.

4. Consider $\mathcal{D} = ((\mu_i, \nu_i))_{i \in \mathcal{I}}$ admissible. From Requirement (34), we have $\mathcal{C}(\mu, \nu) \subset \sum_{i \in \mathcal{I}} \mathcal{C}(\mu_i, \nu_i)$: we now prove that $\sum_{i \in \mathcal{I}} \mathcal{C}(\mu_i, \nu_i) \subset \mathcal{C}(\mu, \nu)$. To establish it, consider a family $(\pi_i)_{i \in \mathcal{I}} \in \prod_{i \in \mathcal{I}} \mathcal{C}(\mu_i, \nu_i)$ and define $\pi = \sum_{i \in \mathcal{I}} \pi_i$. Let $(\Gamma_i)_{i \in \mathcal{I}}$ denote a family of cyclically monotone set such that, for all $i \in \mathcal{I} \setminus \mathcal{I}^-$, $\pi_i$ is concentrated on $\Gamma_i$. For all $i \in \mathcal{I} \setminus \mathcal{I}^-$, $\pi_i - H_i$ is concentrated on $\Gamma_i \cap (\bigcup_{C \in \mathcal{C}_i} C)$. Hence $\pi$ is concentrated on $\Gamma := \mathcal{D} \cup (\bigcup_{i \in \mathcal{I} \setminus \mathcal{I}^-} \bigcup_{C \in \mathcal{C}_i} (\Gamma_i \cap C))$. According to Equation (37), $\cup_{i \in \mathcal{I} \setminus \mathcal{I}^-} C_i$ belongs to $\mathfrak{DR}$. Therefore, $(\Gamma_i \cap C)_{i \in \mathcal{I}, C \in \mathcal{C}_i}$ is an ordered family of cyclically monotone sets. According to Remark 2.32, $\Gamma$ is cyclically monotone. As $\pi$ clearly has marginals $\mu$ and $\nu$, $\pi$ belongs to $\mathcal{C}(\mu, \nu)$, which establishes the inclusion $\sum_{i \in \mathcal{I}} \mathcal{C}(\mu_i, \nu_i) \subset \mathcal{C}(\mu, \nu)$. By Equation (34), we have $\mathcal{C}(\mu, \nu) = \sum_{i \in \mathcal{I}} \mathcal{C}(\mu_i, \nu_i)$.

5. Consider $\mathcal{D} = \{(\mu_i, \nu_i) ; i \in \mathcal{I}\}$ admissible. By Proposition B.9, for all $i \in \mathcal{I} \setminus \mathcal{I}^-$ and $\pi_i \in \mathcal{C}(\mu_i, \nu_i)$, the measure $\pi_i - H_i$ is concentrated on $A_i^*$. According to Equation (38), $(A_i^*)_{i \in \mathcal{I} \setminus \mathcal{I}^-}$ is a family of disjoint subsets. Thus, for all $(\pi_i)_{i \in \mathcal{I} \setminus \mathcal{I}^-} \in \prod_{i \in \mathcal{I} \setminus \mathcal{I}^-} \mathcal{C}(\mu_i, \nu_i)$, the family $(\pi_i - H_i)_{i \in \mathcal{I} \setminus \mathcal{I}^-}$ consists of mutually singular measures. Note, however, that $(\pi_i)_{i \in \mathcal{I} \setminus \mathcal{I}^-}$ is not generally composed of mutually singular measures.

**Lemma C.5.** For all $\mathcal{D} = ((\mu_i, \nu_i))_{i \in \mathcal{I}} \in \mathcal{A}^*$, it holds that $H(\mu, \nu) = \sum_{i \in \mathcal{I}} H_i$ and $\eta(\mu, \nu) = \sum_{i \in \mathcal{I}} \eta_i$.

*Proof.* Consider $\pi \in \mathcal{C}(\mu, \nu)$. According to Equation (34), there exists $(\pi_i)_{i \in \mathcal{I}}$ such that $\pi = \sum_{i \in \mathcal{I}} \pi_i$. Then, $\pi_{\mathcal{I} \setminus \mathcal{I}^-} = \sum_{i \in \mathcal{I}} (\pi_i - H_i)_{\mathcal{I} \setminus \mathcal{I}^-} + \sum_{i \in \mathcal{I}} H_i \geq \sum_{i \in \mathcal{I}} H_i$. Since this inequality holds for all $\pi \in \mathcal{C}(\mu, \nu)$, it follows that $H(\mu, \nu) \geq \sum_{i \in \mathcal{I}} H_i$. To obtain the reverse inequality, we apply Lemma B.4: for every $i \in \mathcal{I}$, there exists $\pi_i \in \mathcal{C}(\mu_i, \nu_i)$ such that $\pi_i_{\mathcal{I} \setminus \mathcal{I}^-} = H_i$. By Point 4 of Remark C.4, $\pi := \sum_{i \in \mathcal{I}} \pi_i$ belongs to $\mathcal{C}(\mu, \nu)$. Therefore, $\sum_{i \in \mathcal{I}} H_i = \sum_{i \in \mathcal{I}} \pi_i_{\mathcal{I} \setminus \mathcal{I}^-} = \pi_{\mathcal{I} \setminus \mathcal{I}^-} \geq H(\mu, \nu)$. Finally, $H(\mu, \nu) = \sum_{i \in \mathcal{I}} H_i$. Applying the first marginal projection yields $\eta(\mu, \nu) = \sum_{i \in \mathcal{I}} \eta_i$. $\square$

**Remark C.6.** Consider $\mathcal{D} = ((\mu_i, \nu_i))_{i \in \mathcal{I}} \in \mathcal{A}^*$.

1. We now prove that the equality $\mathcal{C}(\mu, \nu) = \sum_{i \in \mathcal{I}} \mathcal{C}(\mu_i, \nu)$ can be improved to

$$ \mathcal{C}(\mu, \nu) = \bigoplus_{i \in \mathcal{I}} \mathcal{C}(\mu_i, \nu_i). \quad (39) $$
Assume $\pi \in \mathfrak{C}(\mu, \nu)$ and $(\pi_i)_{i \in \mathcal{I}} \in \prod_{i \in \mathcal{I}} \mathfrak{C}(\mu_i, \nu_i)$ satisfies $\pi = \sum_{i \in \mathcal{I}} \pi_i$. According to Lemma C.5, we have $\pi - H(\mu, \nu) = \sum_{i \in \mathcal{I}} (\pi_i - H_i) = \sum_{i \in \mathcal{I} \setminus \mathcal{I}^c} (\pi_i - H_i)$. For all $i \in \mathcal{I} \setminus \mathcal{I}^c$, restricting the previous equality to $A_i^*$ and applying Point 5 of Remark C.4, we obtain

$$ (\pi - H(\mu, \nu))_{\mathcal{I}^c} = \sum_{j \in \mathcal{I} \setminus \mathcal{I}^c} (\pi_j - H_j)_{\mathcal{I}^c} = \pi_i - H_i. $$

Hence, $\pi_i = (\pi - H(\mu, \nu))_{\mathcal{I}^c} + H_i$. If $i \in \mathcal{I}^c$, $\pi_i = H_i$. Thus, uniqueness holds, and the conclusion of Point 4 of Remark C.4 yields the identity $\mathfrak{C}(\mu, \nu) = \bigoplus_{i \in \mathcal{I}^c} \mathfrak{C}(\mu_i, \nu_i)$.

