Inequalities in Triangular Norm-Based ∗-fuzzy ( L + ) p Spaces

Abstract

In this article, we introduce the ∗-fuzzy ${\left(L^{+}\right)}^p$ spaces for $1\le p<\infty$ 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 ∗-fuzzy ${\left(L^{+}\right)}^p$ spaces.

Function spaces, especially $L^p$ spaces, play an important role in many parts in analysis. The impact of $L^p$ spaces follows from the fact that they offer a partial but useful generalization of the fundamental $L^1$ 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 [4,5,6,7] 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 $(X,\mathcal{C},\mu ,\ast )$ introduced in [3] which was derived from the idea of fuzzy and probabilistic metric spaces [5,6,7,10,11]. Using the concept of fuzzy measurable functions and fuzzy integrable functions we define a special class of function spaces named by ∗-fuzzy ${\left(L^{+}\right)}^p$. After some overview given in Section 2, Section 3 and Section 4 and devoted to the basic information concerning ∗-fuzzy measures and related integration, in Section 5 we define a norm on ∗-fuzzy ${\left(L^{+}\right)}^p$ spaces and show these spaces are complete ∗-fuzzy normed space in the sense of Cheng-Mordeson and others [12,13,14,15]. 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, $\mathcal{C}$ be a $\sigma$-algebra of subsets of X. Unless stated otherwise, all subsets of X are supposed to belong to $\mathcal{C}$. Here, we let $I=[0,1]$.

Definition 1.

([10,11]) A continuous triangular norm (shortly, a $ct$-norm) is a continuous binary operation ∗ from $I^2={[0,1]}^2$ to I such that

(a) 

$\varsigma \ast \tau =\tau \ast \varsigma$ and $\varsigma \ast (\tau \ast \upsilon )=(\varsigma \ast \tau )\ast \upsilon$ for all $\varsigma ,\tau ,\upsilon \in [0,1]$;

(b) 

$\varsigma \ast 1=\varsigma$ for all $\varsigma \in I$;

(c) 

$\varsigma \ast \tau \le \upsilon \ast \iota$ whenever $\varsigma \le \upsilon$ and $\tau \le \iota$ for all $\varsigma ,\tau ,\upsilon ,\iota \in I$.

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

• $\varsigma {\ast }_P\tau =\varsigma \tau$ (: the product t-norm);
• $\varsigma {\ast }_M\tau =min\{\varsigma ,\tau \}$ (: the minimum t-norm);
• $\varsigma {\ast }_L\tau =max\{\varsigma +\tau -1,0\}$ (: the Lukasiewicz t-norm);
• $$\varsigma {\ast }_H\tau =\left\{\begin{array}{c}\begin{array}{cc}0,& \mathrm{if}\varsigma =\tau =0,\\ \frac{1}{\frac{1}{\varsigma }+\frac{1}{\tau }-1},& \mathrm{otherwise},\end{array}\hfill \end{array}\right.$$

  (: the Hamacher product t-norm).

We define

$${\ast }_{i=1}^k{\varsigma }_i={\varsigma }_1\ast {\varsigma }_2\ast \cdots \ast {\varsigma }_k,$$

for $k\in \{2,3,\cdots \}$, which is well defined due to the associativity of the operation ∗. Moreover,

$${\ast }_{i=1}^{\infty }{\varsigma }_i=\underset{k\to \infty }{lim}{\ast }_{i=1}^k{\varsigma }_i,$$

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 $\mathcal{C}$ be a σ-algebra consisting of subsets of X. A fuzzy measure on $\mathcal{C}\times (0,\infty )$ is a fuzzy set $\mu :\mathcal{C}\times (0,\infty )\to I$ such that

(i) 

$\mu (\varnothing ,\tau )=1$, $\forall \tau \in (0,\infty )$;

(ii) 

if ${\mathsf{A}}_i\in \mathcal{C},i=1,2,\cdots$, are pairwise disjoint, then

$$\begin{array}{c}\hfill \mu ({\cup }_{i=1}^{\infty }{\mathsf{A}}_i,\tau )={\ast }_{i=1}^{\infty }\mu ({\mathsf{A}}_i,\tau ),\forall \tau \in (0,\infty ).\end{array}$$

Saying the ${\mathsf{A}}_i$ are pairwise disjoint means that ${\mathsf{A}}_i\cap {\mathsf{A}}_j=\varnothing$, if $i\ne j$.

