*Article* **Inequalities in Triangular Norm-Based** *∗***-Fuzzy (***L***+)***<sup>p</sup>* **Spaces**

#### **Abbas Ghaffari 1, Reza Saadati 2,\* and Radko Mesiar 3,4,\***


Received: 19 October 2020; Accepted: 3 November 2020; Published: 6 November 2020

**Abstract:** In this article, we introduce the <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>* spaces for 1 <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup> on triangular norm-based ∗-fuzzy measure spaces and show that they are complete ∗-fuzzy normed space and investigate some properties in these space. Next, we prove Chebyshev's inequality and Hölder's inequality in <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>* spaces.

**Keywords:** fuzzy measure space; fuzzy integration; t-norm; Chebyshev's inequality; Hölder's inequality

**MSC:** Primary 54C40, 14E20; Secondary 46E25, 20C20

Function spaces, especially *L<sup>p</sup>* spaces, play an important role in many parts in analysis. The impact of *L<sup>p</sup>* spaces follows from the fact that they offer a partial but useful generalization of the fundamental *L*<sup>1</sup> space of integrable functions. The standard analysis, based on sigma-additive measures and Lebesgue–Stieltjess integral, including also several integral inequalities, has been generalized in the past decades into set-valued analysis, including set-valued measures, integrals, and related inequalities. Some subsequent generalizations are based on fuzzy sets [1,2] and include fuzzy measures, fuzzy integrals and several fuzzy integral inequalities. Our aim is the further development of fuzzy set analysis, expanding our original proposal given in [3]. In fact, we use a new model of the fuzzy measure theory (∗-fuzzy measure) which is a dynamic generalization of the classical measure theory. Our model of the fuzzy measure theory created by replacing the non-negative real range and the additivity of classical measures with fuzzy sets and triangular norms. Moreover, the ∗-fuzzy measure theory has been motivated by defining new additivity property using triangular norms. Our approach is related to the idea of fuzzy metric spaces [4–7] and can be apply for decision making problems [8,9].

In this paper, we shall work on a fixed triangular norm-based ∗-fuzzy measure space (*X*, C, *μ*, ∗) introduced in [3] which was derived from the idea of fuzzy and probabilistic metric spaces [5–7,10,11]. Using the concept of fuzzy measurable functions and fuzzy integrable functions we define a special class of function spaces named by <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>*. After some overview given in Sections 2–4 and devoted to the basic information concerning ∗-fuzzy measures and related integration, in Section 5 we define a norm on <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>* spaces and show these spaces are complete <sup>∗</sup>-fuzzy normed space in the sense of Cheng-Mordeson and others [12–15]. This definition of ∗-fuzzy norm helps us to prove Chebyshev's Inequality and Hölder's Inequality.

#### **1.** *∗***–Fuzzy Measure**

First, we recall some basic concepts and notations that will be used throughout the paper. Let *X* be a non-empty set, C be a *σ*-algebra of subsets of *X*. Unless stated otherwise, all subsets of *X* are supposed to belong to C. Here, we let *I* = [0, 1].

**Definition 1.** *([10,11]) A continuous triangular norm (shortly, a ct-norm) is a continuous binary operation* ∗ *from I*<sup>2</sup> = [0, 1] <sup>2</sup> *to I such that*


Some examples of the *ct*-norms are as follows.


```
4.
```

$$
\xi \ast\_H \tau = \begin{cases} \begin{array}{cc} 0, & \text{if } \emptyset = \tau = 0, \\\ \frac{1}{\frac{1}{\xi} + \frac{1}{\tau} - 1}, & \text{otherwise}, \end{array} \end{cases}
$$

(: the Hamacher product *t*-norm).

We define

$$\*\_{i=1}^k \mathfrak{g}\_i = \mathfrak{g}\_1 \* \mathfrak{g}\_2 \* \dots \* \mathfrak{g}\_{k'} $$

for *k* ∈ {2, 3, ···}, which is well defined due to the associativity of the operation ∗. Moreover,

$$\*\_{i=1}^{\\\infty} \zeta\_i = \lim\_{k \to \infty} \*\_{i=1}^k \zeta\_{i'}$$

which is well defined due to the monotonicity and boundedness of the operation ∗.

Now, we introduce the concept of ∗-fuzzy measure.

**Definition 2** ([3])**.** *Let X be a set and* C *be a σ-algebra consisting of subsets of X. A fuzzy measure on* C × (0, ∞) *is a fuzzy set μ* : C × (0, ∞) → *I such that*

*(i) μ*(∅, *τ*) = 1*,* ∀*τ* ∈ (0, ∞)*;*

*(ii) if* A*<sup>i</sup>* ∈ C, *i* = 1, 2, ··· *, are pairwise disjoint, then*

$$
\mu(\cup\_{i=1}^{\infty} \mathbb{A}\_{i\prime} \pi) = \ast\_{i=1}^{\infty} \mu(\mathbb{A}\_{i\prime} \pi), \ \forall \pi \in (0, \infty).
$$

*Saying the* A*<sup>i</sup> are pairwise disjoint means that* A*<sup>i</sup>* ∩ A*<sup>j</sup>* = ∅*, if i* = *j.*

Definition 2 is known as countable ∗-additivity. We say a fuzzy measure *μ* is finitely ∗-additive if, for any *n* ∈ N

$$
\mu(\cup\_{i=1}^n \mathbb{A}\_{i\prime} \pi) = \ast\_{i=1}^n \mu(\mathbb{A}\_{i\prime} \pi), \,\,\forall \pi \in (0, \infty).
$$

whenever A1, ··· ,A*<sup>n</sup>* are in C and are pairwise disjoint. The quadruple (*X*, C, *μ*, ∗) is called a ∗-fuzzy measure space (in short, ∗*-FMS*).

**Example 1.** *Let* (*X*, C, *m*) *be a measurable space. Let* ∗ = ∗*<sup>H</sup> and define*

$$
\mu\_0(A,\tau) = \frac{\tau}{\tau + m(A)}, \ \forall \tau \in (0,\infty),
$$

*then* (*X*, C, *μ*0, ∗) *is a* ∗*-FMS.*

**Example 2.** *Let* (*X*, C, *m*) *be a measurable space. Let* ∗ = ∗*P. Define*

$$\mu\_0(A,\tau) = e^{-\frac{m(A)}{\tau}}, \ \forall \tau \in (0,\infty).$$

*Then, μ*<sup>0</sup> *is a* ∗*-FM on* C × (0, ∞)*.*

#### **2.** *∗***-Fuzzy Measurable Functions**

Now, we review the concept of ∗-fuzzy normed spaces, for more details, we refer to the works in [12–15].

**Definition 3.** *Let X be a vector space,* ∗ *be a ct-norm and the fuzzy set N on X* × (0, ∞) *satisfies the following conditions for all x*, *y* ∈ *X and τ*, *σ* ∈ (0, ∞)*,*


$$\text{(iii)}\quad N(\alpha \mathbf{x}, \tau) = N\left(\mathbf{x}, \frac{\tau}{|\alpha|}\right) \\ \text{for every } \alpha \neq 0.$$


*Then, N is called a* ∗*-fuzzy norm on X and* (*X*, *N*, ∗) *is called* ∗*-fuzzy normed space.*

Assume that (R, <sup>|</sup>.|) is a standard normed space, we define: *<sup>N</sup>*(*x*, *<sup>τ</sup>*) = *<sup>τ</sup> τ* + |*x*| with ∗ = ∗*P*, it is obvious (R, *N*, ∗*P*) is a ∗-fuzzy normed space.

Let (*X*, *N*, ∗) be a ∗-fuzzy normed space. We define the open ball B(*x*,*r*, *τ*) and the closed ball B[*x*,*r*, *τ*] with center *x* ∈ *X* and radius 0 < *r* < 1, *τ* > 0 as follows,

$$\mathcal{B}(\mathbf{x}, r, \tau) = \{ y \in X : N(\mathbf{x} - y, \tau) > 1 - r \},\tag{1}$$

$$\mathcal{B}[\mathbf{x}, \mathbf{r}, \mathbf{r}] = \{ y \in X : N(\mathbf{x} - y, \mathbf{r}) \ge 1 - r \}. \tag{2}$$

Let (*X*, *N*, ∗) be a ∗-fuzzy normed space. A set *E* ⊂ *X* is said to be open if for each *x* ∈ *E*, there is 0 < *rx* < 1 and *τ<sup>x</sup>* > 0 such that B(*x*,*rx*, *τx*) ⊆ *E*. A set *F* ⊆ *X* is said to be closed in *X* in case its complement *<sup>F</sup><sup>c</sup>* <sup>=</sup> *<sup>X</sup>* <sup>−</sup> *<sup>F</sup>* is open in *<sup>X</sup>*.

Let (*X*, *N*, ∗) be a ∗-fuzzy normed space. A subset *E* ∈ *X* is said to be fuzzy bounded if there exist *τ* > 0 and *r* ∈ (0, 1) such that *N*(*x* − *y*, *τ*) > 1 − *r* for all *x*, *y* ∈ *E*.

Let (*X*, *N*, ∗) be a ∗-fuzzy normed space. A sequence {*xn*} ⊂ *X* is fuzzy convergent to an *x* ∈ *X* in ∗-fuzzy normed space (*X*, *N*, ∗) if for any *τ* > 0 and > 0 there exists a positive integer *N* > 0 such that *N*(*xn* − *x*, *τ*) > 1 − whenever *n* ≥ *N*.

Now, we define ∗-fuzzy measurable functions.

**Definition 4.** *Let* (*X*, C) *and* (*Y*, D) *be* ∗*-fuzzy measurable spaces. A mapping f* : *X* → *Y is called* ∗*-fuzzy* (C, <sup>D</sup>)*-measurable if f* <sup>−</sup>1(*E*) ∈ C *for all E* ∈ D*. If X is any* <sup>∗</sup>*-fuzzy normed space, the <sup>σ</sup>-algebra generated by* *the family of open sets in X (or, equivalently, by the family of closed sets in X) is called the Borel σ-algebra on X and is denoted by* B*X.*

#### **3.** *∗***-Fuzzy Integration**

In this section, we recall the concept of ∗-fuzzy integration by using fuzzy simple functions on the ∗-FMS (*X*, C, ∗, *μ*) and add some new results.

