*Article* **Construction of Fuzzy Measures over Product Spaces**

### **Fernando Reche, María Morales \* and Antonio Salmerón**

Department of Mathematics and Center for the Development and Transfer of Mathematical Research to Industry (CDTIME), University of Almería, 04120 Almería, Spain; fernando.reche@ual.es (F.R.); antonio.salmeron@ual.es (A.S.)

**\*** Correspondence: maria.morales@ual.es

Received: 26 August 2020; Accepted: 15 September 2020; Published: 17 September 2020

**Abstract:** In this paper, we study the problem of constructing a fuzzy measure over a product space when fuzzy measures over the marginal spaces are available. We propose a definition of independence of fuzzy measures and introduce different ways of constructing product measures, analyzing their properties. We derive bounds for the measure on the product space and show that it is possible to construct a single product measure when the marginal measures are capacities of order 2. We also study the combination of real functions over the marginal spaces in order to produce a joint function over the product space, compatible with the concept of marginalization, paving the way for the definition of statistical indices based on fuzzy measures.

**Keywords:** fuzzy measures; monotone measures; product spaces

### **1. Introduction**

Fuzzy measures, also known as capacities [1], non-additive measures, or monotone measures [2] emerged as an extension of classic probabilistic measure theory by relaxing the additivity property. Fuzzy measures started to receive significant interest from the scientific community due to Choquet's work on capacities [1], but it was Sugeno [3] who first used the term fuzzy measure in relation to non-additive measures on finite domains.

Examples of fuzzy measures can also be found in contexts related to probability theory, dating back to the works by Dempster [4] and Shafer [5], who studied the substitution of the additivity property by superadditivity and subadditivity, resulting in the so-called belief and plausibility measures, respectively, and showed that they can be regarded as probability intervals.

In this paper, we are interested in the problem of constructing a fuzzy measure over a product space when fuzzy measures over the marginal spaces are available. From a practical point of view, this can be regarded as the extension of particular information, given by the fuzzy measures on the marginal spaces, to a more general setting, determined by the product space. We also study the combination of real functions defined on the marginal spaces, in order to obtain a function over the product space coherent with the initial functions.

The problem of composing fuzzy measures has received remarkable attention in the last years, but most of the works consider measures defined over the same space, and typically over the same *σ*-algebra [6]. With the motivation of reducing the number of parameters involved in the definition of a fuzzy measure, a variety of particular types of fuzzy measures have been studied, like *m*-separable fuzzy measures [7], that take advantage of the structure of the space where the measure is defined in order to obtain a compact representation. *k*-maxitive fuzzy measures [8] have been recently proposed as a way of encoding the interactions between the subsets of the reference set. Previously, the internal structure of fuzzy measures showing partial additivity was studied in [9]. The combination of the elements in the decomposition is carried out using different aggregation measures, including copulas [10].

The problem of combining fuzzy measures from marginal spaces in order to obtain a fuzzy measure over a product space has been approached from different perspectives, fundamentally based on the concept of conditioning [11–18]. Recently, the problem has been studied within the context of game theory as a way of representing coalitions between agents [19]. The measure over the product space is obtained by combining a fuzzy measure over a given *σ*-algebra A with a Lebesgue measure over the Borel *σ*-algebra on the interval [0, 1].

In this paper, we consider a more general setting, in which the measures to be combined are general fuzzy measures over potentially different spaces. The rest of the paper is organized as follows. Section 2 is devoted to give the necessary basic definitions and preliminaries. The original contributions in this paper are presented in Sections 3 and 4, covering, respectively, the combination of fuzzy measures and real functions. The paper ends with conclusions in Section 5.

#### **2. Preliminaries**

**Definition 1.** *Let X be a set and* A *be a non-empty class of subsets of X so that X* ⊂ A *and* ∅ ⊂ A*. We say that μ* : A −→ [0, 1] *is a fuzzy measure if the following conditions hold.*


$$\lim\_{n} \mu(A\_n) = \mu\left(\bigcup\_{n=1}^{\infty} A\_n\right). \tag{1}$$

$$\text{15.} \quad \text{If } \{A\_n\}\_{n \in \mathbb{N}} \in \mathcal{A} \text{ such that } A\_1 \supseteq A\_2 \supseteq \dots \text{ and } \bigcap\_{n=1}^{\infty} A\_n \in \mathcal{A} \text{, then}$$

$$\lim\_{n} \mu(A\_{\mathbb{N}}) = \mu\left(\bigcap\_{n=1}^{\infty} A\_{\mathbb{N}}\right). \tag{2}$$

The triplet (*X*, A, *μ*) is a *measurable space*, and *X* is called the *reference set*.