Definition 2 is known as countable ∗-additivity. We say a fuzzy measure $\mu$ is finitely ∗-additive if, for any $n\in N$

$$\begin{array}{c}\hfill \mu ({\cup }_{i=1}^n{\mathsf{A}}_i,\tau )={\ast }_{i=1}^n\mu ({\mathsf{A}}_i,\tau ),\forall \tau \in (0,\infty ).\end{array}$$

whenever ${\mathsf{A}}_1,\cdots ,{\mathsf{A}}_n$ are in $\mathcal{C}$ and are pairwise disjoint. The quadruple $(X,\mathcal{C},\mu ,\ast )$ is called a ∗-fuzzy measure space (in short, ∗-FMS).

Example 1.

Let $(X,\mathcal{C},m)$ be a measurable space. Let $\ast ={\ast }_H$ and define

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

then $(X,\mathcal{C},{\mu }_0,\ast )$ is a ∗-FMS.

Example 2.

Let $(X,\mathcal{C},m)$ be a measurable space. Let $\ast ={\ast }_P$. Define

$$\begin{array}{c}\hfill {\mu }_0(\mathit{A},\tau )=e^{-{\displaystyle \frac{m\left(\mathit{A}\right)}{\tau }}},\forall \tau \in (0,\infty ).\end{array}$$

Then, ${\mu }_0$ is a ∗-FM on $\mathcal{C}\times (0,\infty )$.

2. ∗-Fuzzy Measurable Functions

Now, we review the concept of ∗-fuzzy normed spaces, for more details, we refer to the works in [12,13,14,15].

Definition 3.

Let X be a vector space, ∗ be a $ct$-norm and the fuzzy set N on $X\times (0,\infty )$ satisfies the following conditions for all $x,y\in X$ and $\tau ,\sigma \in (0,\infty )$,

(i) 

$N(x,\tau )>0$.

(ii) 

$N(x,\tau )=1\iff x=0$.

(iii) 

$N(\alpha x,\tau )=N\left(x,{\displaystyle \frac{\tau }{\left|\alpha \right|}}\right)$ for every $\alpha \ne 0$.

(iv) 

$N(x,\tau )\ast N(y,\sigma )\le N(x+y,\tau +\sigma )$.

(v) 

$N(x,.):(0,\infty )\to (0,1]$ is continuous.

(vi) 

$\underset{\tau \to \infty }{lim}N(x,\tau )=1$ and $\underset{\tau \to 0}{lim}N(x,\tau )=0$.

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

Assume that $(\mathbb{R},|.\left|\right)$ is a standard normed space, we define: $N(x,\tau )={\displaystyle \frac{\tau }{\tau +\left|x\right|}}$ with $\ast ={\ast }_P$, it is obvious $(\mathbb{R},N,{\ast }_P)$ is a ∗-fuzzy normed space.

Let $(X,N,\ast )$ be a ∗-fuzzy normed space. We define the open ball $\mathcal{B}(x,r,\tau )$ and the closed ball $\mathcal{B}[x,r,\tau ]$ with center $x\in X$ and radius $0<r<1$, $\tau >0$ as follows,

$$\begin{array}{c}\hfill \mathcal{B}(x,r,\tau )=\{y\in X:N(x-y,\tau )>1-r\},\end{array}$$

(1)

$$\begin{array}{c}\hfill \mathcal{B}[x,r,\tau ]=\{y\in X:N(x-y,\tau )\ge 1-r\}.\end{array}$$

(2)

Let $(X,N,\ast )$ be a ∗-fuzzy normed space. A set $E\subset X$ is said to be open if for each $x\in E$, there is $0<r_x<1$ and ${\tau }_x>0$ such that $\mathcal{B}(x,r_x,{\tau }_x)\subseteq E$. A set $F\subseteq X$ is said to be closed in X in case its complement $F^c=X-F$ is open in X.

Let $(X,N,\ast )$ be a ∗-fuzzy normed space. A subset $E\in X$ is said to be fuzzy bounded if there exist $\tau >0$ and $r\in (0,1)$ such that $N(x-y,\tau )>1-r$ for all $x,y\in E$.

Let $(X,N,\ast )$ be a ∗-fuzzy normed space. A sequence $\left\{x_n\right\}\subset X$ is fuzzy convergent to an $x\in X$ in ∗-fuzzy normed space $(X,N,\ast )$ if for any $\tau >0$ and $ϵ>0$ there exists a positive integer $N_{ϵ}>0$ such that $N(x_n-x,\tau )>1-ϵ$ whenever $n\ge N_{ϵ}$.