**Definition 5.** *Let* (*X*, C, ∗, *μ*) *be* ∗*-FMS, we define*

$$L\_{+} = \{ f: X \to [0, \infty) \mid f \text{ is } \text{fuzzy } (\mathcal{C}, \mathcal{B}\_{\mathbb{R}}) \text{-measurable function} \} \dots$$

*If <sup>φ</sup> is a simple fuzzy (*(C, <sup>B</sup>R)*-measurable) function in <sup>L</sup>*<sup>+</sup> *with standard representation <sup>φ</sup>* <sup>=</sup> *<sup>n</sup>* ∑ *i*=1 *aiχEi , where ai* > 0 *and Ei* ∈ C *for i* = 1, ..., *n, and Ei* - *Ej* = <sup>∅</sup> *for i* = *j, we define the fuzzy integral of <sup>φ</sup> as*

$$\int\_{X} \phi(\mathbf{x}) d\mu(\mathbf{x}, \boldsymbol{\pi}) = \int\_{X} \sum\_{i=1}^{n} a\_{i} \chi\_{E\_{i}} d\mu(\mathbf{x}, \boldsymbol{\pi}) = \ast\_{i=1}^{n} \mu\left(E\_{i\prime} \frac{\boldsymbol{\pi}}{a\_{i}}\right) \dots$$

In [3], the authors have shown that, with respect to *μ*(*A*, *τ*), *μ* satisfies the following statement;

(i) *μ* : (*A*, .) : (., ∞) → [0, 1] is increasing and continuous.

$$\text{(ii)}\quad \mu\left(A, \frac{\tau}{a+b}\right) \ge \mu\left(A, \frac{\tau}{a}\right) \ast \mu\left(A, \frac{\tau}{b}\right) \text{ for every } a, b > 0, \tau \in (0, \infty).$$

$$\text{(iii)}\quad\lim\_{\tau\_{\mathfrak{n}}\longrightarrow\tau\_{0}}\left(\ast\_{i=1}^{k}\mu(A\_{i\prime}\tau\_{\mathfrak{n}})\right) = \ast\_{i=1}^{k}\lim\_{\tau\_{\mathfrak{n}}\longrightarrow\tau\_{0}}\mu(A\_{i\prime}\tau\_{\mathfrak{n}})\text{ for every }A\_{i}\cap A\_{j}=\mathcal{O}.\text{}$$

$$\text{(iv)}\,\lim\_{\tau \longrightarrow 0} \mu(E,\tau) = 0 \text{ and } \lim\_{\tau \stackrel{\frown}{\longrightarrow} \infty} \mu(E,\tau) = 1.$$

$$\text{(b)}\quad\lim\_{\tau\_n \xrightarrow{} \tau\_0} \lim\_{m \xrightarrow{\longrightarrow} \infty} \left(\mu\left(E^m, \frac{\tau\_n}{a^m}\right)\right) = \lim\_{m \xrightarrow{\longrightarrow} \infty} \lim\_{\tau\_n \xrightarrow{} \tau\_0} \left(\mu\left(E^m, \frac{\tau\_n}{a^m}\right)\right).$$

If *<sup>A</sup>* ∈ C, then *φχ<sup>A</sup>* is also fuzzy simple function *φχ<sup>A</sup>* <sup>=</sup> *<sup>n</sup>* ∑ *i*=1 *aiχA*∩*Ei* , and we define *<sup>φ</sup>*(*x*)*dμ*(*x*, *<sup>τ</sup>*) to be *φχAdμ*(*x*, *<sup>τ</sup>*).

**Theorem 1** ([3])**.** *Let φ and ψ be simple functions in L*+*. Then, we have*


In the next theorem, we prove an important fuzzy integral inequality for fuzzy simple functions.

**Theorem 2.** *Let φ and ψ be fuzzy simple functions in L*+*, then*

$$\int (\phi + \psi)(\mathbf{x}) d\mu(\mathbf{x}, \tau) \ge \left( \int \phi(\mathbf{x}) d\mu(\mathbf{x}, \tau) \right) \* \left( \int \psi(\mathbf{x}) d\mu(\mathbf{x}, \tau) \right).$$

**Proof.** Let *φ* and *ψ* be fuzzy simple functions in *L*+, then we have

$$\int\_{X} (\phi + \psi)(x) d\mu(x, \tau),\tag{3}$$

$$=\int\_{X} \left( \left( \sum\_{i=1}^{n} a\_{i} \chi\_{E\_{i}}(x) \right) + \left( \sum\_{j=1}^{m} b\_{j} \chi\_{\tilde{r}\_{j}}(x) \right) \right) d\mu(x, \tau),$$

$$=\int\_{X} \left( \sum\_{i,j} (a\_{i} + b\_{j}) \chi\_{E\_{i} \cap F\_{j}}(x) \right) d\mu(x, \tau),$$

$$=\quad \ast\_{i=1}^{n} \ast\_{j=1}^{m} \mu\left( \left( E\_{i} \cap F\_{j} \right), \frac{\tau}{(a\_{i} + b\_{j})} \right).$$

On the other hand,

$$\begin{split} & \quad \left( \int\_{X} \phi(\mathbf{x}) d\mu(\mathbf{x}, \tau) \* \int\_{X} \psi(\mathbf{x}) d\mu(\mathbf{x}, \tau) \right), \\ & \quad = \quad \left( \int\_{X} \left( \sum\_{i=1}^{n} a\_{i} \chi\_{E\_{i}}(\mathbf{x}) \right) d\mu(\mathbf{x}, \tau) \right) \* \left( \int\_{X} \left( \sum\_{j=1}^{m} b\_{j} \chi\_{F\_{j}}(\mathbf{x}) \right) d\mu(\mathbf{x}, \tau) \right), \\ & \quad = \quad \left( \*\_{i=1}^{n} \*\_{i=1}^{m} \mu \left( \left( E\_{i} \cap F\_{j} \right), \frac{\tau}{a\_{i}} \right) \right) \* \left( \*\_{j=1}^{m} \*\_{i=1}^{n} \mu \left( \left( E\_{i} \cap F\_{j} \right), \frac{\tau}{b\_{j}} \right) \right), \\ & \quad = \quad \ast\_{i=1}^{n} \*\_{j=1}^{m} \left( \mu \left( \left( E\_{i} \cap F\_{j} \right), \frac{\tau}{a\_{i}} \right) \* \mu \left( \left( E\_{i} \cap F\_{j} \right), \frac{\tau}{b\_{j}} \right) \right), \\ & \quad \leq \quad \ast\_{i=1}^{n} \*\_{j=1}^{m} \left( \mu \left( \left( E\_{i} \cap F\_{j} \right), \frac{\tau}{(a\_{i} + b\_{j})} \right) \right). \end{split}$$

From (3) and (4), we get

$$\int\_{X} (\phi + \psi)(\mathbf{x}) d\mu(\mathbf{x}, \tau) \ge \left( \int\_{X} \phi(\mathbf{x}) d\mu(\mathbf{x}, \tau) \right) \ast \left( \int\_{X} \psi(\mathbf{x}) d\mu(\mathbf{x}, \tau) \right).$$

Now, we extend the concept of fuzzy integral to all functions in *L*+.

**Definition 6.** *Let f be a fuzzy measurable function in L*+*, we define fuzzy integral by*

$$\begin{aligned} &\int\_X f(x)d\mu(x,\tau) \\ &= \inf \left\{ \int\_X \phi(x)d\mu(x,\tau) \mid \, \, 0 \le \phi \le f, \, \phi \text{ is fuzzy simple function} \right\}. \end{aligned}$$

By Theorem 1 (iii), the two definitions of *f* agree when *f* is fuzzy simple function, as the family of fuzzy simple functions over which the infimum is taken includes *f* itself. Moreover, it is obvious from the definition that *<sup>f</sup>* <sup>≥</sup> *<sup>g</sup>* whenever *<sup>f</sup>* <sup>≤</sup> *<sup>g</sup>*, and *c f* <sup>≥</sup> *<sup>c</sup> <sup>f</sup>* for all *<sup>c</sup>* <sup>∈</sup> (0, 1] and *c f* <sup>≤</sup> *<sup>c</sup> f* for all *<sup>c</sup>* <sup>∈</sup> [1, <sup>∞</sup>) and (*f* + *g*) ≥ ( *<sup>f</sup>*) <sup>∗</sup> ( *g*).

**Definition 7.** *If <sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*+*, we say that <sup>f</sup> is fuzzy integrable if f dμ*(*x*, *<sup>τ</sup>*) <sup>&</sup>gt; <sup>0</sup> *for each <sup>τ</sup>* <sup>&</sup>gt; <sup>0</sup>*. Let* (*X*, <sup>C</sup>, *<sup>μ</sup>*, <sup>∗</sup>) *be a* ∗*-FMS. We define*

$$L^+ := \left\{ f: X \to [0, \infty), f \text{ is measurable function and } \int f(\mathbf{x}) d\mu(\mathbf{x}, \mathbf{r}) > 0 \right\}.$$

**Theorem 3** ([3])**. (The fundamental convergence theorem)***. Let* (*X*, C, *μ*, ∗) *be a* ∗*-FMS. Let fn be a sequence in L*<sup>+</sup> *such that fn* −→ *f almost everywhere, then f* <sup>∈</sup> *<sup>L</sup>*<sup>+</sup> *and <sup>f</sup>* <sup>=</sup> lim *<sup>n</sup>*−→<sup>∞</sup> *fn.*

## <sup>∗</sup>*-Fuzzy L*<sup>+</sup> *Spaces*

Here, we are ready to show that every *<sup>L</sup>*<sup>+</sup> is a <sup>∗</sup>-fuzzy normed space. It is clear if we define

*L* := { *f* : *X* −→ R, *f* is fuzzy measurable function},

then (*L*, <sup>+</sup>, .)<sup>R</sup> is a vector space. Moreover, in [3] the authors proved that if *<sup>f</sup>* , *<sup>g</sup>* <sup>∈</sup> *<sup>L</sup>*+, then <sup>|</sup> *<sup>f</sup>* <sup>−</sup> *<sup>g</sup>*| ∈ *<sup>L</sup>*+. Using definition *<sup>L</sup>* and *<sup>L</sup>*<sup>+</sup> we can show *<sup>L</sup>*<sup>+</sup> <sup>⊆</sup> *<sup>L</sup>*. In *<sup>L</sup>*<sup>+</sup> we define *<sup>f</sup>* <sup>≤</sup> *<sup>g</sup>* if and only if *<sup>f</sup>*(*x*) <sup>≤</sup> *<sup>g</sup>*(*x*) and so (*L*+, <sup>≤</sup>) is a cone.