2. We have $\biguplus_{i \in \mathcal{I} \setminus \mathcal{I}^c} C_i \in \mathfrak{DR}(\mu, \nu)$. Since for all $i \in \mathcal{I} \setminus \mathcal{I}^c$, $C_i \subset SQ$, it follows that $\biguplus_{i \in \mathcal{I} \setminus \mathcal{I}^c} C_i \subset SQ$. By Equation (37), $\biguplus_{i \in \mathcal{I} \setminus \mathcal{I}^c} C_i \in \mathfrak{DR}$. Now, for all $\pi \in \mathfrak{C}(\mu, \nu)$, there exists $(\pi_i)_{i \in \mathcal{I}} \in \prod_{i \in \mathcal{I}} \mathfrak{C}(\mu_i, \nu_i)$ such that $\pi = \sum_{i \in \mathcal{I}} \pi_i$. Then $\pi - H(\mu, \nu) = \sum_{i \in \mathcal{I}} (\pi_i - H_i)$, and for all $j \in \mathcal{I}$, $\pi_j - H_j \in \mathfrak{C}(\mu_j - \eta_j, \nu_j - \eta_j)$ is concentrated on $\bigcup_{C \in \mathcal{C}_j} C \subset \bigcup_{C \in \bigcup_{i \in \mathcal{I} \setminus \mathcal{I}^c} C_i} C$. Therefore, $\pi - H(\mu, \nu)$ is concentrated on $\bigcup_{C \in \biguplus_{i \in \mathcal{I} \setminus \mathcal{I}^c} C_i} C$. Thus, we obtain $\bigcup_{i \in \mathcal{I} \setminus \mathcal{I}^c} C_i \in \mathfrak{DR}(\mu, \nu)$.

We now define a relation $\preceq_A^*$ on $\mathcal{A}^*$ to compare different admissible decompositions. This definition, formulated purely in terms of measures, admits a more intuitive counterpart at the level of cyclically monotone transport plans (see Proposition C.8)

**Definition C.7 (Order on decompositions).** Let $\mathcal{D}_1 = ((\mu_i^1, \nu_i^1))_{i \in \mathcal{I}_1}$ and $\mathcal{D}_2 = ((\mu_i^2, \nu_i^2))_{i \in \mathcal{I}_2}$ be elements of $\mathcal{A}^*$. We say that $\mathcal{D}_1$ is a finer decomposition than $\mathcal{D}_2$ if the following two conditions hold:

1. There exists a disjoint covering $(J_{i_1})_{i_1 \in \mathcal{I}_1^c}$ of $\mathcal{I}_1^c$ such that:

$$ \forall i_1 \in \mathcal{I}_1^c, \eta_{i_1}^1 \geq \sum_{j_2 \in J_{i_1}} \eta_{j_2}^2. \qquad (40) $$

2. There exists a partition $(J_{i_2})_{i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^c}$ of $\mathcal{I}_1 \setminus \mathcal{I}_2^c$ such that:

$$ \forall i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^c, \left\{ \begin{array}{l} \mu_{i_2}^2 - \eta_{i_2}^2 = \sum_{j_1 \in J_{i_2}} \mu_{j_1}^1 - \eta_{j_1}^1 \\ \nu_{i_2}^2 - \eta_{i_2}^2 = \sum_{j_1 \in J_{i_2}} \nu_{j_1}^1 - \eta_{j_1}^1 \\ \eta_{i_2}^2 \geq \sum_{j_1 \in J_{i_2}} \eta_{j_1}^1 \end{array} \right. . \qquad (41) $$

In this case, we write $\mathcal{D}_1 \preceq_A^* \mathcal{D}_2$.

The following proposition provides a counterpart of Definition C.7 formulated in terms of cyclically monotone transport plans: $\mathcal{D}_1 \preceq_A^* \mathcal{D}_2$ if and only if each non-equal component $\mathfrak{C}(\mu_{i_2}^2, \nu_{i_2}^2)$ for $\mathcal{D}_2$ can be expressed as a sum of non-equal components of $\mathfrak{C}(\mu_{i_1}^1, \nu_{i_1}^1)$ plus a fixed part, while equal components $\mathfrak{C}(\mu_{i_1}^1, \nu_{i_1}^1)$ for $i_1 \in \mathcal{I}_1^c$ can be expressed as the sum of equal components $\mathfrak{C}(\mu_{i_2}^2, \nu_{i_2}^2)$ with $i_2 \in \mathcal{I}_2^c$ plus a fixed part.

**Proposition C.8.** Let $\mathcal{D}_1 = \{(\mu_{i_1}^1, \nu_{i_1}^1)\}_{i_1 \in \mathcal{I}_1}$ and $\mathcal{D}_2 = \{(\mu_{i_2}^2, \nu_{i_2}^2)\}_{i_2 \in \mathcal{I}_2}$ be two admissible decompositions. Then $\mathcal{D}_1 \preceq_A^* \mathcal{D}_2$ if and only if the following two conditions hold:
1. There exists a disjoint covering $(J_{i_1})_{i_1 \in \mathcal{I}_1^=}$ of $\mathcal{I}_2^=$ and a family of measures $(\theta_{i_1}^1)_{i_1 \in \mathcal{I}_1^=}$ such that

$$ \forall i_1 \in \mathcal{I}_1^=, \mathcal{C}(\mu_{i_1}^1, \nu_{i_1}^1) = \bigoplus_{i_2 \in J_{i_1}} \mathcal{C}(\mu_{i_2}^2, \nu_{i_2}^2) \oplus \mathcal{C}(\theta_{i_1}^1, \theta_{i_1}^1). \quad (42) $$