Now, we define ∗-fuzzy measurable functions.

Definition 4.

Let $(X,\mathcal{C})$ and $(Y,\mathcal{D})$ be ∗-fuzzy measurable spaces. A mapping $f:X\to Y$ is called ∗-fuzzy $(\mathcal{C},\mathcal{D})$-measurable if $f^{-1}\left(E\right)\in \mathcal{C}$ for all $E\in \mathcal{D}$. If X is any ∗-fuzzy normed space, the σ-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 ${\mathcal{B}}_X$.

3. ∗-Fuzzy Integration

In this section, we recall the concept of ∗-fuzzy integration by using fuzzy simple functions on the ∗-FMS $(X,\mathcal{C},\ast ,\mu )$ and add some new results.

Definition 5.

Let $(X,\mathcal{C},\ast ,\mu )$ be ∗-FMS, we define

$$\begin{array}{c}\hfill L_{+}=\left\{f:X\to [0,\infty )\mid f\mathit{is}\mathit{fuzzy}(\mathcal{C},{\mathcal{B}}_{\mathbb{R}})\text{-}\mathit{measurable}\mathit{function}\right\}.\end{array}$$

If ϕ is a simple fuzzy ($(\mathcal{C},{\mathcal{B}}_{\mathbb{R}})$-measurable) function in $L_{+}$ with standard representation $\varphi ={\displaystyle \sum _{i=1}^n}{a}_i{\chi }_{E_i}$, where $a_i>0$ and $E_i\in \mathcal{C}$ for $i=1,...,n$, and $E_i\bigcap E_j=\varnothing$ for $i\ne j$, we define the fuzzy integral of ϕ as

$$\begin{array}{c}\hfill {\int }_X\varphi \left(x\right)d\mu (x,\tau )={\int }_X\sum _{i=1}^n{a}_i{\chi }_{E_i}d\mu (x,\tau )={\ast }_{i=1}^n\mu \left(E_i,\frac{\tau }{a_i}\right).\end{array}$$

In [3], the authors have shown that, with respect to $\mu (A,\tau )$, $\mu$ satisfies the following statement;

(i)

$\mu :(A,.):(.,\infty )\to [0,1]$ is increasing and continuous.

(ii)

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

(iii)

$\underset{{\tau }_n⟶{\tau }_0}{lim}\left({\ast }_{i=1}^k\mu (A_i,{\tau }_n)\right)={\ast }_{i=1}^k\underset{{\tau }_n⟶{\tau }_0}{lim}\mu (A_i,{\tau }_n)$ for every $A_i\cap A_j=\varnothing$.

(iv)

$\underset{\tau ⟶0}{lim}\mu (E,\tau )=0$ and $\underset{\tau ⟶\infty }{lim}\mu (E,\tau )=1$.

(v)

$\underset{{\tau }_n⟶{\tau }_0}{lim}\underset{m⟶\infty }{lim}\left(\mu \left(E^m,{\displaystyle \frac{{\tau }_n}{a^m}}\right)\right)=\underset{m⟶\infty }{lim}\underset{{\tau }_n⟶{\tau }_0}{lim}\left(\mu \left(E^m,{\displaystyle \frac{{\tau }_n}{a^m}}\right)\right)$.

If $A\in \mathcal{C}$, then $\varphi {\chi }_A$ is also fuzzy simple function $\left(\varphi {\chi }_A={\displaystyle \sum _{i=1}^n}{a}_i{\chi }_{A\cap E_i}\right)$, and we define $\int \varphi \left(x\right)d\mu (x,\tau )$ to be $\int \varphi {\chi }_{A}d\mu (x,\tau )$.

Theorem 1

([3]). Let ϕ and ψ be simple functions in $L_{+}$. Then, we have

(i) 

${\int }_{X}0d\mu (x,\tau )=1$.

(ii) 

If $c\in (0,1]$ then ${\int }_X\left(c\varphi \right)\left(x\right)d\mu (x,\tau )\ge c{\int }_X\varphi \left(x\right)d\mu (x,\tau )$, and for $c\in [1,\infty )$ we have ${\int }_X\left(c\varphi \right)\left(x\right)d\mu (x,\tau )\le c{\int }_X\varphi \left(x\right)d\mu (x,\tau )$, $\forall \tau \in (0,\infty )$.