**Note**. Recall that, due to the continuity of t-norm ∗, for any systems {*an*}*n*∈<sup>N</sup> and {*bn*}*n*∈<sup>N</sup> of elements form *I* we have inf{*an* ∗ *bn*} = inf{*an*} ∗ inf{*bn*}.

In the next theorem we define a fuzzy norm on *<sup>L</sup>*<sup>+</sup> and prove that (*L*+, *<sup>N</sup>*, <sup>∗</sup>) is a <sup>∗</sup>-fuzzy normed space.

**Theorem 4.** *Let <sup>N</sup>* : *<sup>L</sup>*<sup>+</sup> <sup>×</sup> (0, <sup>∞</sup>) −→ (0, 1] *be a fuzzy set, such that <sup>N</sup>*(*<sup>f</sup>* , *<sup>τ</sup>*) = *f dμ*(*x*, *<sup>τ</sup>*)*, then* (*L*+, *<sup>N</sup>*, <sup>∗</sup>) *is a* ∗*-fuzzy normed space.*

#### **Proof.**

(FN1) *N*(*f* , *τ*) = *f dμ*(*x*, *τ*) > 0.

(FN2) By theorem 4.5 of [3] we have

$$N(f, \tau) = 1 \Longleftrightarrow \int f d\mu(\mathbf{x}, \tau) = 1 \Longleftrightarrow f = 0$$

almost everywhere.

(FN3) Let *<sup>f</sup>* <sup>=</sup> *<sup>φ</sup>* <sup>=</sup> *<sup>n</sup>* ∑ *i*=1 *aiχEi* and *c* > 0 so,

$$N(c\phi,\tau) = \int c\phi d\mu(\mathbf{x},\tau),\tag{5}$$

$$=\int \sum\_{i=1}^{n} a\_i \chi\_{E\_i} d\mu(\mathbf{x},\tau),$$

$$=\ast\_{i=1}^{n} \mu\left(E\_{i\prime}\frac{\tau}{ca\_i}\right).$$

On the other hand,

$$\begin{split} N\left(\phi,\frac{\mathsf{T}}{\mathsf{c}}\right) &= \int \phi d\mu\left(\mathbf{x},\frac{\mathsf{T}}{\mathsf{c}}\right), \\ &= \int \sum\_{i=1}^{n} a\_{i} \chi\_{E\_{i}} d\mu\left(\mathbf{x},\frac{\mathsf{T}}{\mathsf{c}}\right), \\ &= \ast\_{i=1}^{n} \mu\left(E\_{i'}\frac{\mathsf{T}}{ca\_{i}}\right). \end{split}$$

From (5) and (6) we conclude that

$$N(c\phi, \tau) = N\left(\phi, \frac{\tau}{c}\right). \tag{7}$$

Now, if *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>+</sup> we have {*φn*} ⊆ *<sup>L</sup>*<sup>+</sup> such that *<sup>φ</sup><sup>n</sup>* <sup>↑</sup> *<sup>f</sup>* , then *<sup>c</sup>φ<sup>n</sup>* <sup>↑</sup> *c f* so

$$
\int c\phi\_n d\mu(x,\tau) \downarrow \int c f d\mu(x,\tau) \ldots
$$

By (7), we have *cφndμ*(*x*, *τ*) = *φndμ*(*x*, *τ c* ), and so

$$
\int \phi\_n d\mu(\mathbf{x}, \frac{\mathbf{r}}{c}) \downarrow \int cf d\mu(\mathbf{x}, \mathbf{r}).\tag{8}
$$

On the other hand,

$$
\int \Phi\_n d\mu(\mathbf{x}, \frac{\pi}{c}) \downarrow \int f d\mu(\mathbf{x}, \frac{\pi}{c}) \,\tag{9}
$$

by (8) and (9) we have,

$$\begin{aligned} \int \mathfrak{c} f d\mu(\mathfrak{x}, \mathfrak{r}) &= \int f d\mu(\mathfrak{x}, \frac{\mathfrak{r}}{\mathfrak{c}}), \\ N(\mathfrak{c}f, \mathfrak{r}) &= N(f, \frac{\mathfrak{r}}{\mathfrak{c}}). \end{aligned}$$

(FN4) Let *<sup>f</sup>* <sup>=</sup> *<sup>m</sup>* ∑ *i*=1 *aiχEi* , *<sup>g</sup>* <sup>=</sup> *<sup>n</sup>* ∑ *j*=1 *bjχFj* then,

$$\begin{split} N(\phi + \psi, s + \tau) &= \int (\phi + \psi) d\mu(x, \tau + s), \\ &= \int \sum\_{i,j} (a\_i + b\_j) \chi\_{E\_i \cap F\_j} d\mu(x, \tau + s), \\ &= \*\_{i,j} \mu \left( E\_i \cap F\_{j'} \frac{\tau + s}{a\_i + b\_j} \right). \end{split}$$

On the other hand

$$\begin{split} N(\boldsymbol{\phi}, \boldsymbol{s}) \* N(\boldsymbol{\upmu}, \boldsymbol{\uptau}) &= \left( \int \boldsymbol{\upmu} d\mu(\mathbf{x}, \boldsymbol{s}) \right) \* \left( \int \boldsymbol{\upmu} d\mu(\mathbf{x}, \boldsymbol{\uptau}) \right), \\ &= \left( \int \sum\_{i,j} \boldsymbol{a}\_{i} \chi\_{E\_{i} \cap F\_{j}} d\mu(\mathbf{x}, \boldsymbol{s}) \right) \* \left( \int \sum\_{i,j} b\_{j} \chi\_{E\_{i} \cap F\_{j}} d\mu(\mathbf{x}, \boldsymbol{\uptau}) \right), \\ &= \left( \ast\_{i,j} \mu\left( E\_{i} \cap F\_{j}, \frac{\mathbf{s}}{a\_{i}} \right) \right) \* \left( \ast\_{i,j} \mu\left( E\_{i} \cap F\_{j}, \frac{\mathbf{r}}{b\_{j}} \right) \right), \\ &= \ast\_{i,j} \left( \mu\left( E\_{i} \cap F\_{j}, \frac{\mathbf{s}}{a\_{i}} \right) \* \mu\left( (E\_{i} \cap F\_{j}, \frac{\mathbf{r}}{b\_{j}} \right) \right), \\ &\leq \ast\_{i,j} \left( \min \left\{ \mu\left( E\_{i} \cap F\_{j}, \frac{\mathbf{s}}{a\_{i}} \right), \mu\left( (E\_{i} \cap F\_{j}, \frac{\mathbf{r}}{b\_{j}} \right) \right\} \right). \end{split}$$

Now, we assume *<sup>s</sup> ai* <sup>&</sup>lt; *<sup>τ</sup> bj* . From (10), we conclude

$$N(\phi, s) \* N(\psi, \tau) \le \*\_{i,j} \mu \left( E\_i \cap F\_{j\prime} \frac{s}{a\_i} \right). \tag{11}$$

*Mathematics* **2020**, *8*, 1984

$$\text{Again, from } \frac{s}{a\_i} < \frac{\tau}{b\_j}, \text{ we get } \frac{s}{a\_i} < \frac{\tau + s}{a\_i + b\_j} \text{ because}$$

$$bjs < a\_i \tau,$$

then

$$a\_{\mathbf{i}}\mathbf{s} + b\_{\mathbf{j}}\mathbf{s} < a\_{\mathbf{i}}\mathbf{s} + a\_{\mathbf{i}}\pi\_{\mathbf{i}}$$

and

$$(a\_i + b\_j)s < a\_i(\tau + s)\_{\tau}$$

and so

$$\frac{s}{a\_i} < \frac{\tau + s}{a\_i + b\_j}.$$

Therefore, from (11) we have

$$N(\phi, s) \* N(\psi, \tau) \le \*\_{i, j} \mu \left( E\_i \cap F\_{j \prime} \frac{s}{a\_i} \right),\tag{12}$$

and

$$\*\_{i,j}\mu\left(E\_i \cap F\_{j\prime}\frac{s}{a\_i}\right) \le \*\_{i,j}\mu\left(E\_i \cap F\_{j\prime}\frac{\tau+s}{a\_i+b\_j}\right).\tag{13}$$

From (12) and (13) we have

$$\begin{aligned} N(\phi, s) \* N(\psi, \tau) &\leq \*\_{i,j} \mu \left( E\_i \cap F\_j, \frac{\tau + s}{a\_i + b\_j} \right), \\ &= N \left( \phi + \psi, s + \tau \right). \end{aligned}$$

Now let *<sup>f</sup>* , *<sup>g</sup>* <sup>∈</sup> *<sup>L</sup>*+, then there exist {*φn*} ⊆ *<sup>L</sup>*<sup>+</sup> such that *<sup>φ</sup><sup>n</sup>* <sup>↑</sup> *<sup>f</sup>* . Similarly, there exist {*ψn*} ⊆ *<sup>L</sup>*<sup>+</sup> such that *<sup>ψ</sup><sup>n</sup>* <sup>↑</sup> *<sup>g</sup>*, and *<sup>φ</sup><sup>n</sup>* <sup>+</sup> *<sup>ψ</sup><sup>n</sup>* <sup>↑</sup> *<sup>f</sup>* <sup>+</sup> *<sup>g</sup>*, then

$$\inf \left\{ \int \left( \phi\_n + \psi\_n \right) d\mu(\mathbf{x}, \tau + s) \right\} = \int \left( f + \mathbf{g} \right) d\mu(\mathbf{x}, \tau + s).$$

Also according to (12), we get

$$\int \left(\phi\_n + \psi\_n\right) d\mu(\mathbf{x}, \tau + \mathbf{s}) \ge \int \phi\_n d\mu(\mathbf{x}, \mathbf{s}) \ast \int \psi\_n d\mu(\mathbf{x}, \tau),$$

and

$$\begin{aligned} &\int \left(f+g\right)d\mu(\mathbf{x},\tau+s) = \inf\left\{\int \left(\phi\_{\mathrm{n}}+\psi\_{\mathrm{n}}\right)d\mu(\mathbf{x},\tau+s)\right\}, \\ &\geq \quad \inf\left\{\int \phi\_{\mathrm{n}}d\mu(\mathbf{x},\mathbf{s}) \* \int \psi\_{\mathrm{n}}d\mu(\mathbf{x},\tau)\right\}, \\ &\geq \quad \inf\left\{\int \phi\_{\mathrm{n}}d\mu(\mathbf{x},\mathbf{s})\right\} \* \inf\int \psi\_{\mathrm{n}}d\mu(\mathbf{x},\tau) \\ &= \quad \int f d\mu(\mathbf{x},\mathbf{s}) \* \int \mathrm{g}d\mu(\mathbf{x},\tau), \end{aligned}$$

then

$$\int \left(f+g\right)d\mu(\mathbf{x},\mathbf{r}+\mathbf{s}) \ge \int f d\mu(\mathbf{x},\mathbf{s}) \ast \int g d\mu(\mathbf{x},\mathbf{r}).$$

