*3.1.* %*-Independent Measures*

Consider two measurable spaces (*X*1, A*X*<sup>1</sup> , *μ*1) and (*X*2, A*X*<sup>2</sup> , *μ*2) . Our goal is to construct a fuzzy measure over the product space in a sensible way. We start off by defining a fuzzy measure over the product space, compatible with the marginal measures:

**Definition 11.** *A product fuzzy measure of μ*<sup>1</sup> *and μ*<sup>2</sup> *is a function μ*<sup>12</sup> : A*X*1×*X*<sup>2</sup> −→ [0, 1] *satisfying*


The next step is to guarantee that the composition of measures using the product in Definition 11 is compatible with the concept of independence. More precisely, assuming independence between two fuzzy measures, their product fuzzy measure should be possible to be obtained using exclusively the two original fuzzy measures. In this work, we will assume that two fuzzy measures are independent if they can be composed resulting in a product fuzzy measure within the class R. This is formally defined through the concept of %-independence.

**Definition 12.** *Let* (*X*1, A*X*<sup>1</sup> , *μ*1) *and* (*X*2, A*X*<sup>2</sup> , *μ*2) *be measurable spaces. We say that μ*<sup>1</sup> *and μ*<sup>2</sup> *are* %*-independent fuzzy measures if there exists a product fuzzy measure μ*% <sup>12</sup> *satisfying that for any H* ∈ R *it holds that*

$$
\mu\_{12}^{\odot}(H) = \mu\_1(A) \odot \mu\_2(B),
\tag{18}
$$

*where H* = *A* × *B and* % *is a t-norm.*

From now on we will refer to this measure as the %*-independent product* of *μ*<sup>1</sup> and *μ*2.

The next proposition shows that the %-independent product results in a well defined fuzzy measure on R.

**Proposition 2.** *Let* (*X*1, A*X*<sup>1</sup> , *μ*1) *and* (*X*2, A*X*<sup>2</sup> , *μ*2) *be measurable spaces. The* %*-independent product of μ*<sup>1</sup> *and μ*2*, μ*% <sup>12</sup>*, is a fuzzy measure on* R*.*

**Proof.** Let *H* ∈ R such that *H* = *A* × *B* with *A* ∈ A*X*<sup>1</sup> and *B* ∈ A*X*<sup>2</sup> . We have to show that the conditions in Definition 11 are satisfied by *μ*% 12.


$$
\mu\_{12}^{\ominus}(H\_1) = \mu\_1(A\_1) \odot \mu\_2(B\_1) \le \mu\_1(A\_2) \odot \mu\_2(B\_2) = \mu\_{12}^{\ominus}(H\_2),
$$

which means that *μ*% <sup>12</sup> is monotone. 4. Given *A* ∈ A*X*<sup>1</sup> , it holds that *A* × *X*<sup>2</sup> ∈ R and therefore

$$
\mu\_{12}^\circ(A \times X\_2) = \mu\_1(A) \odot \mu\_2(X\_2) = \mu\_1(A) \odot 1 = \mu\_1(A).
$$

5. Analogously, we can see that *μ*% <sup>12</sup>(*X*<sup>1</sup> × *B*) = *μ*2(*B*).

**Example 3.** *Let* (*X*1, A*X*<sup>1</sup> , *μ*1) *and* (*X*2, A*X*<sup>2</sup> , *μ*2) *be probabilistic spaces. Therefore, μ*<sup>1</sup> *and μ*<sup>2</sup> *are additive measures and* A*X*<sup>1</sup> *and* A*X*<sup>2</sup> *are algebras. By letting* % *be equal to the product between real numbers, denoted by* ×*, we can define the* ×*-independent product of μ*<sup>1</sup> *and μ*<sup>2</sup> *as*

$$
\mu\_{12}^{\times}(H) = \mu\_1(A) \times \mu\_2(B) \tag{19}
$$

*with H* = *A* × *B* ∈ R*. It is known that, in this case, there exists a unique additive product measure defined over the smaller algebra generated by the corresponding rectangles [30].*

Similar examples can be found for different types of fuzzy measures, as for instance, possibility measures [31,32], illustrating that, given a %-independent fuzzy measure, it is possible to find a unique product measure compatible with the initial ones. However, uniqueness does not hold in general, so we will instead pursue the idea of defining bounds where the product fuzzy measure lies.

**Definition 13.** *Let* (*X*1, A*X*<sup>1</sup> , *μ*1) *and* (*X*2, A*X*<sup>2</sup> , *μ*2) *be measurable spaces. We define the* %*-exterior product measure for any H* ∈ A*X*1×*X*<sup>2</sup> *as*

$$\overline{\mu}\_{12}^{\ominus}(H) = \min\_{A \times B \supseteq H} \mu\_1(A) \odot \mu\_2(B) \tag{20}$$

*where* % *is a t-norm.*

**Definition 14.** *Let* (*X*1, A*X*<sup>1</sup> , *μ*1) *and* (*X*2, A*X*<sup>2</sup> , *μ*2) *be measurable spaces. We define the* %*-interior product measure for any H* ∈ A*X*1×*X*<sup>2</sup> *as*

$$\underline{\mu}\_{12}^{\odot}(H) = \max\_{A \times B \subseteq H} \mu\_1(A) \odot \mu\_2(B) \tag{21}$$

*where* % *is a t-norm.*

These definitions are more general than the ones introduced in [13], where the product t-norm is used instead. Figure 1 shows a representation of the %-interior and exterior product measures corresponding, respectively, to the contained rectangle of larger measure and the containing rectangle of lower measure.