(iii) 

If $\varphi \le \psi$, then ${\int }_X\varphi \left(x\right)d\mu (x,\tau )\ge {\int }_X\psi \left(x\right)d\mu (x,\tau )$.

(iv) 

The map $A\to {\int }_A\varphi \left(x\right)d\mu (x,\tau )$ is a fuzzy measure on $\mathcal{C}$, $\forall \tau \in (0,\infty )$.

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

$$\begin{array}{c}\hfill \int (\varphi +\psi )\left(x\right)d\mu (x,\tau )\ge \left(\int \varphi (x)d\mu (x,\tau )\right)\ast \left(\int \psi (x)d\mu (x,\tau )\right).\end{array}$$

Proof. 

Let $\varphi$ and $\psi$ be fuzzy simple functions in $L_{+}$, then we have

$$\begin{array}{ccc}\hfill & & {\int }_X(\varphi +\psi )\left(x\right)d\mu (x,\tau ),\hfill \\ \hfill & =& {\int }_X\left(\left(\sum _{i=1}^n{a}_i{\chi }_{E_i}\left(x\right)\right)+\left(\sum _{j=1}^m{b}_j{\chi }_{F_j}\left(x\right)\right)\right)d\mu (x,\tau ),\hfill \\ \hfill & =& {\int }_X\left(\sum _{i,j}(a_i+b_j){\chi }_{E_i\cap F_j}\left(x\right)\right)d\mu (x,\tau ),\hfill \\ \hfill & =& {\ast }_{i=1}^n{\ast }_{j=1}^m\mu \left(\left(E_i\cap F_j\right),\frac{\tau }{(a_i+b_j)}\right).\hfill \end{array}$$

(3)

On the other hand,

$$\begin{array}{ccc}\hfill & & \left({\int }_X\varphi \left(x\right)d\mu (x,\tau )\ast {\int }_X\psi \left(x\right)d\mu (x,\tau )\right),\hfill \\ & =& \left({\int }_X\left(\sum _{i=1}^n{a}_i{\chi }_{E_i}\left(x\right)\right)d\mu (x,\tau )\right)\ast \left({\int }_X\left(\sum _{j=1}^m{b}_j{\chi }_{F_j}\left(x\right)\right)d\mu (x,\tau )\right),\hfill \\ & =& \left({\ast }_{i=1}^n{\ast }_{j=1}^m\mu \left(\left(E_i\cap F_j\right),\frac{\tau }{a_i}\right)\right)\ast \left({\ast }_{j=1}^m{\ast }_{i=1}^n\mu \left(\left(E_i\cap F_j\right),\frac{\tau }{b_j}\right)\right),\hfill \\ & =& {\ast }_{i=1}^n{\ast }_{j=1}^m\left(\mu \left(\left(E_i\cap F_j\right),\frac{\tau }{a_i}\right)\ast \mu \left(\left(E_i\cap F_j\right),\frac{\tau }{b_j}\right)\right),\hfill \\ & \le & {\ast }_{i=1}^n{\ast }_{j=1}^m\left(\mu \left(\left(E_i\cap F_j\right),\frac{\tau }{(a_i+b_j)}\right)\right).\hfill \end{array}$$

(4)

From (3) and (4), we get

$$\begin{array}{c}\hfill {\int }_X(\varphi +\psi )\left(x\right)d\mu (x,\tau )\ge \left({\int }_X\varphi \left(x\right)d\mu (x,\tau )\right)\ast \left({\int }_X\psi \left(x\right)d\mu (x,\tau )\right).\end{array}$$

 □

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

$$\begin{array}{ccc}& & {\int }_{X}f\left(x\right)d\mu (x,\tau )\hfill \\ & =& inf\left\{{\int }_X\varphi \left(x\right)d\mu (x,\tau )\mid 0\le \varphi \le f,\varphi \mathit{is}\mathit{fuzzy}\mathit{simple}\mathit{function}\right\}.\hfill \end{array}$$

By Theorem 1 (iii), the two definitions of $\int f$ 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 $\int f\ge \int g$ whenever $f\le g$, and $\int cf\ge c\int f$ for all $c\in (0,1]$ and $\int cf\le c\int f$ for all $c\in [1,\infty )$ and $\int (f+g)\ge (\int f)\ast (\int g)$.