In this paper, we will only consider finite spaces that are sufficient to cover a wide range of applications domains [20]. Note that, in this case, the continuity conditions in Equations (1) and (2) always hold. Furthermore, we will also assume that A is the power set of *X*, i.e., the set of all subsets of *X*. In order to simplify the notation, from now on we will write *μ<sup>i</sup>* for *μ*({*xi*}), and *μ<sup>A</sup>* for *μ*(*A*).

It can be shown [21] that a fuzzy measure over a reference set of cardinality *n* is equivalent to *n*! probability functions, each one of them associated with one possible permutation of the elements in the reference set. We will denote by *X<sup>σ</sup>* the ordering of the elements of *X* according to permutation *σ*, so that *<sup>X</sup><sup>σ</sup>* <sup>=</sup> {*x*(1),..., *<sup>x</sup>*(*n*)}.

**Definition 2.** *[21] Let* (*X*, <sup>A</sup>, *<sup>μ</sup>*) *be a measurable space. The probability functions associated with <sup>μ</sup> and <sup>X</sup><sup>σ</sup> are defined as the set P<sup>σ</sup>* = {*pσ*(*x*(1)),..., *pσ*(*x*(*n*))} *such that*

$$p\_{\sigma}(\mathbf{x}\_{(i)}) = \begin{cases} \mu(A\_{(i)}) - \mu(A\_{(i+1)}) & \text{if } i < n\_{\prime} \\ \mu(\mathbf{x}\_{(n)}) & \text{if } i = n\_{\prime} \end{cases} \tag{3}$$

*where A*(*i*) = {*x*(*i*),..., *x*(*n*)}*.*

It is straightforward to show that 0 <sup>≤</sup> *<sup>p</sup>σ*(*xi*) <sup>≤</sup> 1 and <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *pσ*(*xi*) = 1 for any *σ*. **Definition 3.** *[21] Let* (*X*, A, *μ*) *be a measurable space and let P<sup>σ</sup> be the probability function associated with μ and Xσ. The probability measure generated by μ and X<sup>σ</sup> is*

$$P\_{\mathcal{T}}(A) = \sum\_{\mathbf{x} \in A} p\_{\mathcal{T}}(\mathbf{x}). \tag{4}$$

When it is clear from the context, we will use *P<sup>σ</sup>* for both the probability function and probability measure.

Note that there are as many probability measures generated by *μ* as there are possible permutations of the element of the reference set. However, not all those measures are necessarily different.

An important property of the generated probability measure is that it bounds the underlying fuzzy measure in the extreme cases, as stated in the following proposition [22].

**Proposition 1.** *Let* (*X*, A, *μ*) *be a measurable space and let* {*Pσ*}*σ*∈*Sn be the set of all the probability measures generated by μ. Then,*

$$\min\_{\sigma} P\_{\sigma}(A) \le \mu(A) \le \max\_{\sigma} P\_{\sigma}(A). \tag{5}$$

Two types of fuzzy measures that are specially relevant for this paper are belief functions [4,5] and capacities of order 2 [1,23].

**Definition 4.** *Let* (*X*, A, *μ*) *be a measurable space. Given a function m* : P(*X*) → [0, 1] *such that m*(∅) = 0 *and* <sup>∑</sup>*A*∈P(*X*) *<sup>m</sup>*(*A*) = <sup>1</sup>*, we say that <sup>μ</sup> is a belief function if*

$$\mu(A) = \sum\_{B \in \mathcal{P}(X) \mid B \subseteq A} m(B), \quad \forall A \in \mathcal{P}(X), \tag{6}$$

*where* P(*X*) *denotes the power set of X.*

From now on, if *μ* is a belief function we will denote *μ*(*A*) as Bel(*A*).