The next results shows that both %-interior and exterior product measures are indeed product fuzzy measures.

**Proposition 3.** *Let* (*X*1, A*X*<sup>1</sup> , *μ*1) *and* (*X*2, A*X*<sup>2</sup> , *μ*2) *be measurable spaces. Then, the* %*-interior (resp. exterior) product measure is a product fuzzy measure of μ*<sup>1</sup> *and μ*2*.*

**Proof.** We will show that *μ*% <sup>12</sup>(*H*) satifies Definition 11. The proof for the %-exterior product measure is analogous. Note that

$$\underline{\mu}\_{12}^{\odot}(\mathcal{Q}) = \max\_{A \times B \subseteq \mathcal{Q}} \mu\_1(A) \odot \mu\_2(B) = 0\_\prime$$

as *A* × *B* ⊆ ∅, and therefore at least one of them will be equal to ∅. We have also used that, as *μ*<sup>1</sup> and *μ*<sup>2</sup> are fuzzy measures and % is a t-norm, the value of the %-interior product measure for the empty set is zero.

The value of the %-interior product measure for *X*<sup>1</sup> × *X*<sup>2</sup> is

$$\begin{aligned} \mu\_{12}^{\ominus}(\mathcal{X}\_1 \times \mathcal{X}\_2) &= \max\_{A \times B \subseteq \mathcal{X}\_1 \times \mathcal{X}\_2} \mu\_1(A) \odot \mu\_2(B) \\ &= \quad \mu\_1(\mathcal{X}\_1) \odot \mu\_2(\mathcal{X}\_2) \\ &= \quad 1 \odot 1 = 1. \end{aligned}$$

Now we will show that monotonicity also holds. Let *H*<sup>1</sup> ⊆ *H*<sup>2</sup> ⊆ *X*<sup>1</sup> × *X*2, then

$$\begin{aligned} \underline{\mu\_{12}^{\ominus}(H\_1)} &= \max\_{A\_1 \times B\_1 \subseteq H\_1} \mu\_1(A\_1) \odot \mu\_2(B\_1) \\ &= \max\_{A\_1 \times B\_1 \subseteq H\_1 \subseteq H\_2} \mu\_1(A\_1) \odot \mu\_2(B\_1) \\ &\le \max\_{A\_2 \times B\_2 \subseteq H\_2} \mu\_1(A\_2) \odot \mu\_2(B\_2) \\ &= \underline{\mu\_{12}^{\ominus}(H\_2)} \underline{\cdot}(H\_2). \end{aligned}$$

That is, as *H*<sup>1</sup> ⊆ *H*<sup>2</sup> and % is a t-norm (and thus a monotone operator), in the worst case, the interior rectangle of larger measure in *H*<sup>2</sup> has, at least, the same measure as the one in *H*1.

Let us check now the compatibility with marginalization. As *A* × *X*<sup>2</sup> and *X*<sup>1</sup> × *B* belong to R, and taking into account that % is a t-norm, it holds that

$$\begin{array}{rcl} \underline{\mu}\_{12}^{\circ \circ}(A \times X\_2) &=& \mu\_1(A) \odot \mu\_2(X\_2) = \mu\_1(A), \\\underline{\mu}\_{12}^{\circ \circ}(X\_1 \times B) &=& \mu\_1(X\_1) \odot \mu\_2(B) = \mu\_2(B). \end{array}$$

**Figure 1.** A representation of internal and external product measures.

Note that, for the particular case of the class R, both measures turn out to be the same, i.e., for all *H* ∈ R, it holds that

$$
\underline{\mu}\_{12}^{\odot}(H) = \mu\_{12}^{\odot}(H) = \overline{\mu}\_{12}^{\odot}(H). \tag{22}
$$

Finally, the next result states how, in the general case, *μ*% <sup>12</sup> and *<sup>μ</sup>*% <sup>12</sup> conform lower and upper bounds, respectively, for any %-independent product fuzzy measure.

**Proposition 4.** *Let* (*X*1, A*X*<sup>1</sup> , *μ*1) *and* (*X*2, A*X*<sup>2</sup> , *μ*2) *be measurable spaces. Given any* %*-independent product of μ*<sup>1</sup> *and μ*2*, it holds that for all C* ∈ A*X*1×*X*<sup>2</sup> *,*

$$
\underline{\mu}\_{12}^{\odot}(\mathbb{C}) \le \mu\_{12}^{\odot}(\mathbb{C}) \le \overline{\mu}\_{12}^{\odot}(\mathbb{C}).\tag{23}
$$

**Proof.** We will only develop the proof for the lower bound. The upper bound case is analogous.

As *μ*% <sup>12</sup> is a monotone measure, for any *A* × *B* ∈ R, with *A* × *B* ⊆ *C* ∈ A*X*1×*X*<sup>2</sup> it holds that *μ*% <sup>12</sup>(*A* × *B*) ≤ *μ*(*C*), and therefore

$$\mu\_{12}^{\triangleleft}(\mathbb{C}) \ge \max\_{A \times B \subseteq \mathbb{C}} \mu\_{12}^{\triangleleft}(A \times B) = \max\_{A \times B \subseteq \mathbb{C}} \mu\_1(A) \odot \mu\_2(B) = \underline{\mu}\_{12}^{\triangleleft}(\mathbb{C}),$$

taking into account that *μ*% <sup>12</sup> satisfies that *μ*% <sup>12</sup>(*A* × *B*) = *μ*1(*A*) % *μ*2(*B*).