2. There exists a partition $(J_{i_2})_{i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^=}$ of $\mathcal{I}_1 \setminus \mathcal{I}_2^=$, and a family of measures $(\theta_{i_2}^2)_{i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^=}$ such that,

$$ \forall i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^=, \mathcal{C}(\mu_{i_2}^2, \nu_{i_2}^2) = \bigoplus_{i_1 \in J_{i_2}} \mathcal{C}(\mu_{i_1}^1, \nu_{i_1}^1) \oplus \mathcal{C}(\theta_{i_2}^2, \theta_{i_2}^2). \quad (43) $$

*Proof.* Assume that $\mathcal{D}_1 \preceq_{\mathcal{A}^*} \mathcal{D}_2$ and let $(J_{i_1})_{i_1 \in \mathcal{I}_1^=}$ and $(J_{i_2})_{i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^=}$ be as in Definition C.7. First, note that Equation (42) is equivalent to $\eta_{i_1}^1 = \sum_{i_2 \in J_{i_2}} \eta_{i_2}^2 + \theta_{i_1}^1$. Therefore Equation (42) follows directly from Requirement (40). To prove Equation (43), define

$$ E_{i_2} = \begin{cases} \mathcal{C}(\eta_{i_2}^2, \eta_{i_2}^2) \oplus \left( \bigoplus_{i_1 \in J_{i_2}} \mathcal{C}(\mu_{i_1}^1 - \eta_{i_1}^1, \nu_{i_1}^1 - \eta_{i_1}^1) \right) & \text{if } i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^= \\ \mathcal{C}(\mu_{i_2}^2, \nu_{i_2}^2) & \text{otherwise} \end{cases}. \quad (44) $$

By Lemma C.5, we obtain:

$$ \begin{aligned} \mathcal{C}(\mu, \nu) &= \left( \bigoplus_{i_1 \in \mathcal{I}_1} \mathcal{C}(\mu_{i_1}^1, \nu_{i_1}^1) \right) - \left( \bigoplus_{i_1 \in \mathcal{I}_1} \mathcal{C}(\eta_{i_1}^1, \eta_{i_1}^1) \right) + \left( \bigoplus_{i_2 \in \mathcal{I}_2} \mathcal{C}(\eta_{i_2}^2, \eta_{i_2}^2) \right) \\ &= \left( \bigoplus_{i_1 \in \mathcal{I}_1 \setminus \mathcal{I}_2^=} \mathcal{C}(\mu_{i_1}^1 - \eta_{i_1}^1, \nu_{i_1}^1 - \eta_{i_1}^1) \right) \oplus \left( \bigoplus_{i_2 \in \mathcal{I}_2} \mathcal{C}(\eta_{i_2}^2, \eta_{i_2}^2) \right) \\ &= \bigoplus_{i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^=} \left( \mathcal{C}(\eta_{i_2}^2, \eta_{i_2}^2) \oplus \bigoplus_{i_1 \in J_{i_2}} \mathcal{C}(\mu_{i_1}^1 - \eta_{i_1}^1, \nu_{i_1}^1 - \eta_{i_1}^1) \right) \oplus \bigoplus_{i_2 \in \mathcal{I}_2^=} \mathcal{C}(\eta_{i_2}^2, \eta_{i_2}^2) \\ &= \bigoplus_{i_2 \in \mathcal{I}_2} E_{i_2}. \end{aligned} $$

From Equation (41), it follows that $E_{i_2} \subset \mathcal{C}(\mu_{i_2}^2, \nu_{i_2}^2)$. As $\mathcal{C}(\mu, \nu) = \bigoplus_{i_2 \in \mathcal{I}_2} \mathcal{C}(\mu_{i_2}^2, \nu_{i_2}^2)$, it is straightforward that, for all $i_2 \in \mathcal{I}_2$, $E_{i_2} = \mathcal{C}(\mu_{i_2}^2, \nu_{i_2}^2)$. Define, for all $i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^=$, $\theta_{i_2} = \eta_{i_2}^2 - \sum_{i_1 \in J_{i_2}} \eta_{i_1}^1$. Equation (43) then follows from $\mathcal{C}(\mu_{i_2}^2, \nu_{i_2}^2) = E_{i_2}$. The implication from Conditions (42) and (43) to $\mathcal{D}_1 \preceq_{\mathcal{A}^*} \mathcal{D}_2$ is straightforward and left to the reader. $\square$

**Remark C.9.** 1. For any $\mathcal{D} = ((\mu_i, \nu_i))_{i \in \mathcal{I}}$, the measure $\sum_{i \in \mathcal{I}} \eta_i$ represents the information that $\mathcal{D}$ provides about the fixed part of elements of $\mathcal{C}(\mu, \nu)$. Observe that by taking $(J_{i_1})_{i_1 \in \mathcal{I}_1^=}$ as in Equation (40), we obtain:

$$ \sum_{i_1 \in \mathcal{I}_1^=} \eta_{i_1}^1 \ge \sum_{i_1 \in \mathcal{I}_1^=} \sum_{i_2 \in J_{i_1}} \eta_{i_2}^2 = \sum_{i_2 \in \mathcal{I}_2^=} \eta_{i_2}^2. \quad (45) $$

Requirement (40) is strictly stronger than the inequality (45), as it favours decompositions in which the fixed parts are grouped: $(\delta_0, \delta_0)$ is a better decomposition than $(2^{-1}(\delta_0, \delta_0), 2^{-1}(\delta_0, \delta_0))$.
2. The last line of Equation (41) also requires that the fixed part unaccounted for by $\{((\mu_{i_1}^1, \nu_{i_1}^1))\}_{i_1 \in J_{i_2}}$ is less than that of $\{(\mu_{i_2}^2, \nu_{i_2}^2)\}$.

The following definition makes use of the notation $A_k^+$ and $A_k^-$ introduced in Definition 2.26, as well as $A^i$ introduced in Notation C.1.

**Definition C.10.** Let $\mathcal{D} = ((\mu_i, \nu_i))_{i \in \mathcal{I}}$ be an admissible decomposition. For all $i \in \mathcal{I} \setminus \mathcal{I}^-$, define:

$$K_i^+ = \{k \in \mathcal{K}^+; A_k^+ \subset A^i\} \quad \text{and} \quad K_i^- = \{k \in \mathcal{K}^-; A_k^- \subset A^i\}.$$

**Remark C.11.** 1. According to Equation (37), for all $(i, j) \in \mathcal{I} \setminus \mathcal{I}^-$, if $i \neq j$, then $A^i \cap A^j \subset \mathcal{D}$. Therefore, $(K_i^+)$$_{i \in \mathcal{I} \setminus \mathcal{I}^-}$ and $(K_i^-)$$_{i \in \mathcal{I} \setminus \mathcal{I}^-}$ form disjoint family of index sets.