**Definition 5.** *Let* (*X*, A, *μ*) *be a measurable space. We say that μ is a monotone capacity of order 2 if*

$$
\mu(A \cup B) + \mu(A \cap B) \ge \mu(A) + \mu(B), \quad \forall A, B \in \mathcal{A}.\tag{7}
$$

**Definition 6.** *Let* (*X*, A, *μ*) *be a measurable space. We say that μ is an alternating capacity of order 2 if*

$$
\mu(A \cup B) + \mu(A \cap B) \le \mu(A) + \mu(B), \quad \forall A, B \in \mathcal{A}.\tag{8}
$$

The following result [22] links capacities of order 2 with the probability measures generated by the fuzzy measure and will be key in the proof of one of the results in this paper.

**Theorem 1.** *A fuzzy measure μ is a monotone (alternating) capacity of order 2 if and only if*

$$\mu(A) = \min\_{\sigma} P\_{\sigma}(A) \quad \left(\mu(A) = \max\_{\sigma} P\_{\sigma}(A)\right) \tag{9}$$

*for all A* ∈ A*, being* {*Pσ*}*σ*∈*Sn the set of probability measures generated by μ (see Definition 3).*

We will consider two possible scenarios related to a given product space *X*<sup>1</sup> × *X*2, where × denotes the Cartesian product:

• A fuzzy measure defined over the product space is available.

• Two fuzzy measures, respectively, defined over *X*<sup>1</sup> and *X*<sup>2</sup> are available, but no measure over *X*<sup>1</sup> × *X*<sup>2</sup> is known.

In the first scenario, we are interested in particularizing the information contained in the fuzzy measure over the unidimensional spaces. Therefore, we need to define a marginalization operation over the measure on the product space.

In the second scenario, we focus on building a fuzzy measure over the product space, by combining the two measures over the marginal spaces. Thus, we need to define an appropriate way of combining fuzzy measures.

Likewise marginal spaces, in a measurable product space we will assume the product class A*X*1×*X*<sup>2</sup> to be the power set of *X*<sup>1</sup> × *X*2, i.e., A*X*1×*X*<sup>2</sup> = P(*X*<sup>1</sup> × *X*2), which is not the same as P(*X*1) × P(*X*2).

Among the possible elements of a product class, we are particularly interested in those that can be obtained from sets in the marginal space. They are called *rectangles* and are formally defined as follows.

**Definition 7.** *Let* (*X*1, A*X*<sup>1</sup> ) *and* (*X*2, A*X*<sup>2</sup> ) *be two spaces where* A*X*<sup>1</sup> *and* A*X*<sup>2</sup> *are classes defined on X*<sup>1</sup> *and X*2*, respectively. We define the class of rectangles of* A*X*1×*X*<sup>2</sup> *as*

$$\mathcal{R} = \{ H \in \mathcal{A}\_{X\_1 \times X\_2} \mid H = A \times B, \text{ where } A \in \mathcal{A}\_{X\_1}, B \in \mathcal{A}\_{X\_2} \}. \tag{10}$$

Taking into account that we are assuming A*X*1×*X*<sup>2</sup> = P(*X*<sup>1</sup> × *X*2), it is easy to show that R is closed for intersections, but not for unions.

We will make use of the concept of triangular norm and conorm. Both are operators that raised within the context of probabilistic metric spaces [24]. They have also been widely used by the theory of fuzzy sets [25–29] as an extension of classic operations over sets.

**Definition 8** ([24])**.** *An operator T* : [0, 1] <sup>2</sup> −→ [0, 1] *is a triangular norm or t-norm for short, if it satisfies the following conditions.*


**Example 1.** *Some examples of t-norms are*


*Note that any t-norm T is always bounded by T*<sup>0</sup> *and T*<sup>3</sup> *in the following way.*

$$T\_0(\mathbf{x}\_1, \mathbf{x}\_2) \le T(\mathbf{x}\_1, \mathbf{x}\_2) \le T\_3(\mathbf{x}\_1, \mathbf{x}\_2). \tag{11}$$

**Definition 9** ([24])**.** *An operator T* : [0, 1] <sup>2</sup> −→ [0, 1] *is a triangular conorm or t-conorm for short, if it satisfies the following properties.*


Given any t-norm *T*, a t-conorm *S* can always be constructed as

$$S(a,b) = 1 - T(1-a, 1-b). \tag{12}$$

**Example 2.** *Applying Equation* (12) *to the t-norms in Example 1, we obtain the following t-conorms.*

$$\begin{aligned} \text{1. } \quad \mathcal{S}\_0(\mathbf{x}\_1, \mathbf{x}\_2) = \begin{cases} \max\{\mathbf{x}\_1, \mathbf{x}\_2\} & \text{if } \min\{\mathbf{x}\_1, \mathbf{x}\_2\} = 0, \\ 1 & \text{otherwise.} \end{cases} \end{aligned}$$