Up to this point, we have only been able to provide a way to obtain a combined fuzzy measure when working within the class R. Outside it only bounds have been obtained. In the next section we solve this problem for some particular kinds of fuzzy measures.

#### *3.2. Maximin Product*

We will elaborate on an idea introduced in [17] based on an alternative representation of general fuzzy measures. Given any fuzzy measure, it is always possible to construct classes of measures bounding it. Such is the case of

$$\mathcal{M}\_P(\mu) = \{ P \in \mathfrak{P} \mid P(A) \ge \mu(A) \,\,\,\forall A \in \mathcal{A} \}\tag{24}$$

and

$$\mathcal{M}\_{Bel}(\mu) = \{Bel(A) \in \mathfrak{B} \mid \operatorname{Bel}(A) \le \mu(A) \; \forall A \in \mathcal{A} \}, \tag{25}$$

where P denotes the set of all probability measures and B is the set of all belief measures.

It can be shown [17,18] that any fuzzy measure *μ* can be represented using elements of the classes defined above as

$$\mu(A) = \max\_{\beta \in \mathcal{M}\_{\text{Bel}}(\mu)} \min\_{P \in \mathcal{M}\_P(\beta)} P(A)\_\prime \tag{26}$$

and the Choquet integral of any function *h* with respect to *μ* can also be computed as

$$\oint h \circ \mu = \max\_{\boldsymbol{\beta} \in \mathcal{M}\_{Rel}(\mu)} \min\_{P \in \mathcal{M}\_P(\boldsymbol{\beta})} \int h \, dP. \tag{27}$$

In other words, given any fuzzy measure, there is always a probability measure whose value matches it for a given subset of the reference set.

Taking this representation as a basis, we have the conditions to propose a product fuzzy measure over *X*<sup>1</sup> × *X*<sup>2</sup> as follows.

**Definition 15.** *Let* (*X*1, A*X*<sup>1</sup> , *μ*1) *and* (*X*2, A*X*<sup>2</sup> , *μ*2) *be measurable spaces. We define the maximin product measure as*

$$\mu\_1 \otimes \mu\_2(A) = \max\_{\substack{\beta\_i \in \mathcal{M}\_{\text{Rel}}(\mu\_i) \\ i=1,2}} \min\_{\substack{P\_i \in \mathcal{M}\_P(\beta\_i) \\ i=1,2}} P\_1 \otimes P\_2(A), \tag{28}$$

*where P*<sup>1</sup> ⊗ *P*<sup>2</sup> *is the standard product for probability measures.*

The main drawback of this definition, from a practical point of view, is that it involves solving an optimization problem over very general classes of fuzzy measures. However, we will show that an approximation can be easily obtained taking advantage of the representation of a fuzzy measure by means of a set of probability measures. Remarkably, for some particular types of fuzzy measures, we will also show that, rather than just bounds, we can obtain a precise product fuzzy measure.

**Definition 16.** *Let* (*X*1, <sup>A</sup>*X*<sup>1</sup> , *<sup>μ</sup>*1) *and* (*X*2, <sup>A</sup>*X*<sup>2</sup> , *<sup>μ</sup>*2) *be measurable spaces and <sup>P</sup>μ*<sup>1</sup> *<sup>σ</sup>*<sup>1</sup> *and <sup>P</sup>μ*<sup>2</sup> *<sup>σ</sup>*<sup>2</sup> *be the probability functions associated with Xσ*<sup>1</sup> <sup>1</sup> *and Xσ*<sup>2</sup> <sup>2</sup> *, respectively. We define the lower product p-measure as*

$$\underline{\mathbf{m}}\_{12}(\mathcal{C}) = \min\_{\sigma\_1, \sigma\_2} \left[ P\_{\sigma\_1}^{\mu\_1} \odot P\_{\sigma\_2}^{\mu\_2}(\mathcal{C}) \right],\tag{29}$$

*where* ⊗ *is the standard probabilistic product.*

**Definition 17.** *Given the conditions in Definition 16, we define the upper product p-measure as*

$$\overline{\mathfrak{m}}\_{12}(\mathbb{C}) = \max\_{\sigma\_1, \sigma\_2} \left[ P\_{\sigma\_1}^{\mu\_1} \otimes P\_{\sigma\_2}^{\mu\_2}(\mathbb{C}) \right],\tag{30}$$

*where* ⊗ *is the standard probabilistic product.*

The construction of these measures comprises the following steps, assuming measurable spaces (*X*1, A*X*<sup>1</sup> , *μ*1) and (*X*2, A*X*<sup>2</sup> , *μ*2) with cardinalities *n*<sup>1</sup> and *n*2, respectively.


In fact, there is no need to know the value of the probabilities for all the possible subsets for all the permutations, as the probability measures are additive, and therefore it suffices to know the values for the elementary events (singletones).