2. As $\bigcup_{i \in \mathcal{I} \setminus \mathcal{I}^-} C_i \in \mathcal{D}\Re(\mu, \nu)$, we have $C(\mu, \nu) \leq \bigcup_{i \in \mathcal{I} \setminus \mathcal{I}^-} C_i$. Thus, for all $k \in \mathcal{K}^+$, there exists $i \in \mathcal{I} \setminus \mathcal{I}^-$ and $C \in C_i$ such that $\text{cl}(A_k^+) \subset C$. Hence, $A_k^* \subset \text{cl}(A_k^+) \subset C \subset A^i$, which yields $k \in K_i^+$. Therefore, $\mathcal{K}^+ \subset \bigcup_{i \in \mathcal{I} \setminus \mathcal{I}^-} K_i^+$. Similarly, $\mathcal{K}^- \subset \bigcup_{i \in \mathcal{I} \setminus \mathcal{I}^-} K_i^-$. By the previous point, we have

$$\left\{ \begin{array}{l} \mathcal{K}^+ = \bigcup_{i \in \mathcal{I} \setminus \mathcal{I}^-} K_i^+ \\ \mathcal{K}^- = \bigcup_{i \in \mathcal{I} \setminus \mathcal{I}^-} K_i^- \end{array} \right. . \qquad (46)$$

**Lemma C.12.** Consider $\mathcal{D} = ((\mu_i, \nu_i))_{i \in \mathcal{I}} \in \mathcal{A}^*$. Then, for all $i \in \mathcal{I}$:

$$\left\{ \begin{array}{l} \mu_i - \eta_i = \sum_{k \in K_i^+} \tilde{\mu}_k^+ + \sum_{k \in K_i^-} \tilde{\mu}_k^- \\ \nu_i - \eta_i = \sum_{k \in K_i^+} \tilde{\nu}_k^+ + \sum_{k \in K_i^-} \tilde{\nu}_k^- \end{array} \right. .$$

*Proof.* Consider $\pi \in \mathcal{C}(\mu, \nu)$, define $\tilde{\varphi}$ and $\tilde{\mathfrak{P}}$ as in Theorem A.4, and let $((\tilde{\pi}_k^+))_{k \in \mathcal{K}^+}, (\tilde{\pi}_k^-))_{k \in \mathcal{K}^-}$, (id, id)$_{\#\eta}$ denote $\tilde{\varphi}^{-1}(\pi) \in \tilde{\mathfrak{P}}$. By Theorem A.4, $\pi - H(\mu, \nu) = \sum_{k \in \mathcal{K}^+} \tilde{\pi}_k^+ + \sum_{k \in \mathcal{K}^-} \tilde{\pi}_k^-$. According to Equation (46), $\pi - H(\mu, \nu) = \sum_{i \in \mathcal{I} \setminus \mathcal{I}^-} (\sum_{k \in K_i^+} \tilde{\pi}_k^+ + \sum_{k \in K_i^-} \tilde{\pi}_k^-)$. Let $(\pi_i)_{i \in \mathcal{I}} \in \prod_{i \in \mathcal{I}} \mathcal{C}(\mu_i, \nu_i)$ be the family satisfying $\pi = \sum_{i \in \mathcal{I}} \pi_i$. By Lemma C.5, $\pi - H(\mu, \nu) = \sum_{i \in \mathcal{I} \setminus \mathcal{I}^-} \pi_i - H_i$. Therefore,

$$\sum_{i \in \mathcal{I} \setminus \mathcal{I}^-} \pi_i - H_i = \sum_{i \in \mathcal{I} \setminus \mathcal{I}^-} \left( \sum_{k \in K_i^+} \tilde{\pi}_k^+ + \sum_{k \in K_i^-} \tilde{\pi}_k^- \right). \qquad (47)$$

Moreover, for all $i \in \mathcal{I} \setminus \mathcal{I}^-$, $\pi_i - H_i$ is concentrated on $A_i^*$ and $\sum_{k \in K_i^+} \tilde{\pi}_k^+ + \sum_{k \in K_i^-} \tilde{\pi}_k^-$ is concentrated on $(\bigcup_{k \in K_i^+} A_k^+) \cup (\bigcup_{k \in K_i^-} A_k^-) \subset A_i^*$. As $(A_i^*)_{i \in \mathcal{I} \setminus \mathcal{I}^-}$ is a family of disjoint set, by restricting Equation (47) to $A_i^*$, we obtain

$$\pi_i - H_i = \sum_{k \in K_i^+} \tilde{\pi}_k^+ + \sum_{k \in K_i^-} \tilde{\pi}_k^-.$$

Projecting onto the first and second marginals yields the desired equalities. $\square$

We now prove that $\leq_{\mathcal{A}^*}$ defines a partial order on $\mathcal{A}^*$. Strictly speaking, this is false. If $\mathcal{D} = ((\mu_i, \nu_i))_{i \in \mathcal{I}} \in \mathcal{A}^*$ and $\phi$ is a permutation of $\mathcal{I}$, then the family $\mathcal{D}'$ defined as $\mathcal{D}' = ((\mu_{\phi(i)}, \nu_{\phi(i)}))_{i \in \mathcal{I}}$ belongs to $\mathcal{A}^*$: we have $\mathcal{D} \leq_{\mathcal{A}^*} \mathcal{D}'$ and $\mathcal{D}' \leq_{\mathcal{A}^*} \mathcal{D}$, whereas $\mathcal{D}$ and $\mathcal{D}'$ are distinct. We overcome this obstacle by introducing an equivalence relation $\sim$ as follows.
**Definition C.13.** 1. Let $E$ be an arbitrary set and $(x_i)_{i \in \mathcal{I}}$, $(y_j)_{j \in \mathcal{J}}$ two families of elements of $E$. We say that $(y_j)_{j \in \mathcal{J}}$ is a reordering of $(x_i)_{i \in \mathcal{I}}$ if there exists a bijection $\phi: \mathcal{I} \to \mathcal{J}$ such that $x_{\phi(i)} = y_i$ for all $i \in \mathcal{I}$. In this case, we write $(x_i)_{i \in \mathcal{I}} \sim (y_j)_{j \in \mathcal{J}}$. The reader may easily verify that $\sim$ is an equivalence relation on families of element of $E$.