(FN5) Let *<sup>f</sup>* <sup>=</sup> *<sup>k</sup>* ∑ *i*=1 *aiχEi* , then

$$\begin{aligned} N(f, \tau\_n) &= \int \sum\_{i=1}^k a\_i \chi\_{E\_i} d\mu(x, \tau\_n) \\ &= \*\_{i=1}^k \mu\left(E\_i, \frac{\tau\_n}{a\_i}\right), \end{aligned}$$

and

$$\lim\_{\tau\_{\mathfrak{n}} \xrightarrow{\prod} \tau\_0} N(f, \tau\_{\mathfrak{n}}) = \lim \ast\_{i=1}^k \mu\left(E\_{i\prime} \frac{\tau\_{\mathfrak{n}}}{a\_i}\right).$$

$$\text{According to Definition 5 (iii), we get}$$

$$\begin{aligned} \lim\_{\tau\_n \longrightarrow \tau\_0} N(f, \tau\_n) &= \lim\_{\tau\_n \longrightarrow \tau\_0} \ast\_{i=1}^k \mu \left( E\_i, \frac{\tau\_n}{a\_i} \right), \\ &= \ast\_{i=1}^k \lim\_{\tau\_n \longrightarrow \tau\_0} \left( E\_i, \frac{\tau\_n}{a\_i} \right), \end{aligned}$$

and by Definition 5 (i),

$$\begin{aligned} \lim\_{\tau\_0 \to \tau\_0} N(f, \tau\_0) &= \ast\_{i=1}^k \lim\_{\tau\_0 \to \tau\_0} \left( E\_{i\prime} \frac{\tau\_n}{a\_i} \right), \\ &= \ast\_{i=1}^k \mu \left( E\_{i\prime} \frac{\tau\_0}{a\_i} \right), \\ &= \int f d\mu(\mathbf{x}, \tau\_0), \\ &= N(f, \tau\_0). \end{aligned}$$

Now, let *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*+, then

$$\begin{aligned} N(f, \tau\_n) &= \int f d\mu(\mathbf{x}, \tau\_n), \\ &= \inf \left\{ \int \phi\_m d\mu(\mathbf{x}, \tau\_n) |\phi\_m \uparrow f\rangle, \\ &= \lim\_{m \to \infty} \int \phi\_m d\mu(\mathbf{x}, \tau\_n). \end{aligned}$$

and

$$\begin{split} \lim\_{\tau\_{\mathfrak{u}} \longrightarrow \tau\_{0}} N(f, \tau\_{\mathfrak{u}}) &= \lim\_{\tau\_{\mathfrak{u}} \longrightarrow \tau\_{0}} \lim\_{\mathfrak{m} \longrightarrow \infty} \int \phi\_{m} d\mu(\mathbf{x}, \tau\_{\mathfrak{u}}), \\ &= \lim\_{\tau\_{\mathfrak{u}} \longrightarrow \tau\_{0}} \lim\_{\mathfrak{m} \longrightarrow \infty} \int \sum\_{i=1}^{k} a\_{i}^{m} \chi\_{E\_{i}^{m}} d\mu(\mathbf{x}, \tau\_{\mathfrak{u}}), \\ &= \lim\_{\tau\_{\mathfrak{u}} \longrightarrow \tau\_{0}} \lim\_{\mathfrak{m} \longrightarrow \infty} \ast\_{i=1}^{k} \mu\left(E\_{i}^{m}, \frac{\tau\_{\mathfrak{u}}}{a\_{i}^{\mathfrak{m}}}\right). \end{split}$$

According to Definition 5 (v), we get

$$\begin{aligned} \lim\_{\tau\_n \longrightarrow \tau\_0} N(f, \tau\_n) &= \lim\_{\tau\_n \longrightarrow \tau\_0} \lim\_{m \longrightarrow \infty} \ast\_{i=1}^k \mu\left(E\_i^m, \frac{\tau\_n}{a\_i^m}\right), \\ &= \lim\_{m \longrightarrow \infty} \lim\_{\tau\_n \longrightarrow \tau\_0} \ast\_{i=1}^k \mu\left(E\_i^m, \frac{\tau\_n}{a\_i^m}\right). \end{aligned}$$

and by Definition 5 (iii), we get

$$\begin{aligned} \lim\_{\tau\_n \longrightarrow \tau\_0} N(f, \tau\_n) &= \lim\_{m \longrightarrow \infty} \lim\_{\tau\_n \longrightarrow \tau\_0} \ast\_{i=1}^k \mu\left(E\_i^m, \frac{\tau\_n}{a\_i^m}\right), \\ &= \lim\_{m \longrightarrow \infty} \ast\_{i=1}^k \lim\_{\tau\_n \longrightarrow \tau\_0} \mu\left(E\_i^m, \frac{\tau\_n}{a\_i^m}\right). \end{aligned}$$

Using Definition 5 (i), we get

$$\begin{split} \lim\_{\tau\_{0}\longrightarrow\mp\pi\_{0}} N(f,\tau\_{n}) &= \lim\_{m\to\infty} \ast\_{i=1}^{k} \mathop{\rm l\!}\_{\tau\_{0}\longrightarrow\mp\pi\_{0}} \mu\left(E\_{i}^{m}, \frac{\tau\_{n}}{a\_{i}^{m}}\right), \\ &= \lim\_{m\to\infty} \ast\_{i=1}^{k} \mu\left(E\_{i}^{m}, \frac{\tau\_{0}}{a\_{i}^{m}}\right), \\ &= \lim\_{m\to\infty} \int \Phi\_{m} d\mu(\mathbf{x}, \tau\_{0}), \\ &= \inf \left\{ \int \Phi\_{m} d\mu(\mathbf{x}, \tau\_{0}) \right\}, \\ &= \int f d\mu(\mathbf{x}, \tau\_{0}), \\ &= N(f,\tau\_{0}). \end{split}$$

,

(FN6) Let *<sup>f</sup>* <sup>=</sup> *<sup>k</sup>* ∑ *i*=1 *aiχEi* , then

$$\begin{aligned} N(f,\tau) &= \int f d\mu(\mathbf{x},\tau), \\ &= \int \sum\_{i=1}^{n} a\_i \chi\_{E\_i} d\mu(\mathbf{x},\tau), \\ &= \*\_{i=1}^{k} \mu\left(E\_{i\prime} \frac{\tau}{a\_i}\right). \end{aligned}$$

and

$$\lim\_{\tau \longrightarrow \tau\_0} N(f, \tau) = \lim\_{\tau \longrightarrow \tau\_0} \ast\_{i=1}^k \mu\left(E\_{i\prime} \frac{\tau}{a\_i}\right).$$

According to Definition 5 (iii), we have

$$\begin{aligned} \lim\_{\tau \longrightarrow 0} N(f, \tau) &= \lim\_{\tau \longrightarrow 0} \ast\_{i=1}^k \mu \left( E\_{i\prime} \frac{\tau}{a\_i} \right), \\ &= \ast\_{i=1}^k \lim\_{\tau \longrightarrow 0} \mu \left( E\_{i\prime} \frac{\tau}{a\_i} \right), \end{aligned}$$

and by Definition 5 (iv),

$$\begin{aligned} \lim\_{\tau \longrightarrow 0} N(f, \tau) &= \star\_{i=1}^k \lim\_{\tau \longrightarrow 0} \mu \left( E\_{i\prime} \frac{\tau}{a\_i} \right), \\ &= \star\_{i=1}^k 0, \\ &= 0. \end{aligned}$$

Now let *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*+, so

$$N(f, \tau) = \int f d\mu(\mathbf{x}, \tau) = \inf \left\{ \int \phi\_{\mathrm{mf}} d\mu(\mathbf{x}, \tau) \right\},$$

$$= \lim\_{m \to \infty} \left\{ \int \phi\_{\mathrm{mf}} d\mu(\mathbf{x}, \tau) \right\},$$

$$= \lim\_{m \to \infty} \left\{ N(\phi\_{\mathrm{mf}}, \tau) \right\}.$$

Then,

$$\begin{aligned} \lim\_{\tau \longrightarrow 0} N(f, \tau) &= \lim\_{\tau \longrightarrow 0} \lim\_{m \longrightarrow \infty} \left\{ N(\phi\_{m}, \tau) \right\}, \\ &= \lim\_{\tau \longrightarrow 0} \lim\_{m \longrightarrow \infty} \ast\_{i=1}^{k} \mu \left( E\_{i}^{m}, \frac{\tau}{a\_{i}^{m}} \right). \end{aligned}$$

According to Definition 5 (v), we get

$$\begin{aligned} \lim\_{\tau \longrightarrow 0} N(f, \tau) &= \lim\_{\tau \longrightarrow 0} \lim\_{m \to \infty} \ast\_{i=1}^k \mu\left(E\_i^m, \frac{\tau}{a\_i^m}\right), \\ &= \lim\_{m \to \infty} \lim\_{\tau \longrightarrow 0} \ast\_{i=1}^k \mu\left(E\_i^m, \frac{\tau}{a\_i^m}\right), \end{aligned}$$

and from Definition 5 (iii), we get

$$\begin{aligned} \lim\_{\tau \longrightarrow 0} N(f, \tau) &= \lim\_{m \to \infty} \lim\_{\tau \longrightarrow 0} \ast\_{i=1}^k \mu\left(E\_i^m, \frac{\tau}{a\_i^m}\right), \\ &= \lim\_{m \to \infty} \ast\_{i=1}^k \lim\_{\tau \longrightarrow 0} \mu\left(E\_i^m, \frac{\tau}{a\_i^m}\right). \end{aligned}$$

From Definition 5 (iv), we get

$$\begin{aligned} \lim\_{\tau \longrightarrow 0} N(f, \tau) &= \lim\_{m \to \infty} \ast\_{i=1}^k \lim\_{\tau \longrightarrow 0} \mu \left( E\_i^m \wedge \frac{\tau}{a\_i^m} \right), \\ &= \lim\_{m \to \infty} \ast\_{i=1}^k 0, \\ &= 0. \end{aligned}$$

Similarly,

$$\lim\_{\tau \longrightarrow \infty} N(f, \tau) = 1.$$

We have proved (*L*+, *<sup>N</sup>*, <sup>∗</sup>) is a <sup>∗</sup>-fuzzy normed space. Define *<sup>M</sup>* : *<sup>L</sup>*<sup>+</sup> <sup>×</sup> *<sup>L</sup>*<sup>+</sup> <sup>×</sup> (0, <sup>∞</sup>) −→ (0, 1] by

$$M(f, \mathfrak{g}, \tau) = N\left(|f - \mathfrak{g}|, \tau\right) = \int |f - \mathfrak{g}| d\mu(\mathfrak{x}, \tau),$$

then *<sup>M</sup>* is a fuzzy metric on *<sup>L</sup>*<sup>+</sup> and (*L*+, *<sup>M</sup>*, <sup>∗</sup>) is called the <sup>∗</sup>-fuzzy metric induced by the <sup>∗</sup>-fuzzy normed space (*L*+, *<sup>N</sup>*, <sup>∗</sup>).