Definition 7.

If $f\in L_{+}$, we say that f is fuzzy integrable if $\int fd\mu (x,\tau )>0$ for each $\tau >0$. Let $(X,\mathcal{C},\mu ,\ast )$ be a ∗-FMS. We define

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

Theorem 3

([3]). (The fundamental convergence theorem). Let $(X,\mathcal{C},\mu ,\ast )$ be a ∗-FMS. Let $f_n$ be a sequence in $L^{+}$ such that $f_n⟶f$ almost everywhere, then $f\in L^{+}$ and $\int f=\underset{n⟶\infty }{lim}\int f_n$.

∗-Fuzzy $L^{+}$ Spaces

Here, we are ready to show that every $L^{+}$ is a ∗-fuzzy normed space. It is clear if we define

$$L:=\{f:X⟶\mathbb{R},f\mathrm{is}\mathrm{fuzzy}\mathrm{measurable}\mathrm{function}\},$$

then ${(L,+,.)}_{\mathbb{R}}$ is a vector space. Moreover, in [3] the authors proved that if $f,g\in L^{+}$, then $|f-g|\in L^{+}$. Using definition L and $L^{+}$ we can show $L^{+}\subseteq L$. In $L^{+}$ we define $f\le g$ if and only if $f\left(x\right)\le g\left(x\right)$ and so $(L^{+},\le )$ is a cone.

Note. Recall that, due to the continuity of t-norm ∗, for any systems ${\left\{a_n\right\}}_{n\in \mathbb{N}}$ and ${\left\{b_n\right\}}_{n\in \mathbb{N}}$ of elements form I we have $inf\{a_n\ast b_n\}=inf\left\{a_n\right\}\ast inf\left\{b_n\right\}$.

In the next theorem we define a fuzzy norm on $L^{+}$ and prove that $(L^{+},N,\ast )$ is a ∗-fuzzy normed space.

Theorem 4.

Let $N:L^{+}\times (0,\infty )⟶(0,1]$ be a fuzzy set, such that $N(f,\tau )=\int fd\mu (x,\tau )$, then $(L^{+},N,\ast )$ is a ∗-fuzzy normed space.

Proof. 

(FN1)

$N(f,\tau )=\int fd\mu (x,\tau )>0$.

(FN2)

By theorem 4.5 of [3] we have

$$N(f,\tau )=1⟺\int fd\mu (x,\tau )=1⟺f=0$$

almost everywhere.

(FN3)

Let $f=\varphi ={\displaystyle \sum _{i=1}^n}{a}_i{\chi }_{E_i}$ and $c>0$ so,

$$\begin{array}{cc}\hfill N(c\varphi ,\tau )=& \int c\varphi d\mu (x,\tau ),\hfill \\ \hfill =& \int {\displaystyle \sum _{i=1}^n}{a}_i{\chi }_{E_i}d\mu (x,\tau ),\hfill \\ \hfill =& {\ast }_{i=1}^n\mu \left(E_i,{\displaystyle \frac{\tau }{c{a}_i}}\right).\hfill \end{array}$$

(5)

On the other hand,

$$\begin{array}{cc}\hfill N\left(\varphi ,{\displaystyle \frac{\tau }{c}}\right)& =\int \varphi d\mu \left(x,{\displaystyle \frac{\tau }{c}}\right),\hfill \\ \hfill & =\int \sum _{i=1}^n{a}_i{\chi }_{E_i}d\mu \left(x,{\displaystyle \frac{\tau }{c}}\right),\hfill \\ \hfill & ={\ast }_{i=1}^n\mu \left(E_i,{\displaystyle \frac{\tau }{c{a}_i}}\right).\hfill \end{array}$$

(6)

From (5) and (6) we conclude that

$$\begin{array}{c}\hfill N(c\varphi ,\tau )=N\left(\varphi ,{\displaystyle \frac{\tau }{c}}\right).\end{array}$$

(7)

Now, if $f\in L^{+}$ we have $\left\{{\varphi }_n\right\}\subseteq L^{+}$ such that ${\varphi }_n\uparrow f$, then $c{\varphi }_n\uparrow cf$ so

$$\begin{array}{c}\hfill \int c{\varphi }_{n}d\mu (x,\tau )\downarrow \int cfd\mu (x,\tau ).\end{array}$$