**Example 4.** *Consider two measurable spaces,* (*X*1, A*X*<sup>1</sup> , *μ*1) *and* (*X*2, A*X*<sup>2</sup> , *μ*2) *, both with cardinality 3. Table 1 shows an example of a fuzzy measure with all its possible values specified, as well as the associated probability measures corresponding to each permutation of the subsets of the reference set.*

*As both reference sets have cardinality 3, each one of them has three proper sets (i.e., excluding the total and empty sets). Thus, <sup>X</sup>*<sup>1</sup> <sup>×</sup> *<sup>X</sup>*<sup>2</sup> *contains 9 elements and* 29 <sup>−</sup> <sup>2</sup> <sup>=</sup> <sup>510</sup> *proper subsets. For the sake of readability, we will use the notation* (*x*1*i*, *x*2*j*) = *zij, meaning that*

$$X\_1 \times X\_2 = \{z\_{11}, z\_{12}, z\_{13}, z\_{21}, z\_{22}, z\_{23}, z\_{31}, z\_{32}, z\_{33}\} \dots$$

*Let us see how to compute, from the data in Table 1, the product p-measures for some subsets:*

• *Consider the unitary subset z*11*.*

$$\min\_{\pi\_{12}} (\{z\_{11}\}) = \min\_{\sigma\_1, \sigma\_2} P\_{\sigma\_1}^{\mu\_1}(\{x\_{11}\}) P\_{\sigma\_2}^{\mu\_2}(\{x\_{21}\}) = 0.1 \cdot 0.1 = 0.01.$$

*The calculation for unitary subsets is easy, as it requires just to search the permutation with lower value in both components, which in this case are Pμ*<sup>1</sup> (3,1,2) *and Pμ*<sup>2</sup> (2,3,1) *.*

• *Now we will consider a set from class* R*, namely,* {*z*22, *z*23}*,*

$$\begin{split} \min\_{\mathsf{M}12} (\{z\_{22}, z\_{23}\}) &= \min\_{\sigma\_1, \sigma\_2} [P\_{\sigma\_1}^{\mu\_1}(\{x\_{12}\})P\_{\sigma\_2}^{\mu\_2}(\{x\_{22}\}) + P\_{\sigma\_1}^{\mu\_1}(\{x\_{12}\})P\_{\sigma\_2}^{\mu\_2}(\{x\_{23}\})]] \\ &= \min\_{\sigma\_1, \sigma\_2} [P\_{\sigma\_1}^{\mu\_1}(\{x\_{12}\})(P\_{\sigma\_2}^{\mu\_2}(\{x\_{22}\}) + P\_{\sigma\_2}^{\mu\_2}(\{x\_{23}\}))] \\ &= \min\_{\sigma\_1, \sigma\_2} [P\_{\sigma\_1}^{\mu\_1}(\{x\_{12}\})P\_{\sigma\_2}^{\mu\_2}(\{x\_{22}, x\_{23}\})] \\ &= \quad 0.2 \cdot 0.6 = 0.12. \end{split}$$

*The calculations here are analogous to the unitary set case, as it is enough to find the permutations returning the minimum value for the projections. In this case, they are Pμ*<sup>1</sup> (3,2,1) *for* {*x*12} *and Pμ*<sup>2</sup> (1,3,2) *for* {*x*22, *x*23}*.*

• *Consider now a subset outside the class* R*, for instance* {*z*11, *z*22}*. We obtain*

$$\begin{split} \mathfrak{m}\_{12}(\{z\_{11}, z\_{22}\}) &= \min\_{\sigma\_1, \sigma\_2} [P\_{\sigma\_1}^{\mu\_1}(\{x\_{11}\})P\_{\sigma\_2}^{\mu\_2}(\{x\_{21}\}) + P\_{\sigma\_1}^{\mu\_1}(\{x\_{12}\})P\_{\sigma\_2}^{\mu\_2}(\{x\_{22}\})] \\ &= \ 0.3 \cdot 0.1 + 0.2 \cdot 0.6 = 0.15. \end{split}$$

*This case is more complicated, as it requires exploring all the possible combination of products and finding the permutation returning the lowest value. In this case, they are Pμ*<sup>1</sup> (3,2,1) *and Pμ*<sup>2</sup> (2,3,1) *.*


**Table 1.** Probability measures generated by two sample fuzzy measures.

*The calculations are similar for the upper product p-measure, resulting in*

$$\begin{aligned} \overline{\mathfrak{m}}\_{12}(\{z\_{11}\}) &=& 0.12, \\ \overline{\mathfrak{m}}\_{12}(\{z\_{22}, z\_{23}\}) &=& 0.36, \\ \overline{\mathfrak{m}}\_{12}(\{z\_{11}, z\_{22}\}) &=& 0.32. \end{aligned}$$

Note that the product p-measures provide an interval of measure over the product space, but they are obtained in a rather different way than the exterior and interior product measures defined in Equations (20) and (21). We will analyze now some remarkable properties of product p-measures, first of all checking that they are actually fuzzy measures.

**Proposition 5.** m<sup>12</sup> *and* m<sup>12</sup> *are fuzzy measures.*

**Proof.** Consider m12(*C*) = min *σ*1,*σ*<sup>2</sup> *Pμ*1 *<sup>σ</sup>*<sup>1</sup> <sup>⊗</sup> *<sup>P</sup>μ*<sup>2</sup> *<sup>σ</sup>*<sup>2</sup> (*C*). As ⊗ is the standard probabilistic product and we are assuming the reference set to be finite, it holds that there exist two permutations *σ*<sup>1</sup> and *σ*<sup>2</sup> such that

$$\underline{\underline{m}}\_{12}(\mathcal{C}) = \sum\_{(\underline{x}\_{1i},\underline{x}\_{2j}) \in \mathcal{C}} p\_{\sigma\_1}(\underline{x}\_{1i}) p\_{\sigma\_2}(\underline{x}\_{2j}) \dots$$