2. Let $\overline{\mathcal{A}}^*$ be the quotient set of $\mathcal{A}^*$ by the previously defined relation $\sim$. For all $\mathcal{D} \in \mathcal{A}^*$ we denote $\overline{\mathcal{D}}$ the equivalence class of $\mathcal{D}$ under $\sim$.

3. Given $\overline{\mathcal{D}}_1, \overline{\mathcal{D}}_2 \in \mathcal{A}^*$, we write $\overline{\mathcal{D}}_1 \leq_{\overline{\mathcal{A}}^*} \overline{\mathcal{D}}_2$ if $\mathcal{D}_1 \leq_{\mathcal{A}^*} \mathcal{D}_2$. Observe that this relation is well defined as $\leq_{\mathcal{A}^*}$ is class invariant: if $\mathcal{D}_1 \leq_{\mathcal{A}^*} \mathcal{D}_2$, $\mathcal{D}_1 \sim \mathcal{D}'_1$ and $\mathcal{D}_2 \sim \mathcal{D}'_2$, then $\mathcal{D}'_1 \leq_{\mathcal{A}^*} \mathcal{D}'_2$.

In order to prove that $\leq_{\overline{\mathcal{A}}^*}$ defines a partial order on $\overline{\mathcal{A}}^*$, we need a technical lemma.

**Lemma C.14.** Assume that $(\gamma_i)_{i \in \mathcal{I}} \in (\mathcal{M}_+(\mathbb{R}) \setminus \{0\})^\mathcal{I}$ is such that $\sum_{i \in \mathcal{I}} \gamma_i(\mathbb{R}) < +\infty$ and there exists a partition $(J_i)_{i \in \mathcal{I}}$ of $\mathcal{I}$ satisfying the following condition:

$$ \forall i \in \mathcal{I}, \gamma_i = \sum_{j \in J_i} \gamma_j. \qquad (48) $$

Then, there exists a bijection $\phi: \mathcal{I} \to \mathcal{I}$ such that, for all $i \in \mathcal{I}$, $J_i = \{\phi(i)\}$.

**Proof.** Since $(J_i)_{i \in \mathcal{I}}$ is a partition of $\mathcal{I}$, there exists $\varphi: \mathcal{I} \to \mathcal{I}$ such that, for all $i \in \mathcal{I}$, $i \in J_{\varphi(i)}$. To derive a contradiction, assume there exists $i \in \mathcal{I}$ such that $\gamma_i \neq \gamma_{\varphi(i)}$. Since $\gamma_{\varphi(l)} = \sum_{j \in \varphi(l)} \gamma_j \geq \gamma_l$ for each $l \in \mathcal{I}$, we obtain $+\infty > \sum_{j \in \{\varphi^l(i); l \geq 1\}} \gamma_j(\mathbb{R}) \geq \mathrm{card}(\{\varphi^l(i); l \geq 1\})\gamma_i(\mathbb{R})$. Thus $\mathrm{card}(\{\varphi^l(i); l \geq 1\})\gamma_i(\mathbb{R}) < +\infty$, which implies that $k := \inf(\{l \geq 0; \varphi^{l+1}(i) \in \{i, \dots, \varphi^l(i)\}\})$ is finite. Let $j \geq 0$ be such that $j \in [0, k-1]$ and $\varphi^{j+1}(i) = \varphi^{k+1}(i)$. Since $\varphi^k(i) \neq \varphi^j(i)$ and $(\varphi^k(i), \varphi^j(i)) \in J_{\varphi^{k+1}(i)} \times J_{\varphi^{j+1}(i)} = J_{\varphi^{j+1}(i)}^2$, we obtain $\gamma_{\varphi^{j+1}(i)} = \sum_{l \in J_{\varphi^{j+1}(i)}} \gamma_l \geq \gamma_{\varphi^j(i)} + \gamma_{\varphi^k(i)} \geq \gamma_{\varphi^i(i)} + \gamma_{\varphi^{j+1}(i)}$. This implies $\gamma_{\varphi^j(i)} = 0$, which is a contradiction. Therefore $\gamma_i = \gamma_{\varphi(i)}$, which implies $\gamma_i = \gamma_i + \sum_{l \in J_{\varphi(i)} \setminus \{i\}} \gamma_l$. Thus $J_{\varphi(i)} \setminus \{i\} = \emptyset$, i.e., $J_{\varphi(i)} = \{i\}$. It is now sufficient to prove that $\varphi$ is a bijection: surjectivity is a consequence from $J_i \neq \emptyset$, while injectivity follows from $J_{\varphi(i)} = \{i\}$. $\square$

**Theorem C.15.** The relation $\leq_{\overline{\mathcal{A}}^*}$ constitutes a partial order on $\overline{\mathcal{A}}^*$. Furthermore, $(\overline{\mathcal{A}}^*, \leq_{\overline{\mathcal{A}}^*})$ admits a minimum element, given by $\overline{\mathcal{D}}_K^* = \{(\tilde{\mu}_k^+, \tilde{\nu}_k^+)\}_{k \in \mathcal{K}^+} \cup \{(\tilde{\mu}_k^-, \tilde{\nu}_k^-)\}_{k \in \mathcal{K}^-} \cup \{(\eta, \eta)\}$.

**Proof.** The reflexivity and transitivity of $\leq_{\mathcal{A}^*}$ follow directly from the definitions and are therefore omitted. Reflexivity and transitivity of $\leq_{\overline{\mathcal{A}}^*}$ directly follows. We now establish that $\leq_{\mathcal{A}^*}$ is antisymmetric. Let $\mathcal{D}_1 = \{(\mu_i^1, \nu_i^1)\}_{i \in \mathcal{I}_1}$ and $\mathcal{D}_2 = \{(\mu_i^2, \nu_i^2)\}_{i \in \mathcal{I}_2}$ be two admissible decompositions such that $\mathcal{D}_1 \leq_{\mathcal{A}^*} \mathcal{D}_2$ and $\mathcal{D}_2 \leq_{\mathcal{A}^*} \mathcal{D}_1$. Let $(J_{i_2}^2)_{i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^-}$ be a partition of $\mathcal{I}_1 \setminus \mathcal{I}_1^-$ and $(J_{i_1}^1)_{i_1 \in \mathcal{I}_1 \setminus \mathcal{I}_1^-}$ a partition of $\mathcal{I}_2 \setminus \mathcal{I}_2^-$ such that, for all $(i_1, i_2) \in (\mathcal{I}_1 \setminus \mathcal{I}_1^-) \times (\mathcal{I}_2 \setminus \mathcal{I}_2^-)$,