**Theorem 5** ([3])**.** *If <sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>+</sup> *and <sup>ε</sup>* <sup>&</sup>gt; <sup>0</sup>*, there is an integrable fuzzy simple function <sup>φ</sup>* <sup>=</sup> *<sup>n</sup>* ∑ *j*=1 *ajχEJ such that* <sup>|</sup> *<sup>f</sup>* <sup>−</sup> *<sup>φ</sup>*|*dμ*(*x*, *<sup>τ</sup>*) <sup>&</sup>gt; <sup>1</sup> <sup>−</sup> *<sup>ε</sup> for each <sup>τ</sup>* <sup>&</sup>gt; <sup>0</sup> *(that is, the integrable simple functions are dense in L*+*).*

Now, we show *L*<sup>+</sup> is a complete space.

**Theorem 6.** *<sup>L</sup>*<sup>+</sup> *is a* <sup>∗</sup>*-fuzzy Banach space.*

**Proof.** Let { *fn*} ⊆ *<sup>L</sup>*<sup>+</sup> is a Cauchy sequence, then { *fn*(*x*)} ⊂ <sup>R</sup><sup>+</sup> is a Cauchy sequence for every *x* ∈ *X* and R is complete so there exist *y* ∈ R such that *fn*(*x*) −→ *y*. We get *f* : *X* −→ R, *f*(*x*) = *y* according to corollary 3.16 [3], *<sup>f</sup>* is fuzzy measurable so *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>+</sup> and according to Theorem (3), *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>+</sup> so, lim *<sup>n</sup>*−→<sup>∞</sup> *fn*(*x*) = *<sup>f</sup>*(*x*) almost everywhere or lim *<sup>n</sup>*−→<sup>∞</sup> *fn* <sup>=</sup> *<sup>f</sup>* .

## **4.** *<sup>∗</sup>***-Fuzzy (***L***+)***<sup>p</sup>* **Spaces**

In this section, by the concept of fuzzy measurable functions and fuzzy integrable functions we define a class of function spaces.

**Definition 8.** *Let* (*X*, C, ∗) *be a* ∗*-fuzzy measure space. We define*

$$\begin{aligned} & (L^+)^p \\ &= \left\{ f: \mathbf{X} \longrightarrow \mathbb{R}^+ \text{ in which } f \text{ is fuzzy measurable function and } \int f^p d\mu(\mathbf{x}, \mathbf{r}) > 0, \ p \ge 1 \right\}. \end{aligned}$$

There is an order on ((*L*+)*<sup>p</sup>*, <sup>≤</sup>) such that *<sup>f</sup>* , *<sup>g</sup>* <sup>∈</sup> (*L*+)*<sup>p</sup>* we have *<sup>f</sup>* <sup>≤</sup> *<sup>g</sup>* if and only if *<sup>f</sup>*(*x*) <sup>≤</sup> *<sup>g</sup>*(*x*). Furthermore, if *<sup>f</sup>* , *<sup>g</sup>* <sup>∈</sup> (*L*+)*<sup>p</sup>* then <sup>|</sup> *<sup>f</sup>* <sup>−</sup> *<sup>g</sup>*| ∈ (*L*+)*<sup>p</sup>*, and <sup>|</sup> *<sup>f</sup>* <sup>−</sup> *<sup>g</sup>*<sup>|</sup> *<sup>p</sup>* <sup>≤</sup> *<sup>f</sup> <sup>p</sup>* or *<sup>g</sup><sup>p</sup>* hence <sup>|</sup> *<sup>f</sup>* <sup>−</sup> *<sup>g</sup>*<sup>|</sup> *pdμ*(*x*, *<sup>τ</sup>*) <sup>≥</sup> max[ *f pdμ*(*x*, *τ*), *gpdμ*(*x*, *τ*)].

In the next theorem we prove <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>* is a <sup>∗</sup>- fuzzy normed space.

**Theorem 7.** *Define Np* : (*L*+)*<sup>p</sup>* <sup>×</sup> (0, <sup>∞</sup>) −→ (0, 1] *by Np*(*<sup>f</sup>* , *<sup>τ</sup>*) = *<sup>f</sup> pdμ*(*x*, *<sup>τ</sup>*) *then* ((*L*+)*<sup>p</sup>*, *Np*, <sup>∗</sup>) *is a* <sup>∗</sup> *fuzzy normed space.*

#### **Proof.**


$$\begin{split} N\_p(c\phi, \tau) &= \int (c\phi)^p d\mu\_\prime \\ &= \int \left(\sum\_{i=1}^n c a\_i \chi\_{E\_i}\right)^p d\mu\_\prime \\ &= \*\_{i=1}^n \mu\left(E\_{i\prime} \frac{\tau}{c^p a\_i^p}\right). \end{split} \tag{14}$$

On the other hand,

$$\begin{split} N\_p(\boldsymbol{\phi}, \frac{\boldsymbol{\tau}}{c^p}) &= \int \boldsymbol{\phi}^p d\mu(\mathbf{x}, \frac{\boldsymbol{\tau}}{c^p}), \\ &= \int \left( \sum\_{i=1}^n a\_i \chi\_{E\_i} \right)^p d\mu(\mathbf{x}, \frac{\boldsymbol{\tau}}{c^p}), \\ &= \int \sum\_{i=1}^n a\_i^p \chi\_{E\_i} d\mu(\mathbf{x}, \frac{\boldsymbol{\tau}}{c^p}), \\ &= \ast\_{i=1}^n \mu\left( E\_i, \frac{\boldsymbol{\tau}}{c^p a\_i^p} \right). \end{split} \tag{15}$$

From (14) and (15) we conclude that

$$N\_{\mathcal{P}}(cf,\tau) = N\_{\mathcal{P}}\left(f, \frac{\tau}{c}\right).$$

Now let *<sup>f</sup>* <sup>∈</sup> (*L*+)*<sup>p</sup>*, then we have

$$N\_p(cf, \tau) = \int (cf)^p d\mu(\mathbf{x}, \tau) = \inf \left\{ \int (c\phi\_n)^p d\mu(\mathbf{x}, \tau) : (c\phi\_n)^p \uparrow (cf)^p \right\}.\tag{16}$$

On the other hand,

$$N\_p(f, \frac{\pi}{c}) = \int f^p d\mu(\mathbf{x}, \frac{\pi}{c})\tag{17}$$

$$= -\inf \left\{ \int \phi\_n^p d\mu(\mathbf{x}, \frac{\pi}{c}) : \phi\_n^p \uparrow f\_n^p \right\}.$$

From (14) and (15) we get

$$\int (c\phi\_n)^p d\mu(\mathbf{x}, \mathbf{r}) = N\_p(c\phi\_{n\prime}\mathbf{r}) = N\_p(\phi\_{n\prime}\frac{\mathbf{r}}{c}) = \int \phi\_n^p d\mu(\mathbf{x}, \frac{\mathbf{r}}{c}).$$

Using (16) and (17) we get

$$N\_{\mathbb{P}}(cf,\pi) = N\_{\mathbb{P}}(f,\frac{\pi}{c}).$$

(FN4) Let *f* = *φ* and *g* = *ψ* be simple functions. Then,

$$N\_p\left(\boldsymbol{\phi} + \boldsymbol{\psi}, \boldsymbol{s} + \tau\right) = N\_p\left(\sum\_{i=1}^n a\_i \chi\_{E\_i} + \sum\_{j=1}^m b\_j \chi\_{F\_j}, \mathbf{s} + \tau\right), \tag{18}$$

$$= N\_p\left(\sum\_{i,j} (a\_i + b\_j) \chi\_{E\_i \cap F\_j}, \mathbf{s} + \tau\right),$$

$$= \int \left(\sum\_{i,j} (a\_i + b\_j) \chi\_{E\_i \cap F\_j}\right)^p d\mu(\mathbf{x}, \mathbf{s} + \tau),$$

$$= \int \sum\_{i,j} (a\_i + b\_j)^p \chi\_{E\_i \cap F\_j} d\mu(\mathbf{x}, \mathbf{s} + \tau),$$

$$= \star\_{i,j} \mu\left(E\_i \cap F\_j, \frac{\mathbf{s} + \tau}{(a\_i + b\_j)^p}\right).$$

On the other hand,

*Np*(*φ*,*s*) <sup>∗</sup> *Np*(*ψ*, *<sup>τ</sup>*) = *φpdμ*(*x*,*s*) ∗ *ψpdμ*(*x*, *τ*) , (19) = *n* ∑ *i*=1 *aiχEi*∩*Fj <sup>p</sup> dμ*(*x*,*s*) ∗ *m* ∑ *j*=1 *bjχEi*∩*Fj <sup>p</sup> dμ*(*x*, *τ*) , = *n* ∑ *i*= *a p <sup>i</sup> χEi*∩*Fj dμ*(*x*,*s*) ∗ *m* ∑ *j*=1 *b p <sup>j</sup> χEi*∩*Fj dμ*(*x*, *τ*) , = ∗*i*,*<sup>j</sup> μ Ei* <sup>∩</sup> *Fj*, *<sup>s</sup> a p i* ∗ ∗*i*,*<sup>j</sup> μ Ei* <sup>∩</sup> *Fj*, *<sup>τ</sup> b p j* , = ∗*i*,*<sup>j</sup> μ Ei* <sup>∩</sup> *Fj*, *<sup>s</sup> a p i* ∗ *μ Ei* <sup>∩</sup> *Fj*, *<sup>τ</sup> b p j* , ≤ ∗*i*,*j μ Ei* <sup>∩</sup> *Fj*, min *<sup>s</sup> a p i* , *τ b p j* ≤ ∗*i*,*jμ Ei* <sup>∩</sup> *Fj*, *<sup>s</sup>* <sup>+</sup> *<sup>τ</sup>* (*ai* + *bj*)*<sup>p</sup>* .