Given *C* ⊂ *H*, it follows from Definition 16 that there exist two permutations *τ*<sup>1</sup> and *τ*<sup>2</sup> such that

$$\begin{aligned} \underline{m}\_{12}(H) &= \sum\_{(\mathbf{x}\_{1i}, \mathbf{x}\_{2j}) \in H} p\_{\tau\_1}(\mathbf{x}\_{1i}) p\_{\tau\_2}(\mathbf{x}\_{2j}) \\ &= \sum\_{(\mathbf{x}\_{1i}, \mathbf{x}\_{2j}) \in C} p\_{\tau\_1}(\mathbf{x}\_{1i}) p\_{\tau\_2}(\mathbf{x}\_{2j}) + \sum\_{(\mathbf{x}\_{1i}, \mathbf{x}\_{2j}) \in H - C} p\_{\tau\_1}(\mathbf{x}\_{1i}) p\_{\tau\_2}(\mathbf{x}\_{2j}). \end{aligned}$$

As *σ*<sup>1</sup> and *σ*<sup>2</sup> are the permutations that minimize the product probability of *C*, it holds that

$$\begin{aligned} \underline{\mathfrak{m}}\_{12}(H) &\geq \sum\_{(\mathbf{x}\_{\mathbf{i}\mathbf{i}}, \mathbf{x}\_{\mathbf{j}\mathbf{i}}) \in \mathbb{C}} p\_{\sigma\_{1}}(\mathbf{x}\_{1i}) p\_{\sigma\_{2}}(\mathbf{x}\_{2j}) + \sum\_{(\mathbf{x}\_{\mathbf{i}\mathbf{i}}, \mathbf{x}\_{\mathbf{j}\mathbf{i}}) \in H - \mathbb{C}} p\_{\tau\_{1}}(\mathbf{x}\_{1i}) p\_{\tau\_{2}}(\mathbf{x}\_{2j}) \\ &= \underline{\mathfrak{m}}\_{12}(\mathbf{C}) + \sum\_{(\mathbf{x}\_{\mathbf{i}\mathbf{i}}, \mathbf{x}\_{\mathbf{j}\mathbf{i}}) \in H - \mathbb{C}} p\_{\tau\_{1}}(\mathbf{x}\_{1i}) p\_{\tau\_{2}}(\mathbf{x}\_{2j}) \end{aligned}$$

and thus the measure is monotone. The proof for the upper product p-measure is analogous.

However, m<sup>12</sup> and m<sup>12</sup> are not, in general, product fuzzy measures of *μ*<sup>1</sup> and *μ*<sup>2</sup> since they can often fail to be consistent with the marginalization, i.e.

$$\underline{\mathfrak{m}}\_{12}(A \times X\_2) = \min\_{\sigma\_1, \sigma\_2} P\_{\sigma\_1}^{\mu\_1} \odot P\_{\sigma\_2}^{\mu\_2}(A \times X\_2) = \min\_{\sigma\_1} P\_{\sigma\_1}^{\mu\_1}(A),$$

that is not guaranteed to be equal to *μ*1(*A*). The same happens to m12. However, these measures conform a bound of a product measure, as stated in the next proposition.

**Proposition 6.** *Let* (*X*1, A*X*<sup>1</sup> , *μ*1) *and* (*X*2, A*X*<sup>2</sup> , *μ*2) *be measurable spaces and let μ*<sup>×</sup> <sup>12</sup> *be the* ×*-independent product of μ*<sup>1</sup> *and μ*2*. Then, for any H* = *A* × *B* ∈ R *it holds that*

$$
\underline{m}\_{12}(H) \le \mu\_{12}^{\times}(H) \le \overline{m}\_{12}(H). \tag{31}
$$

*Moreover, for any C* ∈ A*X*1×*X*<sup>2</sup> *,*

$$
\underline{\mu}\_{12}^{\times}(\mathbb{C}) \le \overline{\mathfrak{m}}\_{12}(\mathbb{C}),
\tag{32}
$$

$$
\underline{\mathfrak{m}}\_{12}(\mathbb{C}) \le \overline{\mu}\_{12}^{\times}(\mathbb{C}).
\tag{32}
$$

**Proof.** According to Proposition 1,

$$\min\_{\sigma\_1} P\_{\sigma\_1}^{\mu\_1}(A) \le \mu\_1(A) \le \max\_{\sigma\_1} P\_{\sigma\_1}^{\mu\_1}(A).$$

and

$$\min\_{\sigma\_2} P\_{\sigma\_2}^{\mu\_2}(B) \le \mu\_2(B) \le \max\_{\sigma\_2} P\_{\sigma\_2}^{\mu\_2}(B).$$

Multiplying both inequalities, we obtain

$$\min\_{\sigma\_1} P^{\mu\_1}\_{\sigma\_1}(A) \min\_{\sigma\_2} P^{\mu\_2}\_{\sigma\_2}(B) \le \mu\_1(A) \mu\_2(B) \le \max\_{\sigma\_1} P^{\mu\_1}\_{\sigma\_1}(A) \max\_{\sigma\_2} P^{\mu\_2}\_{\sigma\_2}(B),$$

$$\Rightarrow \min\_{\sigma\_1, \sigma\_2} P^{\mu\_1}\_{\sigma\_1}(A) P^{\mu\_2}\_{\sigma\_2}(B) \le \mu\_1(A) \mu\_2(B) \le \max\_{\sigma\_1, \sigma\_2} P^{\mu\_1}\_{\sigma\_1}(A) P^{\mu\_2}\_{\sigma\_2}(B)$$

$$\Rightarrow \underline{\mathfrak{m}}\_{12}(H) \le \mu\_{12}^{\times}(H) \le \overline{\mathfrak{m}}\_{12}(H),$$

as *H* = *A* × *B* and *μ*<sup>×</sup> <sup>12</sup>(*H*) = *μ*1(*A*)*μ*2(*B*). This proves Equation (31).