$$ \left\{ \begin{array}{l} \mu_{i_2}^2 - \eta_{i_2}^2 = \sum_{j_1 \in J_{i_2}^2} \mu_{j_1}^1 - \eta_{j_1}^1 \\ \nu_{i_2}^2 - \eta_{i_2}^2 = \sum_{j_1 \in J_{i_2}^2} \nu_{j_1}^1 - \eta_{j_1}^1 \\ \eta_{i_2}^2 \ge \sum_{j_1 \in J_{i_2}^2} \eta_{j_1}^1 \end{array} \right. \quad \text{and} \quad \left\{ \begin{array}{l} \mu_{j_1}^1 - \eta_{j_1}^1 = \sum_{j_2 \in J_{j_1}^1} \mu_{j_2}^2 - \eta_{j_2}^2 \\ \nu_{j_1}^1 - \eta_{j_1}^1 = \sum_{j_2 \in J_{j_1}^1} \nu_{j_2}^2 - \eta_{j_2}^2 \\ \eta_{j_1}^1 \ge \sum_{j_2 \in J_{j_1}^1} \eta_{j_2}^2 \end{array} \right. . \qquad (49) $$
Consider $i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^-$ and define $\tilde{J}_{i_2} = \bigsqcup_{j_1 \in J_{i_2}^1} J_{j_1}^1$. By Equation (49),

$$ \left\{ \begin{array}{l} \mu_{i_2}^2 - \eta_{i_2}^2 = \sum_{j_2 \in \tilde{J}_{i_2}} \mu_{j_2}^2 - \eta_{j_2}^2 \\ \nu_{i_2}^2 - \eta_{i_2}^2 = \sum_{j_2 \in \tilde{J}_{i_2}} \nu_{j_2}^2 - \eta_{j_2}^2 \end{array} \right. \qquad (50) $$

and

$$ \eta_{i_2}^2 \ge \sum_{j_1 \in J_{i_2}} \eta_{j_1}^1 \ge \sum_{j_2 \in \tilde{J}_{i_2}} \eta_{j_2}^2. \qquad (51) $$

From Proposition C.8, it follows

$$ \mathcal{C}(\mu_{i_2}^2 - \eta_{i_2}^2, \nu_{i_2}^2 - \eta_{i_2}^2) = \bigoplus_{j_2 \in \tilde{J}_{i_2}} \mathcal{C}(\mu_{j_2}^2 - \eta_{j_2}^2, \nu_{j_2}^2 - \eta_{j_2}^2). $$

By Point 5 of Remark C.4, $\tilde{J}_{i_2} = \{i_2\}$. Since $(J_{j_1}^1)_{j_1 \in J_{i_2}}$ is a family of non-empty disjoint subsets, there exists $\phi(i_2) \in \mathcal{I}_1 \setminus \mathcal{I}_1^-$ such that $J_{i_2}^2 = \{\phi(i_2)\}$ and $J_{\phi(i_2)}^1 = \{i_2\}$. By the two first lines of Equation (49), we obtain

$$ \left\{ \begin{array}{l} \mu_{i_2}^2 - \eta_{i_2}^2 = \mu_{\phi(i_2)}^1 - \eta_{\phi(i_2)}^1 \\ \nu_{i_2}^2 - \eta_{i_2}^2 = \nu_{\phi(i_2)}^1 - \eta_{\phi(i_2)}^1 \end{array} \right. , $$

while Equation (51) gives $\eta_{i_2}^2 \ge \eta_{\phi(i_2)}^1 \ge \eta_{i_2}^2$. Finally, we conclude that $(\mu_{i_2}^2, \nu_{i_2}^2) = (\mu_{\phi(i_2)}^1, \nu_{\phi(i_2)}^1)$. Observe that $\phi$ defines a bijection from $\mathcal{I}_2 \setminus \mathcal{I}_2^-$ to $\mathcal{I}_1 \setminus \mathcal{I}_1^-$: this follows from the fact that $(J_{i_2}^2)_{i_2 \in \mathcal{I}_2^-}$ is a partition of $\mathcal{I}_1 \setminus \mathcal{I}_1^-$ and the equalities $J_{i_2}^2 = \{\phi(i_2)\}$ for $i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^-$.
In particular $\sum_{i_1 \in \mathcal{I}_1 \setminus \mathcal{I}_1^-} \eta_{i_1}^1 = \sum_{i_2 \in \mathcal{I}_2 \setminus \mathcal{I}_2^-} \eta_{i_2}^2$. As $\mathcal{D}_1 \preceq_{A^*} \mathcal{D}_2$ and $\mathcal{D}_2 \preceq_{A^*} \mathcal{D}_1$, there exists a disjoint covering of $(J_{i_1}^1)_{i_1 \in \mathcal{I}_1^-}$ of $\mathcal{I}_2^-$ and a disjoint covering $(J_{i_2}^2)_{i_2 \in \mathcal{I}_2^-}$ of $\mathcal{I}_1 \setminus \mathcal{I}_1^-$ such that,

$$ \left\{ \begin{array}{l} \forall i_1 \in \mathcal{I}_1^-, \eta_{i_1}^1 \ge \sum_{j_2 \in J_{i_1}^1} \eta_{j_2}^2 \\ \forall i_2 \in \mathcal{I}_1^-, \eta_{i_2}^2 \ge \sum_{j_1 \in J_{i_2}^2} \eta_{j_1}^1 \end{array} \right. . $$