(FN5) Let *<sup>f</sup>* <sup>=</sup> *<sup>k</sup>* ∑ *i*=1 *aiχEi* , then

$$\begin{aligned} N\_p(f, \tau\_n) &= \int \left( \sum\_{i=1}^k a\_i \chi\_{E\_i} \right)^p d\mu(\mathbf{x}, \tau\_n) \\ &= \*\_{i=1}^k \mu \left( E\_{i\prime} \frac{\tau\_n}{(a\_i)^p} \right) ,\end{aligned}$$

and so

$$\lim\_{\tau\_n \longrightarrow \tau\_0} N\_p(f, \tau\_n) = \lim \ast\_{i=1}^k \mu\left(E\_i, \frac{\tau\_n}{(a\_i)^p}\right).$$

Using Definition 5 (iii), we get

$$\begin{aligned} \lim\_{\tau\_{\mathfrak{n}} \longrightarrow \tau\_{\mathfrak{l}}} \mathcal{N}\_{\mathbb{P}}(f, \tau\_{\mathfrak{n}}) &= \lim\_{\tau\_{\mathfrak{n}} \longrightarrow \tau\_{\mathfrak{l}}} \ast\_{i=1}^{k} \mu\left(E\_{i\prime}, \frac{\tau\_{\mathfrak{n}}}{(a\_{i})^{p}}\right) \\ &= \ast\_{i=1}^{k} \lim\_{\tau\_{\mathfrak{n}} \longrightarrow \tau\_{\mathfrak{l}}} \mu\left(E\_{i\prime}, \frac{\tau\_{\mathfrak{l}}}{(a\_{i})^{p}}\right), \end{aligned}$$

and according to Definition 5 (i),

$$\begin{aligned} \lim\_{\tau\_{\mathfrak{u}} \longrightarrow \tau\_{\mathfrak{u}}} N\_p(f, \tau\_{\mathfrak{u}}) &= \ast\_{i=1}^k \lim\_{\tau\_{\mathfrak{u}} \longrightarrow \tau\_{\mathfrak{u}}} \mu \left( E\_{i\prime} \frac{\tau\_{\mathfrak{u}}}{(a\_i)^p} \right), \\ &= \ast\_{i=1}^k \mu \left( E\_{i\prime} \frac{\tau\_{\mathfrak{u}}}{(a\_i)^p} \right) \\ &= \int f^p d\mu(\mathbf{x}, \tau\_{\mathfrak{u}}), \\ &= N\_p(f, \tau\_{\mathfrak{u}}). \end{aligned}$$

Now let *<sup>f</sup>* <sup>∈</sup> (*L*+)*<sup>p</sup>*, we have

$$\begin{aligned} N\_p(f, \tau\_\mathfrak{n}) &= \int f^p d\mu(\mathbf{x}, \tau\_\mathfrak{n}) \\ &= \inf \left\{ \int (\phi\_m)^p d\mu(\mathbf{x}, \tau\_\mathfrak{n}) |\phi\_m \uparrow f \right\}, \\ &= \lim\_{m \to \infty} \int (\phi\_\mathfrak{m})^p d\mu(\mathbf{x}, \tau\_\mathfrak{n}). \end{aligned}$$

Then,

$$\begin{split} \lim\_{\tau\_{\mathfrak{u}} \longrightarrow \mathsf{r}\_{0}} N\_{\mathbb{P}}(f, \tau\_{\mathfrak{u}}) &= \lim\_{\mathsf{r}\_{\mathfrak{u}} \longrightarrow \mathsf{r}\_{0}} \lim\_{\mathsf{m} \longrightarrow \infty} \int (\phi\_{\mathfrak{m}})^{p} d\mu(\mathbf{x}, \tau\_{\mathfrak{u}}), \\ &= \lim\_{\mathsf{r}\_{\mathfrak{u}} \longrightarrow \mathsf{r}\_{0}} \lim\_{\mathsf{m} \longrightarrow \infty} \int \left( \sum\_{i=1}^{k} (a\_{i}^{\mathsf{m}} \chi\_{\mathbb{E}\_{i}^{\mathsf{m}}})^{p} d\mu(\mathbf{x}, \tau\_{\mathfrak{u}}) \right), \\ &= \lim\_{\mathsf{r}\_{\mathfrak{u}} \longrightarrow \mathsf{r}\_{0}} \lim\_{\mathsf{m} \longrightarrow \infty} \ast\_{i=1}^{k} \mu\left( E\_{i}^{\mathsf{m}}, \frac{\mathsf{r}\_{\mathfrak{u}}}{(a\_{i}^{\mathsf{m}})^{p}} \right). \end{split}$$

Using Definition 5 (v), we get

$$\begin{aligned} \lim\_{\tau\_{\mathfrak{n}} \longrightarrow \tau\_{\mathfrak{0}}} N\_p(f, \tau\_{\mathfrak{n}}) &= \lim\_{\tau\_{\mathfrak{n}} \longrightarrow \tau\_{\mathfrak{0}}} \lim\_{m \longrightarrow \infty} \ast\_{i=1}^k \mu\left(E\_i^m, \frac{\tau\_{\mathfrak{n}}}{(a\_i^m)^p}\right), \\ &= \lim\_{\mathfrak{m} \longrightarrow \infty} \lim\_{\tau\_{\mathfrak{n}} \longrightarrow \tau\_{\mathfrak{0}}} \ast\_{i=1}^k \mu\left(E\_i^m, \frac{\tau\_{\mathfrak{n}}}{(a\_i^m)^p}\right)' \end{aligned}$$

and according to Definition 5 (iii)

$$\begin{aligned} \lim\_{\tau\_{\pi} \longrightarrow \tau\_{0}} N\_{\mathbb{P}}(f, \tau\_{\pi}) &= \lim\_{m \to \infty} \lim\_{\tau\_{\pi} \longrightarrow \tau\_{0}} \ast\_{i=1}^{k} \mu\left(E\_{i}^{m}, \frac{\tau\_{\pi}}{(a\_{i}^{m})^{\mathcal{P}}}\right), \\ &= \lim\_{m \to \infty} \ast\_{i=1}^{k} \lim\_{\tau\_{\pi} \longrightarrow \tau\_{0}} \mu\left(E\_{i}^{m}, \frac{\tau\_{\pi}}{(a\_{i}^{m})^{\mathcal{P}}}\right). \end{aligned}$$

By Definition 5 (i), we have

$$\begin{split} \lim\_{\tau\_{\pi} \to \tau\_{0}} N\_{p}(f, \tau\_{n}) &= \lim\_{m \to \infty} \ast\_{i=1}^{k} \lim\_{\tau\_{\pi} \to \tau\_{0}} \mu \left( E\_{i}^{m}, \frac{\tau\_{n}}{(a\_{i}^{m})^{p}} \right), \\ &= \lim\_{m \to \infty} \ast\_{i=1}^{k} \mu \left( E\_{i}^{m}, \frac{\tau\_{0}}{(a\_{i}^{m})^{p}} \right), \\ &= \lim\_{m \to \infty} \int (\phi\_{m})^{p} d\mu(\mathbf{x}, \tau\_{0}), \\ &= \inf \left\{ \int (\phi\_{m})^{p} d\mu(\mathbf{x}, \tau\_{0}) \right\}, \\ &= \int f^{p} d\mu(\mathbf{x}, \tau\_{0}), \\ &= N\_{p}(f, \tau\_{0}). \end{split}$$

(FN6) Let *<sup>f</sup>* <sup>=</sup> *<sup>k</sup>* ∑ *i*=1 *aiχEi* , then

$$\begin{aligned} N\_p(f, \tau) &= \int f^p d\mu(\mathbf{x}, \tau), \\ &= \int \left( \sum\_{i=1}^k a\_i \chi\_{E\_i} \right)^p d\mu(\mathbf{x}, \tau), \\ &= \*\_{i=1}^k \mu \left( E\_{i\prime} \frac{\tau}{(a\_i)^p} \right), \end{aligned}$$

and so

$$\lim\_{\tau \longrightarrow \tau\_0} N\_p(f, \tau) = \lim\_{\tau \longrightarrow \tau\_0} \ast\_{i=1}^k \mu\left(E\_{i\prime} \frac{\tau}{(a\_i)^p}\right).$$

Using Definition 5 (iii),

$$\begin{aligned} \lim\_{\tau \longrightarrow 0} N\_p(f, \tau) &= \lim\_{\tau \longrightarrow 0} \ast\_{i=1}^k \mu \left( E\_{i\prime} \frac{\tau}{(a\_i)^p} \right), \\ &= \ast\_{i=1}^k \lim\_{\tau \longrightarrow 0} \mu \left( E\_{i\prime} \frac{\tau}{(a\_i)^p} \right) \end{aligned}$$

and by Definition 5 (iv), we have

$$\begin{aligned} \lim\_{\tau \longrightarrow 0} N\_p(f, \tau) &= \ast\_{i=1}^k \lim\_{\tau \longrightarrow 0} \mu \left( E\_{i\prime} \frac{\tau}{(a\_i)^p} \right), \\ &= \ast\_{i=1}^k 0, \\ &= 0. \end{aligned}$$

,

.