Now consider two sets *C* ∈ A*X*1×*X*<sup>2</sup> and *G* ∈ R with *G* ⊆ *C*. As m<sup>12</sup> is a fuzzy measure, and thus monotone, it holds that m12(*G*) ≤ m12(*C*), which together with Equation (31) yields

$$
\mu\_{12}^{\times}(G) \le \overline{\mathfrak{m}}\_{12}(G) \le \overline{\mathfrak{m}}\_{12}(\mathbb{C}).\tag{33}
$$

As Equation (33) holds for any *G* ∈ R with *G* ⊆ *C*, in particular we can write

$$\max\_{\substack{G \in \mathcal{R} \\ G \subseteq \mathcal{C}}} \mu\_{12}^{\times}(G) \le \overline{\mathfrak{m}}\_{12}(\mathcal{C}).\tag{34}$$

Note that, according to Definition 14, the left hand side of inequality (34) is the interior product measure constructed with the product t-norm, i.e., *μ*× <sup>12</sup>(*C*), which means that *<sup>μ</sup>*<sup>×</sup> <sup>12</sup>(*C*) <sup>≤</sup> <sup>m</sup>12(*C*). The remaining inequality is proven in a similar way.

The next proposition relates the concept of product p-measure with standard probabilistic product.

**Proposition 7.** *Let* (*X*1, A*X*<sup>1</sup> , *P*1) *and* (*X*2, A*X*<sup>2</sup> , *P*2) *be probabilistic spaces. Then,* m<sup>12</sup> *and* m<sup>12</sup> *are equal to the standard probabilistic product.*

**Proof.** If *P*<sup>1</sup> and *P*<sup>2</sup> are probability measures (and thus fuzzy measures after all), it follows from Definitions 2 and 3 that all the associated probability measures are the same, and therefore m<sup>12</sup> = m<sup>12</sup> and they are equal to the standard probabilistic product.

The next theorem states that, for a particular class of fuzzy measures, constructing a single product measure from two marginal measures is indeed possible.

**Theorem 2.** *Let* (*X*1, A*X*<sup>1</sup> , *μ*1) *and* (*X*2, A*X*<sup>2</sup> , *μ*2) *be measurable spaces such that μ*<sup>1</sup> *and μ*<sup>2</sup> *are monotone (alternating) capacities of order 2, then* m<sup>12</sup> *(*m12*) is a product fuzzy measure of μ*<sup>1</sup> *and μ*2*.*

**Proof.** According to Proposition 5, both m<sup>12</sup> and m<sup>12</sup> are fuzzy measures. We only have to prove that they are consistent with the marginalization, which is straightforward taking into account that, if *μ*<sup>1</sup> and *μ*<sup>2</sup> are capacities of order 2, it follows from Theorem 1 that

> *<sup>μ</sup>i*(*A*) = min*<sup>σ</sup> <sup>P</sup>μ<sup>i</sup> <sup>σ</sup>* (*A*), *i* = 1, 2 (for monotone capacities), *<sup>μ</sup>i*(*A*) = max *<sup>σ</sup> <sup>P</sup>μ<sup>i</sup> <sup>σ</sup>* (*A*), *i* = 1, 2 (for alternating capacities).

Thus, <sup>m</sup>12(*<sup>A</sup>* <sup>×</sup> *<sup>X</sup>*2) = min*<sup>σ</sup> <sup>P</sup>μ*<sup>1</sup> *<sup>σ</sup>* (*A*) = *<sup>μ</sup>*1(*A*) and <sup>m</sup>12(*X*<sup>1</sup> <sup>×</sup> *<sup>B</sup>*) = min*<sup>σ</sup> <sup>P</sup>μ*<sup>2</sup> *<sup>σ</sup>* (*B*) = *μ*2(*B*). The result for alternating capacities is analogously obtained.

**Corollary 1.** *If μ*<sup>1</sup> *and μ*<sup>2</sup> *are monotone (alternating) capacities of order 2 and A* × *B* ∈ R*, then*

$$\begin{aligned} \underline{m}\_{12}(A \times B) &= \mu\_1(A)\mu\_2(B) \qquad \text{(for monotone capacities)},\\ \overline{m}\_{12}(A \times B) &= \mu\_1(A)\mu\_2(B) \qquad \text{(for alternating capacities)}. \end{aligned} \tag{35}$$

These results indicate that the composition of capacities of order 2 can be constructed using the product p-measures, so that if both marginals are monotone capacities of order 2, the product measure is m12, while it is m<sup>12</sup> if the marginals are alternating capacities of order 2.

#### **4. Functions over Product Spaces**

After studying the problem of composing fuzzy measures, the next step is to consider the construction of functions over product spaces, from functions defined on the marginal spaces.

#### *4.1. Composition of Functions*

Our starting point consists of two non-negative functions *h*<sup>1</sup> and *h*<sup>2</sup> defined over the reference sets *X*<sup>1</sup> and *X*2, respectively. We propose the use of t-norms for carrying out the composition of the functions, and therefore we will assume, without loss of generality, that they take values on [0, 1].