Thus

$$ \sum_{i_2 \in \mathcal{I}_2} \left( \eta_{i_2}^2 - \sum_{j_1 \in J_{i_2}^2} \eta_{j_1}^1 \right) = \sum_{i_2 \in \mathcal{I}_2^-} \eta_{i_2}^2 - \sum_{i_2 \in \mathcal{I}_2^-} \sum_{j_1 \in J_{i_2}^2} \eta_{j_1}^1 = \sum_{i_2 \in \mathcal{I}_2^-} \eta_{i_2}^2 - \sum_{i_1 \in \mathcal{I}_1^-} \eta_{i_1}^1 = 0, \qquad (52) $$

which implies

$$ \eta_{i_2}^2 = \sum_{j_1 \in J_{i_2}^2} \eta_{j_1}^1 \qquad (53) $$

for every $i_2 \in \mathcal{I}_2^-$. Similarly, for all $i_1 \in \mathcal{I}_1^-$, $\eta_{i_1}^1 = \sum_{i_2 \in J_{i_1}^1} \eta_{i_2}^2$. From the previous equalities, the sets $J_{i_1}^1$ and $J_{i_2}^2$ are non-empty. Now, define $\tilde{J}_{i_2}^2 = \bigsqcup_{i_1 \in J_{i_2}^2} J_{i_1}^1$. For all $i_2 \in \mathcal{I}_2^-$, $\eta_{i_2}^2 = \sum_{j_2 \in \tilde{J}_{i_2}^2} \eta_{j_2}^2$. Since $(J_{i_1})_{i_1 \in \mathcal{I}_1^-}$ is a partition of $\mathcal{I}_2^-$ and $(J_{i_2}^2)_{i_2 \in \mathcal{I}_2^-}$ is a partition of $\mathcal{I}_1^-$, it directly follows that $(\tilde{J}_{i_2}^2)_{i_2 \in \mathcal{I}_2^-}$ is a partition of $\mathcal{I}_2^-$. According to Lemma C.14, there exists bijection $\psi: \mathcal{I}_2^- \to \mathcal{I}_2^-$ such that $\tilde{J}_{i_2}^2 = \{\psi(i_2)\}$ for all $i_2 \in \mathcal{I}_2^-$. This implies that, for all $i_2 \in \mathcal{I}_2^-$, there exists a unique $\phi(i_2) \in \mathcal{I}_2^-$ such that $J_{i_2}^2 = \{\phi(i_2)\}$. From Equation
(53), it follows that $\eta_{i_2}^2 = \eta_{\phi(i_2)}^1$, i.e., $(\mu_{i_2}^2, \nu_{i_2}^2) = (\mu_{\phi(i_2)}^1, \nu_{\phi(i_2)}^1)$. Observe that $\phi$ is a bijection from $\mathcal{I}_2^=$ to $\mathcal{I}_1^=$: this follows directly from the fact that $(J_{i_2}^2)_{i_2 \in \mathcal{I}_2^=} \text{ is a partition of } \mathcal{I}_1^=$ and the equalities $J_{i_2} = \{\phi(i_2)^2\}$ for $i_2 \in \mathcal{I}_2^=$. We have thus established the existence of a bijection $\phi : (\mathcal{I}_2 \setminus \mathcal{I}_2^=) \cup \mathcal{I}_1^= \to (\mathcal{I}_1 \setminus \mathcal{I}_1^=) \cup \mathcal{I}_1^=$ such that, for all $i_2 \in \mathcal{I}_2$, we have $(\mu_{i_2}^2, \nu_{i_2}^2) = (\mu_{\phi(i_1)}^1, \nu_{\phi(i_1)}^1)$. Therefore $\mathcal{D}_1 \sim \mathcal{D}_2$, and antisymmetry of $\overline{\mathcal{A}}^*$ follows directly.

We now prove that $\overline{\mathcal{D}}_K^*$ is a minimal element of $(\overline{\mathcal{A}}^*, \preceq_{\overline{\mathcal{A}}^*})$, that is, $\mathcal{D}_K^* \le \mathcal{D}$ for every $\mathcal{D} \in \mathcal{A}^*$. By Point 3 of Remark C.4, $\mathcal{D}_K \in \mathcal{A}^*$. By Remark B.6, for all $k \in \mathcal{K}^+$ (resp. $k \in \mathcal{K}^-$), $\eta(\tilde{\mu}_k^+, \tilde{\nu}_k^+) = 0$ (resp. $\eta(\tilde{\mu}_k^-, \tilde{\nu}_k^-) = 0$). Consider $\mathcal{D} = ((\mu_i, \nu_i))_{i \in \mathcal{I}} \in \mathcal{A}^*$. For every $i \in \mathcal{I} \setminus \mathcal{I}^=$, define $J_i = K_i^- \uplus K_i^+$. According to Equation (46), $(J_i)_{i \in \mathcal{I} \setminus \mathcal{I}^=} \text{ is a partition of } \mathcal{K}^- \uplus \mathcal{K}^+$. By Lemma C.12, for all $i \in \mathcal{I} \setminus \mathcal{I}^=$,

$$ \left\{ \begin{array}{l} \mu_i - \eta_i = \sum_{k \in K_i^+} (\tilde{\mu}_k^+ - 0) + \sum_{k \in K_i^-} (\tilde{\mu}_k^- - 0) \\ \nu_i - \eta_i = \sum_{k \in K_i^+} (\tilde{\nu}_k^+ - 0) + \sum_{k \in K_i^-} (\tilde{\nu}_k^- - 0) \end{array} \right. , $$

and

$$ \eta_i \ge \sum_{k \in K_i^+} 0 + \sum_{k \in K_i^-} 0. $$

This establishes Requirement (41). Condition (40) follows directly from Lemma C.5: indeed $\eta = \sum_{i \in \mathcal{I}} \eta_i \ge \sum_{i \in \mathcal{I}^=} \eta_i$. Therefore $\mathcal{D}_K \preceq_{\mathcal{A}^*} \mathcal{D}$, which establishes our claim. $\square$

We are now ready to prove Theorem 2.47.

**Definition C.16.** We define $\mathcal{A}^\Delta$ as the set of elements $\mathcal{D} = ((\mu_i, \nu_i))_{i \in \mathcal{I}} \in \mathcal{A}^*$ such that there exists a family $(J_i)_{i \in \mathcal{I} \setminus \mathcal{I}^=} \text{ and families } ((a_i^j, b_i^j))_{j \in J_i} \in (\mathbb{R} \times \mathbb{R})^{J_i}$ with $a_i^j < b_i^j$ such that the following two conditions are satisfied:

1. For every $i \in \mathcal{I}^=$, $\mu_i$ is concentrated on $\bigcap_{l \in \mathcal{I} \setminus \mathcal{I}^=} \bigcap_{j \in J_i} a_l^j, b_l^j$ [$^c$].