By (7), we have $\int c{\varphi }_{n}d\mu (x,\tau )=\int {\varphi }_{n}d\mu (x,{\displaystyle \frac{\tau }{c}})$, and so

$$\begin{array}{c}\hfill \int {\varphi }_{n}d\mu (x,{\displaystyle \frac{\tau }{c}})\downarrow \int cfd\mu (x,\tau ).\end{array}$$

(8)

On the other hand,

$$\begin{array}{c}\hfill \int {\varphi }_{n}d\mu (x,{\displaystyle \frac{\tau }{c}})\downarrow \int fd\mu (x,{\displaystyle \frac{\tau }{c}}),\end{array}$$

(9)

by (8) and (9) we have,

$$\begin{array}{c}\hfill \int cfd\mu (x,\tau )=\int fd\mu (x,{\displaystyle \frac{\tau }{c}}),\\ \hfill N(cf,\tau )=N(f,{\displaystyle \frac{\tau }{c}}).\end{array}$$

(FN4)

Let $f={\displaystyle \sum _{i=1}^m}{a}_i{\chi }_{E_i}$, $g={\displaystyle \sum _{j=1}^n}{b}_j{\chi }_{F_j}$ then,

$$\begin{array}{cc}\hfill N(\varphi +\psi ,s+\tau )=& \int (\varphi +\psi )d\mu (x,\tau +s),\hfill \\ \hfill =& \int \sum _{i,j}(a_i+b_j){\chi }_{E_i\cap F_j}d\mu (x,\tau +s),\hfill \\ \hfill =& {\ast }_{i,j}\mu \left(E_i\cap F_j,{\displaystyle \frac{\tau +s}{a_i+b_j}}\right).\hfill \end{array}$$

On the other hand

$$\begin{array}{cc}\hfill N(\varphi ,s)\ast N(\psi ,\tau )=& \left(\int \varphi d\mu (x,s)\right)\ast \left(\int \psi d\mu (x,\tau )\right),\hfill \\ \hfill =& \left(\int \sum _{i,j}{a}_i{\chi }_{E_i\cap F_j}d\mu (x,s)\right)\ast \left(\int \sum _{i,j}{b}_j{\chi }_{E_i\cap F_j}d\mu (x,\tau )\right),\hfill \\ \hfill =& \left({\ast }_{i,j}\mu (E_i\cap F_j,{\displaystyle \frac{s}{a_i}})\right)\ast \left({\ast }_{i,j}\mu (E_i\cap F_j,{\displaystyle \frac{\tau }{b_j}})\right),\hfill \\ \hfill =& {\ast }_{i,j}\left(\mu (E_i\cap F_j,{\displaystyle \frac{s}{a_i}})\ast \mu ((E_i\cap F_j,{\displaystyle \frac{\tau }{b_j}})\right),\hfill \\ \hfill \le & {\ast }_{i,j}\left(min\left\{\mu (E_i\cap F_j,{\displaystyle \frac{s}{a_i}}),\mu ((E_i\cap F_j,{\displaystyle \frac{\tau }{b_j}})\right\}\right).\hfill \end{array}$$

(10)

Now, we assume ${\displaystyle \frac{s}{a_i}}<{\displaystyle \frac{\tau }{b_j}}$. From (10), we conclude

$$\begin{array}{c}\hfill N(\varphi ,s)\ast N(\psi ,\tau )\le {\ast }_{i,j}\mu \left(E_i\cap F_j,{\displaystyle \frac{s}{a_i}}\right).\end{array}$$

(11)

Again, from ${\displaystyle \frac{s}{a_i}}<{\displaystyle \frac{\tau }{b_j}},$ we get ${\displaystyle \frac{s}{a_i}}<{\displaystyle \frac{\tau +s}{a_i+b_j}}$ because

$$\begin{array}{c}\hfill bjs<a_i\tau ,\end{array}$$

then

$$\begin{array}{c}\hfill a_{i}s+b_{j}s<a_{i}s+a_i\tau ,\end{array}$$

and

$$\begin{array}{c}\hfill \left(a_i+b_j\right)s<a_i\left(\tau +s\right),\end{array}$$

and so

$$\begin{array}{c}\hfill {\displaystyle \frac{s}{a_i}}<{\displaystyle \frac{\tau +s}{a_i+b_j}}.\end{array}$$

Therefore, from (11) we have

$$\begin{array}{c}\hfill N(\varphi ,s)\ast N(\psi ,\tau )\le {\ast }_{i,j}\mu \left(E_i\cap F_j,{\displaystyle \frac{s}{a_i}}\right),\end{array}$$