**Definition 18.** *Let h*<sup>1</sup> *and h*<sup>2</sup> *be functions defined on X*<sup>1</sup> *and X*2*, respectively, and taking values on interval* [0, 1]*. We say that h*- <sup>12</sup> : *X*<sup>1</sup> × *X*<sup>2</sup> −→ [0, 1] *is the* --composition *of h*<sup>1</sup> *and h*<sup>2</sup> *if* ∀(*x*1, *x*2) ∈ *X*<sup>1</sup> × *X*2*,*

$$h\_{12}^\*(\mathbf{x}\_1, \mathbf{x}\_2) = h\_1(\mathbf{x}\_1) \star h\_2(\mathbf{x}\_2),\tag{36}$$

*where is a t-norm.*

**Example 5.** *Consider the functions h*<sup>1</sup> *and h*<sup>2</sup> *defined on X*<sup>1</sup> *and X*<sup>2</sup> *given in Table 2. Using the t-norm* min*, we find that the* min*-composition of h*<sup>1</sup> *and h*2*, denoted as h*min <sup>12</sup> *, is the function specified in Table 3.*

**Table 2.** An example of two functions defined over the marginal spaces.


**Table 3.** Min-composition of the functions in Table 2.


*Note that h*- <sup>12</sup> *is not symmetric. A possible interpretation of the composed function is the worst value of a pair* (*x*1*i*, *x*2*j*) *regarding both marginal spaces simultaneously. For instance, a value of h*- <sup>12</sup>(*x*12, *x*21) *equal to 0.4 would indicate that both h*1(*x*12) ≥ 0.4 *and h*2(*x*21) ≥ 0.4*.*

**Proposition 8.** *Let h*<sup>1</sup> *and h*<sup>2</sup> *be functions defined on X*<sup>1</sup> *and X*2*, respectively, and taking values on* [0, 1]*. Then, the α-cuts generated by h*min <sup>12</sup> *, belong to the class of rectangles,* R*.*

**Proof.** The *α*-cuts of any function *h* in *X*<sup>1</sup> × *X*<sup>2</sup> are (see Equation (15))

$$H\_{\mathfrak{a}} = \left\{ (\mathbf{x}\_1, \mathbf{x}\_2) \in X\_1 \times X\_2 | h(\mathbf{x}\_1, \mathbf{x}\_2) \ge a \right\}.$$

Considering the min t-norm, and thus *h* = *h*min <sup>12</sup> , we find that

$$\begin{aligned} H\_{\mathbb{A}} &=& \{ (\mathbf{x}\_1, \mathbf{x}\_2) \in X\_1 \times X\_2 | h\_{12}^{\min}(\mathbf{x}\_1, \mathbf{x}\_2) \ge a \} \\ &=& \{ (\mathbf{x}\_1, \mathbf{x}\_2) \in X\_1 \times X\_2 | \min\{h\_1(\mathbf{x}\_1), h(\mathbf{x}\_2)\} \ge a \} \\ &=& \{ (\mathbf{x}\_1, \mathbf{x}\_2) \in X\_1 \times X\_2 | h\_1(\mathbf{x}\_1) \ge a \text{ y } h\_2(\mathbf{x}\_2) \ge a \} \\ &=& \{ \mathbf{x}\_1 \in X\_1 | h\_1(\mathbf{x}\_1) \ge a \} \times \{ \mathbf{x}\_2 \in X\_2 | h\_2(\mathbf{x}\_2) \ge a \} \in \mathcal{R}. \end{aligned}$$

Proposition 8 guarantees that, when composing functions using the min t-norm, the resulting function generates *α*-cuts belonging to class R, which facilitates the calculation of a product measure as the interior and exterior product measures are the same in this case.

#### *4.2. Marginalization*

The marginalization operation acts in the opposite direction to composition, i.e., from a function defined on a product space, by applying marginalization we should obtain functions defined over the marginal spaces. In what follows, we pursue the definition of a well founded marginalization process.

**Definition 19.** *Let h be a function defined on X*<sup>1</sup> × *X*<sup>2</sup> *and taking values on* [0, 1]*. We define the ⊕*-marginals *of h as*

$$h^{\boxplus}\_{X\_1}(\mathbf{x}\_{1i}) \quad = \bigoplus\_{\mathbf{x}\_{2j} \in X\_2} h(\mathbf{x}\_{1i}, \mathbf{x}\_{2j}) = h(\mathbf{x}\_{1i}, \mathbf{x}\_{21}) \oplus h(\mathbf{x}\_{1i}, \mathbf{x}\_{22}) \oplus \dots \oplus h(\mathbf{x}\_{1i}, \mathbf{x}\_{2m}), \tag{37}$$

$$h\_{X\_2}^{\ominus}(\mathbf{x}\_{2\circ}) \quad = \bigoplus\_{\mathbf{x}\_{1\circ} \in X\_1} h(\mathbf{x}\_{1\circ}, \mathbf{x}\_{2\circ}) = h(\mathbf{x}\_{11}, \mathbf{x}\_{2\circ}) \oplus h(\mathbf{x}\_{12}, \mathbf{x}\_{2\circ}) \oplus \dots \oplus h(\mathbf{x}\_{1n}, \mathbf{x}\_{2\circ}), \tag{38}$$

*where* ⊕ *is a t-conorm, n is the cardinality of X*<sup>1</sup> *and m is the cardinality of X*2*.*

For instance, if we consider the max t-conorm, the marginalization process would result in

$$h\_{X\_1}^{\max}(\mathbf{x}\_{1i}) = \max\{h(\mathbf{x}\_{1i}, \mathbf{x}\_{21}), h(\mathbf{x}\_{1i}, \mathbf{x}\_{22}), \dots, h(\mathbf{x}\_{1i}, \mathbf{x}\_{2n})\}.$$

**Example 6.** *Consider a function h defined on X*<sup>1</sup> × *X*<sup>2</sup> *as specified in Table 4. The corresponding marginals when the max t-conorm is used are shown on the last column and row. Table 5 illustrates the opposite process, where the marginals have been combined usind the* min *t-conorm.*

**Table 4.** Specification of a function *h*(*x*1*i*, *x*2*j*) and its marginals using the max t-conorm.


**Table 5.** Composition of the marginals in Table 4 using the min t-conorm.


We can notice how, in Example 6, the function obtained by composing the marginals bounds from above the original function over the product space. The next proposition shows that this property holds in rather general settings.