2. For all $i \in \mathcal{I} \setminus \mathcal{I}^=$, $\mu_i$ and $\nu_i$ are concentrated on $\bigcup_{j \in J_i} [a_i^j, b_i^j]$, and:

$$ \forall (i_1, i_2) \in (\mathcal{I} \setminus \mathcal{I}^=)^2, \forall (j_1, j_2) \in J_{i_1} \times J_{i_2} : (i_1, j_1) \ne (i_2, j_2) \implies (b_{i_2}^{j_2} \le a_{i_1}^{j_1} \text{ or } b_{i_1}^{j_1} \le a_{i_2}^{j_2}). \quad (54) $$

**Theorem C.17.** The decomposition $\overline{\mathcal{D}}_K$ is the minimum element of $\overline{\mathcal{A}}^\Delta$ with respect to the order $\preceq_{\mathcal{A}^*}$.

*Proof.* Consider $\mathcal{D} = ((\mu_i, \nu_i))_{i \in \mathcal{I}} \in \mathcal{A}^\Delta$, and let the families $((a_i^j, b_i^j))_{j \in J_i}$ for $i \in \mathcal{I} \setminus \mathcal{I}^=$ be as in Definition C.16. For every $i \in \mathcal{I} \setminus \mathcal{I}^=$, define

$$ B_i = \bigcup_{j \in J_i} ([a_i^j, b_i^j]^2 \setminus \{(a_i^j, a_i^j) (b_i^j, b_i^j)\}). $$

For every $i \in \mathcal{I}$, the measures $\mu_i$ and $\nu_i$ are concentrated on $\bigcup_{j \in J_i} [a_i^j, b_i^j]$: it directly follows that $C_i \preceq ([a_i^j, b_i^j]^2)_{j \in J_i}$. Therefore, for every $k \in K_i^+$, we have $A_k^+ \subset A_i^* \subset B_i$: as $(id, id)_{\#} \eta_k^+$ is concentrated on
$A_k^+$, we obtain $(\mathrm{id}, \mathrm{id})_{\#} \eta_k^+ \llcorner_{B_i} = (\mathrm{id}, \mathrm{id})_{\#} \eta_k^+$. Similarly, for all $k \in \mathcal{K}_i^-$, $(\mathrm{id}, \mathrm{id})_{\#} \eta_k^- \llcorner_{B_i} = (\mathrm{id}, \mathrm{id})_{\#} \eta_k^-$. Therefore,

$$ H(\mu, \nu)_{\llcorner B_i} = \left( \sum_{k \in \mathcal{K}^+} (\mathrm{id}, \mathrm{id})_{\#} \eta_k^+ + \sum_{k \in \mathcal{K}^+} (\mathrm{id}, \mathrm{id})_{\#} \eta_k^+ + (\mathrm{id}, \mathrm{id})_{\#} \mu^- \right)_{\llcorner B_i} \quad (55) \\ \geq \sum_{k \in \mathcal{K}_i^+} (\mathrm{id}, \mathrm{id})_{\#} \eta_k^+ + \sum_{k \in \mathcal{K}_i^-} (\mathrm{id}, \mathrm{id})_{\#} \eta_k^- $$

Moreover, for every $l \in \mathcal{I} \setminus \mathcal{I}^=$ with $l \neq i$, $H_l$ is concentrated on $\cup_{j \in B_l} [a_l^j, b_l^j] \subset B_i^c$. For every $l \in \mathcal{I}^=$, $H_l$ is concentrated on $\cap_{k \notin \mathcal{I}^=} \cap_{j \in J_k} (]a_k^j, b_k^j]^2)^c \subset \cap_{j \in J_i} (]a_i^j, b_i^j]^2)^c \subset B_i^c$. Therefore, by applying Lemma C.5, we obtain

$$ H_i \geq H_i \llcorner_{B_i} = \sum_{l \in \mathcal{I} \setminus \{i\}} H_l \llcorner_{B_i} + H_i \llcorner_{B_i} = H(\mu, \nu)_{\llcorner B_i} \quad (56) $$

From Equation (55), it follows that $H_i \geq \sum_{k \in K_i^+} (\mathrm{id}, \mathrm{id})_{\#} \eta_k^+ + \sum_{k \in K_i^-} (\mathrm{id}, \mathrm{id})_{\#} \eta_k^-$, i.e.,

$$ \eta_i \geq \sum_{k \in K_i^+} \eta_k^+ + \sum_{k \in K_i^-} \eta_k^- \quad (57) $$

Moreover, according to Lemma C.12, we have

$$ \left\{ \begin{array}{l} \mu_i - \eta_i = \sum_{k \in K_i^+} (\mu_k^+ - \eta_k^+) + \sum_{k \in K_i^-} (\mu_k^- - \eta_k^-) \\ \nu_i - \eta_i = \sum_{k \in K_i^+} (\nu_k^+ - \eta_k^+) + \sum_{k \in K_i^-} (\nu_k^- - \eta_k^-) \end{array} \right. . $$

Combined with Equation (57), this establishes Condition (40). Moreover,

$$ \sum_{i \in \mathcal{I}^=} \eta_i = \eta - \sum_{i \in \mathcal{I} \setminus \mathcal{I}^=} \eta_i \leq \eta - \sum_{i \in \mathcal{I} \setminus \mathcal{I}^=} \left( \sum_{k \in K_i^+} \eta_k^+ + \sum_{k \in K_i^-} \eta_k^- \right) = \eta - \sum_{k \in \mathcal{K}^+} \eta_k^+ - \sum_{k \in \mathcal{K}^-} \eta_k^- = \mu^=, $$

which establishes Condition (40). Thus $\mathcal{D}_K \preceq_{A^*} \mathcal{D}$, thereby establishing the claim. $\square$

**Remark C.18.** Observe that $\overline{\mathcal{A}} \subset \overline{\mathcal{A}^\Delta} \subset \overline{\mathcal{A}}^*$. Since $\overline{\mathcal{D}}_K \in \overline{\mathcal{A}}$ is the minimum of $\overline{\mathcal{A}^\Delta}$ with respect to $\preceq_{A^*}$, the proof of Theorem 2.47 boils down to the following condition:

$$ \forall (\mathcal{D}_1, \mathcal{D}_2) \in \mathcal{A}^2 : \mathcal{D}_1 \preceq_{A^*} \mathcal{D}_2 \implies \mathcal{D}_1 \preceq_{\mathcal{A}} \mathcal{D}_2. $$

This condition follows directly from Proposition C.8.

**Acknowledgment.** I would like to express my deep gratitude to Nicolas Juillet for introducing me to this research problem and for his constant support and valuable suggestions throughout the writing process.

**Copyright notice:** This research was funded, in whole or in part, by the Agence nationale de la recherche (ANR), Grant ANR-23-CE40-0017. A CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission, in accordance with the grant's open access conditions.
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