Now, let *<sup>f</sup>* <sup>∈</sup> (*L*+)*<sup>p</sup>*, then

$$\begin{aligned} N\_P(f,\tau) &= \int f^p d\mu(\mathbf{x},\tau) = \inf \left\{ \int (\phi\_m)^p d\mu(\mathbf{x},\tau) : \phi\_m \uparrow f \right\}, \\ &= \lim\_{m \to \infty} \left\{ \int (\phi\_m)^p d\mu(\mathbf{x},\tau) \right\}, \end{aligned}$$

and so

$$\begin{aligned} \lim\_{\tau \to 0} N\_{\mathbb{P}}(f, \tau) &= \lim\_{\tau \to 0} \lim\_{m \to \infty} \left\{ N\_{\mathbb{P}}(\phi\_{m\_{\tau}} \tau) \right\}, \\ &= \lim\_{\tau \to 0} \lim\_{m \to \infty} \star\_{i=1}^{k} \_{\mu} \left( E\_{i}^{m}, \frac{\tau}{(a\_{i}^{m})^{p}} \right). \end{aligned}$$

Using Definition 5 (v), we get

$$\lim\_{\tau \longrightarrow 0} N\_p(f, \tau) = \lim\_{\tau \longrightarrow 0} \lim\_{m \longrightarrow \infty} \ast\_{i=1}^k \mu\left(E\_i^m, \frac{\tau}{(a\_i^m)^p}\right),$$

$$= \lim\_{m \longrightarrow \infty} \lim\_{\tau \longrightarrow 0} \ast\_{i=1}^k \mu\left(E\_i^m, \frac{\tau}{(a\_i^m)^p}\right),$$

and by Definition 5 (iii), we have

$$\begin{aligned} \lim\_{\tau \longrightarrow 0} N\_p(f, \tau) &= \lim\_{m \to \infty} \lim\_{\tau \longrightarrow 0} \ast\_{i=1}^k \mu\left(E\_i^m, \frac{\tau}{(a\_i^m)^p}\right), \\ &= \lim\_{m \to \infty} \ast\_{i=1}^k \lim\_{\tau \longrightarrow 0} \mu\left(E\_i^m, \frac{\tau}{(a\_i^m)^p}\right). \end{aligned}$$

from Definition 5 (iv), we get

$$\lim\_{\tau \longrightarrow 0} N\_p(f, \tau) = \lim\_{\tau \longrightarrow 0} \ast\_{i=1}^k 0,$$

$$= 0.$$

We proved ((*L*+)*<sup>p</sup>*, *Np*, <sup>∗</sup>) is a <sup>∗</sup>-fuzzy normed space. Now, define the fuzzy set *<sup>M</sup>* : (*L*+)*<sup>p</sup>* <sup>×</sup> (*L*+)*<sup>p</sup>* <sup>×</sup> (0, <sup>∞</sup>) −→ (0, 1] by

$$M(f, \mathfrak{g}, \tau) = N\_p \left( |f - \mathfrak{g}|, \tau \right) = \int |f - \mathfrak{g}|^p d\mu(\mathfrak{x}, \tau).$$

Then, *<sup>M</sup>* is a fuzzy metric on <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>* and ((*L*+)*<sup>p</sup>*, *<sup>M</sup>*, <sup>∗</sup>) is called the <sup>∗</sup>-fuzzy metric space induced by the <sup>∗</sup>-fuzzy normed space ((*L*+)*<sup>p</sup>*, *Np*, <sup>∗</sup>). Now, we study further properties of <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>*.

**Theorem 8.** *For* <sup>1</sup> <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>*, the set of simple functions <sup>g</sup>* <sup>=</sup> *<sup>n</sup>* ∑ *i*=1 *aiχEi where μ*(*Ei*, *τ*) > 0 *for all <sup>i</sup>* ∈ {1, 2, ..., *<sup>n</sup>*} *and for all <sup>τ</sup>* <sup>&</sup>gt; <sup>0</sup>*, is dense in* <sup>∗</sup>*-fuzzy* (*L*+)*p.*

**Proof.** Clearly simple functions *<sup>g</sup>* <sup>=</sup> *<sup>n</sup>* ∑ *i*=1 *aiχEi* are in <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>*. Let *<sup>f</sup>* <sup>∈</sup> (*L*+)*<sup>p</sup>*, by theorem 3.20 in [3] we can choose a sequence { *fn*} of simple functions such that *fn* ↑ *f* almost everywhere, and so (*<sup>f</sup>* <sup>−</sup> *fn*)*<sup>p</sup>* <sup>↓</sup> 0.

We assert (*<sup>f</sup>* <sup>−</sup> *fn*)*<sup>p</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>+</sup> because

$$(f - f\_n)^p \le f^p\_{\phantom{p}\prime}$$

and so

$$\int (f - f\_n)^p d\mu(\mathbf{x}, \boldsymbol{\tau}) \ge \int f^p d\mu(\mathbf{x}, \boldsymbol{\tau}) > 0,$$

then (*<sup>f</sup>* <sup>−</sup> *fn*)*<sup>p</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>+</sup> and (*<sup>f</sup>* <sup>−</sup> *fn*)*<sup>p</sup>* −→ 0. Using the fundamental convergence Theorem 3, we get

$$\lim\_{n \to \infty} \int (f - f\_n)^p d\mu(\mathbf{x}, \tau) = \int 0 d\mu(\mathbf{x}, \tau) = 1.1$$

Then, lim *<sup>n</sup>*−→<sup>∞</sup> *Np*(*<sup>f</sup>* <sup>−</sup> *fn*, *<sup>τ</sup>*) = 1 i.e., *fn Np* −→ *f* .

In the next theorem we prove that <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>* spaces are complete.

**Theorem 9.** *For* <sup>1</sup> <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>*,* <sup>∗</sup>*-fuzzy* (*L*+)*<sup>p</sup> is a* <sup>∗</sup>*-fuzzy Banach space.*

**Proof.** Let { *fn*} ⊆ (*L*+)*<sup>p</sup>* be a Cauchy sequence, then for every *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*, { *fn*(*x*)} ⊆ <sup>R</sup> is a Cauchy sequence in R and since R is complete, there exist *y* ∈ R such that *fn*(*x*) −→ *y*, we define *f* : *X* −→ R by *<sup>f</sup>*(*x*) = *<sup>y</sup>*. Since *fn* −→ *<sup>f</sup>* almost everywhere, so (*fn*)*<sup>p</sup>* −→ (*f*)*<sup>p</sup>* almost everywhere, and (*fn*)*<sup>p</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>+</sup> by the fundamental converge Theorem <sup>3</sup> we have (*f*)*<sup>p</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>+</sup> and lim (*fn*)*pdμ*(*x*, *τ*) = (*f*)*pdμ*(*x*, *τ*), hence *<sup>f</sup>* <sup>∈</sup> (*L*+)*<sup>p</sup>*.

## **5. Inequalities on** *<sup>∗</sup>***-Fuzzy (***L***+)***<sup>p</sup>*

In this section, we are ready to prove some important inequalities on <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>*.

**Lemma 1** ([16])**.** *If a* ≥ 0*, b* ≥ 0*, and* 0 < *λ* < 1*, then*

$$a^{\lambda}b^{1-\lambda} \le \lambda a + (1-\lambda)b\_r$$

*we have equality if and only if a* = *b.*

**Theorem 10** (Hölder's Inequality)**.** *Suppose* <sup>1</sup> <sup>&</sup>lt; *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup> *and* <sup>1</sup> *p* + 1 *<sup>q</sup>* <sup>=</sup> <sup>1</sup>*. If <sup>f</sup> and <sup>g</sup> are fuzzy measurable functions on X then,*

$$\mathcal{N}(f\mathbb{g},\tau) \ge \mathcal{N}\_p\left(f, (p)^{\frac{1}{p}}\tau\right) \ast \mathcal{N}\_q\left(\mathcal{g}, (q)^{\frac{1}{q}}\tau\right).$$

**Proof.** We apply Lemma <sup>1</sup> with (*f*(*x*))*<sup>p</sup>* <sup>=</sup> *<sup>a</sup>*, *<sup>b</sup>* = (*g*(*x*))*q*, and *<sup>λ</sup>* <sup>=</sup> <sup>1</sup> *p* to obtain

$$\left( \left( f(\mathbf{x}) \right)^p \right)^{\frac{1}{p}} \cdot \left( (\mathbf{g}(\mathbf{x}))^q \right)^{1 - \frac{1}{p}} \le \frac{1}{p} (f(\mathbf{x}))^p + (1 - \frac{1}{p}) (\mathbf{g}(\mathbf{x}))^q,$$

then

$$f(\mathbf{x}). \mathbf{g}(\mathbf{x}) \le \left( (\frac{1}{p})^{\frac{1}{p}} f(\mathbf{x}) \right)^p + \left( (\frac{1}{q})^{\frac{1}{q}} \mathbf{g}(\mathbf{x}) \right)^q.$$

Takeing integral of both sides, we get

$$\begin{split} \int f(\mathbf{x}) \, g(\mathbf{x}) d\mu(\mathbf{x}, \tau) &\geq \int \left[ \left( (\frac{1}{p})^{\frac{1}{p}} f(\mathbf{x}) \right)^{p} + \left( (\frac{1}{q})^{\frac{1}{q}} g(\mathbf{x}) \right)^{q} \right] d\mu(\mathbf{x}, \tau), \\ &\geq \left( \int \left( (\frac{1}{p})^{\frac{1}{p}} f(\mathbf{x}) \right)^{p} d\mu(\mathbf{x}, \tau) \right) \* \left( \int \left( (\frac{1}{q})^{\frac{1}{q}} g(\mathbf{x}) \right)^{q} d\mu(\mathbf{x}, \tau) \right), \\ &= N\_{p} \left( (\frac{1}{p})^{\frac{1}{p}} f, \tau \right) \* N\_{q} \left( (\frac{1}{q})^{\frac{1}{q}} g, \tau \right), \\ &= N\_{p} \left( f, (p)^{\frac{1}{p}} \tau \right) \* N\_{q} \left( g, (q)^{\frac{1}{q}} \tau \right). \end{split}$$

Then,

$$N\_1\left(f,\emptyset,\tau\right) \ge N\_p\left(f,\left(p\right)^{\frac{1}{p}}\tau\right) \ast N\_q\left(\mathcal{g},\left(q\right)^{\frac{1}{q}}\tau\right).$$

In the next theorem we compare two <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>* spaces.