**Proposition 9.** *Let h be a function defined on X*<sup>1</sup> × *X*<sup>2</sup> *and taking values on* [0, 1]*, and let h*<sup>⊕</sup> *<sup>X</sup>*<sup>1</sup> *and h*<sup>⊕</sup> *<sup>X</sup>*<sup>2</sup> *be the* ⊕*-marginals of h. Then, for any arbitrary t-conorm* ⊕ *it holds that*

$$h(\mathbf{x}\_{1i}, \mathbf{x}\_{2j}) \le \min \{ h^{\ominus}\_{\mathbf{X}\_1}(\mathbf{x}\_{1i}), h^{\ominus}\_{\mathbf{X}\_2}(\mathbf{x}\_{2j}) \}, \quad \forall (\mathbf{x}\_{1i}, \mathbf{x}\_{2j}) \in X\_1 \times X\_2. \tag{39}$$

**Proof.** According to Definition 19,

$$h^{\ominus}\_{\mathbf{X}\_1}(\mathbf{x}\_{1i}) = h(\mathbf{x}\_{1i}, \mathbf{x}\_{21}) \oplus h(\mathbf{x}\_{1i}, \mathbf{x}\_{22}) \oplus \dots \oplus h(\mathbf{x}\_{1i}, \mathbf{x}\_{2m}).\tag{40}$$

As the max t-conorm bounds any other conorm from below,

$$h\_{X\_1}^{\ominus}(\mathbf{x}\_{1i}) \ge \max\{h(\mathbf{x}\_{1i}, \mathbf{x}\_{21}), h(\mathbf{x}\_{1i}, \mathbf{x}\_{22}), \dots, h(\mathbf{x}\_{1i}, \mathbf{x}\_{2m})\} \\ = h\_{X\_1}^{\max}(\mathbf{x}\_{1i}). \tag{41}$$

Likewise, it holds that

$$h\_{X\_2}^{\boxplus}(\mathbf{x}\_{2j}) \ge h\_{X\_2}^{\max}(\mathbf{x}\_{2j}).\tag{42}$$

On the other hand, it is clear that

$$h(\mathbf{x}\_{1i}, \mathbf{x}\_{2j}) \le \min \{ h\_{X\_1}^{\max}(\mathbf{x}\_{1i}), h\_{X\_2}^{\max}(\mathbf{x}\_{2j}) \},\tag{43}$$

as *h*max *<sup>X</sup>*<sup>1</sup> (*x*1*i*) is the maximum value of *<sup>h</sup>*(*x*1*i*, *<sup>x</sup>*2*j*) for a fixed value *<sup>x</sup>*1*<sup>i</sup>* of *<sup>X</sup>*1, and *<sup>h</sup>*max *<sup>X</sup>*<sup>2</sup> (*x*2*j*) is the maximum value of *h*(*x*1*i*, *x*2*j*) for a fixed value *x*2*<sup>j</sup>* of *X*2. Therefore, combining Equation (43) with Equations (41) and (42), we obtain Equation (39).

**Corollary 2.** *Assuming the conditions in Proposition 9, given an arbitrary t-conorm* ⊕ *it holds that, for any* (*x*1*i*, *x*2*j*) ∈ *X*<sup>1</sup> × *X*<sup>2</sup>

$$h(\mathbf{x}\_{1i}, \mathbf{x}\_{2j}) \le h\_{12}^{\min}(\mathbf{x}\_{1i}, \mathbf{x}\_{2j}). \tag{44}$$

#### **5. Conclusions**

In this paper, we have studied the problem of constructing fuzzy measures over product domains, when fuzzy measures over the marginal spaces are available. We have proposed a definition of independence of fuzzy measures and different ways of constructing product measures that are consistent with the defined concept of independence. Even though, in general, we have only been able to give bounds for the measure on the product space when we work outside the class of rectangles R, we show in Theorem 2 that it is possible to construct a single product measure if the marginal measures are capacities of order 2.

Our proposal for combining real functions over the marginal spaces in order to produce a joint function over the product space satisfies that the resulting function yields *α*-cuts within the class of rectangles R, if the min t-norm is used. The importance of this property is that within the class R, we are able to compute a unique product fuzzy measure, as the interior and exterior product measures, which conform the bound of the product fuzzy measure, are the same in this case.

The results in the paper show that we are able to handle marginal spaces endowed with a fuzzy measure and a real function, and work on the product space with product measures and functions containing the information in the marginal case. Likewise, the marginal functions can be measured using, for instance, Choquet integral; the joint function can also be measured in the same way, by integrating with respect to the product measure. This provides the basic tools for defining statistical indices, as for instance indices of association, based on fuzzy measures.

**Author Contributions:** Investigation, F.R., M.M. and A.S.; Writing–original draft, F.R., M.M. and A.S.; Writing–review–editing, F.R., M.M. and A.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Spanish Ministry of Science and Innovation through grant TIN2016- 77902-C3-3-P and by ERDF-FEDER funds.

**Conflicts of Interest:** The authors declare no conflicts of interest.

### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