A similar boundary condition as expressed in Equation (11) for t-norms, holds for t-conorms:

$$\mathcal{S}\_3(\mathbf{x}\_1, \mathbf{x}\_2) \le \mathcal{S}(\mathbf{x}\_1, \mathbf{x}\_2) \le \mathcal{S}\_0(\mathbf{x}\_1, \mathbf{x}\_2). \tag{13}$$

A thorough study of the use of t-norms and t-conorms in the context of fuzzy measures and fuzzy sets can be found in [28].

Functions can be integrated with respect to a fuzzy measure using Choquet integral, which is a generalization of Lebesgue integral to non-additive monotone measures [1]. In the particular case of additive measures, Choquet and Lebesgue integrals coincide. It is formally defined as follows.

**Definition 10.** *Let* (*X*, A, *μ*) *be a measurable space, and let h be a measurable real function of X. The Choquet integral of h with respect to μ is*

$$\oint\_{A} h \circ \mu = \int\_{-\infty}^{0} \left( \mu(H\_{\mathfrak{A}} \cap A) - 1 \right) da + \int\_{0}^{\infty} \mu(H\_{\mathfrak{A}} \cap A) \, da \tag{14}$$

*where A* ∈ A *and H<sup>α</sup> are the α*-cuts *of h, defined as*

$$H\_{\mathfrak{a}} = \{ \mathbf{x} \in \mathcal{X} / h(\mathbf{x}) \ge a \}. \tag{15}$$

If the reference set is finite, the integral can be expressed as

$$\oint h \circ \mu = h(\mathbf{x}\_{(1)})\mu(A\_{(1)}) + \sum\_{i=2}^{n} \mu(A\_{(i)}) [h(\mathbf{x}\_{(i)}) - h(\mathbf{x}\_{(i-1)})],\tag{16}$$

where *<sup>X</sup><sup>σ</sup>* is an ordering such that *<sup>h</sup>*(*x*(1)) <sup>≤</sup> *<sup>h</sup>*(*x*(2)) <sup>≤</sup> ... <sup>≤</sup> *<sup>h</sup>*(*x*(*n*)) and the sets *<sup>A</sup>*(*i*) are of the form {*x*(*i*), *x*(*i*+1),..., *x*(*n*)}.

#### **3. Combining Fuzzy Measures**

The main difficulty when combining fuzzy measures from marginal spaces in order to obtain a fuzzy measure over a product space is that, unlike probability measures, we cannot follow a procedure based on extending the measures, as additivity is required [11–18].

For instance, for sets of the form *A* × *B*, where *A* ∈ A*X*<sup>1</sup> and *B* ∈ A*X*<sup>2</sup> , we could define *μ*(*A* × *B*) = *μX*<sup>1</sup> (*A*) ⊗ *μX*<sup>2</sup> (*B*) for some appropriate operator ⊗. In the case of probability measures, this would suffice as, due to additivity, the measure can easily be extended to arbitrary sets of *X*<sup>1</sup> × *X*<sup>2</sup> using integrals [30]. More precisely, in the case of additive measures, the product measure for sets of the form *A* × *B*, is given by *μX*<sup>1</sup> (*A*)*μX*<sup>2</sup> (*B*), while for the rest of sets *Q* ⊆ *X*<sup>1</sup> × *X*2, the product measure is computed as

$$
\mu(Q) = \int \mu\_{X\_2}(Q\_{\mathbb{X}\_1}) \mu\_{X\_1}(d\mathbf{x}\_1) = \int \mu\_{X\_1}(Q\_{\mathbb{X}\_2}) \mu\_{X\_2}(d\mathbf{x}\_2) \tag{17}
$$

where *Qx*<sup>1</sup> = {*x*<sup>2</sup> ∈ *X*2| (*x*1, *x*2) ∈ *Q*} and *Qx*<sup>2</sup> = {*x*<sup>1</sup> ∈ *X*1| (*x*1, *x*2) ∈ *Q*}.

The same construction is not always possible In the case of non-additive measures, because the integrals in Equation (17) can be different [14]. It happens, for instance, if we use Choquet integral [1].