**Theorem 11.** *If* <sup>0</sup> <sup>&</sup>lt; *<sup>p</sup>* <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; *<sup>r</sup>* <sup>&</sup>lt; <sup>∞</sup>*, then* (*L*+)*<sup>q</sup>* <sup>⊆</sup> (*L*+)*<sup>p</sup>* + (*L*+)*<sup>r</sup> , that is, each <sup>f</sup>* <sup>∈</sup> (*L*+)*<sup>q</sup> is the sum of a function in* <sup>∗</sup>*-fuzzy* (*L*+)*<sup>p</sup> and a function in* <sup>∗</sup>*-fuzzy* (*L*+)*<sup>r</sup> .*

**Proof.** If *<sup>f</sup>* <sup>∈</sup> (*L*+)*<sup>q</sup>*, let *<sup>E</sup>* <sup>=</sup> {*<sup>x</sup>* : *<sup>f</sup>*(*x*) <sup>&</sup>gt; <sup>1</sup>} and set *<sup>g</sup>* <sup>=</sup> *<sup>f</sup> <sup>χ</sup><sup>E</sup>* and *<sup>h</sup>* <sup>=</sup> *<sup>f</sup> <sup>χ</sup>E<sup>c</sup>* , then

$$\begin{aligned} f &= f.1, \\ &= f(\chi\_E + \chi\_{E^c}), \\ &= f\chi\_E + f\chi\_{E^c}, \\ &= g + h. \end{aligned}$$

However,

$$\mathfrak{g}^p = (f\chi\_E)^p = f^p\chi\_E \le f^q\chi\_E$$

then,

$$\int \mathcal{g}^p d\mu \ge \int f^q \chi\_E d\mu > 0,$$

then,

$$\mathbf{g} \in (L^{+})^{P}.$$

On the other hand,

$$h^r = (f\chi\_{\mathcal{E}^c})^r = f^r\chi\_{\mathcal{E}^c} \le f^q\chi\_{\mathcal{E}^c}.$$

then,

$$\int h^r d\mu \ge \int f^q \chi\_{E^c} d\mu > 0,$$

and so

*<sup>h</sup>* <sup>∈</sup> (*L*+)*<sup>r</sup>* .

Now, we apply Hölder's inequality Theorem 10 to prove next theorem.

**Theorem 12.** *If* <sup>0</sup> <sup>&</sup>lt; *<sup>p</sup>* <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; *<sup>r</sup>* <sup>&</sup>lt; <sup>∞</sup>*, then L<sup>p</sup>* <sup>∩</sup> *<sup>L</sup><sup>r</sup>* <sup>⊆</sup> *<sup>L</sup><sup>q</sup> and*

$$\begin{aligned} N\_q(f, \tau) &\geq N\_p \left( f, \left( \frac{p}{\lambda q} \right)^{\frac{1}{p}} \tau \right) \ast N\_r \left( f, \left( \frac{r}{(1-\lambda)q} \right)^{\frac{1}{r}} \tau \right), \\\\ \text{1) is defined by } \lambda &= \frac{\frac{1}{q} - \frac{1}{r}}{\frac{1}{p} - \frac{1}{r}}. \end{aligned}$$

*where λ* ∈ (0, 1) *is defined by λ* = *<sup>p</sup>* <sup>−</sup> <sup>1</sup> *r* **Proof.** From *f qdμ*(*x*, *τ*) = *f <sup>λ</sup>q*. *f* (1−*λ*)*qdμ*(*x*, *τ*) and Hölder's inequality Theorem 10, we have

$$\begin{split} \int f^q d\mu(\mathbf{x},\tau) &= \int f^{\lambda q} \cdot f^{q(1-\lambda)} d\mu(\mathbf{x},\tau), \\ &\geq \left( \int \left( \frac{(\lambda q)}{p} \frac{\lambda q}{r} f^{\lambda q} \right)^{\frac{p}{q}} d\mu(\mathbf{x},\tau) \right) \ast \left( \int \left( \frac{(1-\lambda)q}{r} \right)^{q} \frac{(1-\lambda)q}{r} f^{q(1-\lambda)} d\mu(\mathbf{x},\tau) \right)^{\frac{r}{(1-\lambda)q}}, \\ &\geq \left( \int \frac{\lambda q}{p} f^{p} d\mu(\mathbf{x},\tau) \right) \ast \left( \int \left( \frac{(1-\lambda)q}{r} \right) f^{r} d\mu(\mathbf{x},\tau) \right), \\ &= \left( \int \left( \frac{\lambda q}{p} \right)^{\frac{1}{p}} f \right)^{p} d\mu(\mathbf{x},\tau) \right) \ast \left( \int \left( \left( \frac{(1-\lambda)q}{r} \right)^{\frac{1}{p}} f \right)^{r} d\mu(\mathbf{x},\tau) \right), \\ &= N\_{p} \left( \left( \frac{\lambda q}{p} \right)^{\frac{1}{p}} f, \tau \right) \ast N\_{\mathbf{r}} \left( \left( \frac{(1-\lambda)q}{r} \right)^{\frac{1}{p}} f, \tau \right), \\ &= N\_{p} \left( f, \left( \frac{p}{\lambda q} \right)^{\frac{1}{p}} \tau \right) \ast N\_{\mathbf{r}} \left( f, \left( \frac{r}{(1-\lambda)q} \right)^{\frac{1}{p}} \tau \right). \end{split}$$

then,

$$N\_q(f, \tau) \ge N\_p \left( f, \left( \frac{p}{\lambda q} \right)^{\frac{1}{p}} \tau \right) \ast N\_r \left( f, \left( \frac{r}{(1-\lambda)q} \right)^{\frac{1}{r}} \tau \right).$$

Another application of Hölder's inequality Theorem 10 helps us to prove next theorem.

**Theorem 13.** *If <sup>μ</sup>*(*X*, *<sup>τ</sup>*) <sup>&</sup>gt; <sup>0</sup> *and* <sup>0</sup> <sup>&</sup>lt; *<sup>p</sup>* <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; <sup>∞</sup>*, then Lp*(*μ*) <sup>⊃</sup> *<sup>L</sup>q*(*μ*) *and,*

$$N\_p(f, \tau) \ge N\_q\left(f, (\frac{q}{p})^{\frac{p}{q}}\tau\right) \* \mu\left(X, (\frac{q}{q-p})^{\frac{q-p}{q}}\tau\right).$$

**Proof.** By Theorem 7 and Hölder's inequality Theorem 10, we get

$$\begin{split} N\_{p}(f,\tau) &= \int f^{p} .1 d\mu(\mathbf{x},\tau), \\ &\geq N\_{\frac{q}{p}}\left(f^{p}, (\frac{q}{p})^{\frac{p}{q}}\tau\right) \* N\_{\frac{q}{q-p}}\left(1, (\frac{q}{q-p})^{\frac{q-p}{q}}\tau\right), \\ &= \int (f^{p})^{\frac{p}{p}} d\mu\left(\mathbf{x}, (\frac{q}{p})^{\frac{p}{q}}\tau\right) \* \int 1 d\mu\left(\mathbf{x}, (\frac{q}{q-p})^{\frac{q-p}{q}}\tau\right), \\ &= \int f^{q} d\mu\left(\mathbf{x}, (\frac{q}{p})^{\frac{p}{q}}\tau\right) \* \mu\left(\mathbf{X}, (\frac{q}{q-p})^{\frac{q-p}{q}}\tau\right), \\ &= N\_{\eta}\Big(f, (\frac{q}{p})^{\frac{p}{q}}\tau\Big) \* \mu\left(\mathbf{X}, (\frac{q}{q-p})^{\frac{q-p}{q}}\tau\right). \end{split}$$

Finally, we prove the Chebyshev's Inequality in <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>* spaces.

**Theorem 14** (Chebyshev's Inequality)**.** *If <sup>f</sup>* <sup>∈</sup> (*L*+)*<sup>p</sup>*(<sup>0</sup> <sup>&</sup>lt; *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>) *then for any <sup>a</sup>* <sup>&</sup>gt; <sup>0</sup>*, Np*(*<sup>f</sup>* , *<sup>τ</sup>*) <sup>≤</sup> *Np*(*χEa* , *<sup>τ</sup> <sup>a</sup>* ) *with respect to Ea* = {*x* : *f*(*x*) > *a*}*.*

*Mathematics* **2020**, *8*, 1984

**Proof.** We have,

$$f^p > (f\chi\_{E\_a})^p = f^p \chi\_{E\_{a'}}$$

then

$$\int f^p d\mu(\mathbf{x}, \boldsymbol{\tau}) \le \int f^p d\mu(\mathbf{x}, \boldsymbol{\tau}) \chi\_{E\_a} = \int\_{E\_a} f^p d\mu(\mathbf{x}, \boldsymbol{\tau}),\tag{20}$$

and on *Ea* we have

$$\int\_{E\_d} f^p d\mu(\mathbf{x}, \tau) \le \int\_{E\_d} a^p d\mu(\mathbf{x}, \tau) = \int a^p \chi\_{E\_d} d\mu(\mathbf{x}, \tau). \tag{21}$$

By (20) and (21) we get

$$\begin{aligned} \int f^p d\mu(\mathbf{x}, \tau) &\leq \int a^p \chi\_{E\_x} d\mu(\mathbf{x}, \tau), \\ &= \int \left(a \chi\_{E\_x}\right)^p d\mu(\mathbf{x}, \tau). \end{aligned}$$

Then,

$$\begin{aligned} N\_p(f, \tau) &\leq N\_p(a \chi\_{E\_{a'}}\tau), \\ &= N\_p(\chi\_{E\_{a'}}\frac{\tau}{a}). \end{aligned}$$

#### **6. Conclusions**

We have considered an uncertainty measure *μ* based on the concept of fuzzy sets and continuous triangular norms named by ∗-fuzzy measure. In fact, we worked on a new model of the fuzzy measure theory (∗-fuzzy measure) which is a dynamic generalization of the classical measure theory. ∗-fuzzy measure theory has gotten by replacing the non-negative real range and the additivity of classical measures with fuzzy sets and triangular norms. Moreover, the ∗-fuzzy measure theory has been motivated by defining new additivity property using triangular norms. Our approach can be apply for decision making problems [8,9].

We have restricted fuzzy measurable functions and fuzzy integrable functions and defined important classes of function spaces named by <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>*. Moreover, we have got a norm on <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>* spaces and proved that <sup>∗</sup>-fuzzy (*L*+)*<sup>p</sup>* spaces are <sup>∗</sup>-fuzzy Banach spaces. Finally, we have proved Chebyshev's Inequality and Hölder's Inequality.

**Author Contributions:** Formal analysis, A.G. and R.M.; Methodology, A.G. and R.S.; Project administration, R.M.; Resources, A.G.; Supervision, R.S.; Writing—review & editing, R.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work of the third author on this paper was supported by grants APVV-18-0052 and by the project of Grant Agency of the Czech Republic (GACR) No. 18-06915S.

**Acknowledgments:** The authors are thankful to the anonymous referees for giving valuable comments and suggestions which helped to improve the final version of this paper.

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

#### **References**


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