(12)

and

$$\begin{array}{c}\hfill {\ast }_{i,j}\mu \left(E_i\cap F_j,{\displaystyle \frac{s}{a_i}}\right)\le {\ast }_{i,j}\mu \left(E_i\cap F_j,{\displaystyle \frac{\tau +s}{a_i+b_j}}\right).\end{array}$$

(13)

From (12) and (13) we have

$$\begin{array}{c}\hfill N(\varphi ,s)\ast N(\psi ,\tau )\le {\ast }_{i,j}\mu \left(E_i\cap F_j,{\displaystyle \frac{\tau +s}{a_i+b_j}}\right),\hfill \\ \hfill =N\left(\varphi +\psi ,s+\tau \right).\end{array}$$

Now let $f,g\in L^{+}$, then there exist $\left\{{\varphi }_n\right\}\subseteq L^{+}$ such that ${\varphi }_n\uparrow f$. Similarly, there exist $\left\{{\psi }_n\right\}\subseteq L^{+}$ such that ${\psi }_n\uparrow g$, and ${\varphi }_n+{\psi }_n\uparrow f+g$, then

$$\begin{array}{c}\hfill inf\left\{\int \left({\varphi }_n+{\psi }_n\right)d\mu (x,\tau +s)\right\}=\int \left(f+g\right)d\mu (x,\tau +s).\end{array}$$

Also according to (12), we get

$$\begin{array}{c}\hfill \int \left({\varphi }_n+{\psi }_n\right)d\mu (x,\tau +s)\ge \int {\varphi }_{n}d\mu (x,s)\ast \int {\psi }_{n}d\mu (x,\tau ),\end{array}$$

and

$$\begin{array}{ccc}& & \int \left(f+g\right)d\mu (x,\tau +s)=inf\left\{\int ({\varphi }_n+{\psi }_n)d\mu (x,\tau +s)\right\}\hfill \\ & \ge & inf\left\{\int {\varphi }_{n}d\mu (x,s)\ast \int {\psi }_{n}d\mu (x,\tau )\right\},\hfill \\ & \ge & inf\left\{\int {\varphi }_{n}d\mu (x,s)\right\}\ast inf\int {\psi }_{n}d\mu (x,\tau )\hfill \\ & =& \int fd\mu (x,s)\ast \int gd\mu (x,\tau ),\hfill \end{array}$$

then

$$\begin{array}{c}\hfill \int (f+g)d\mu (x,\tau +s)\ge \int fd\mu (x,s)\ast \int gd\mu (x,\tau ).\end{array}$$

(FN5)

Let $f={\displaystyle \sum _{i=1}^k}{a}_i{\chi }_{E_i}$, then

$$\begin{array}{cc}\hfill N(f,{\tau }_n)=& \int \sum _{i=1}^k{a}_i{\chi }_{E_i}d\mu (x,{\tau }_n),\hfill \\ \hfill =& {\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{{\tau }_n}{a_i}}\right),\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}N(f,{\tau }_n)=lim{\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{{\tau }_n}{a_i}}\right).\end{array}$$

According to Definition 5 (iii), we get

$$\begin{array}{cc}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}N(f,{\tau }_n)=& \underset{{\tau }_n⟶{\tau }_0}{lim}{\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{{\tau }_n}{a_i}}\right),\hfill \\ \hfill =& {\ast }_{i=1}^k\underset{{\tau }_n⟶{\tau }_0}{lim}\left(E_i,{\displaystyle \frac{{\tau }_n}{a_i}}\right),\hfill \end{array}$$

and by Definition 5 (i),

$$\begin{array}{cc}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}N(f,{\tau }_n)=& {\ast }_{i=1}^k\underset{{\tau }_n⟶{\tau }_0}{lim}\left(E_i,{\displaystyle \frac{{\tau }_n}{a_i}}\right),\hfill \\ \hfill =& {\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{{\tau }_0}{a_i}}\right),\hfill \\ \hfill =& \int fd\mu (x,{\tau }_0),\hfill \\ \hfill =& N(f,{\tau }_0).\hfill \end{array}$$

Now, let $f\in L^{+}$, then

$$\begin{array}{cc}\hfill N(f,{\tau }_n)=& \int fd\mu (x,{\tau }_n),\hfill \\ \hfill =& inf\left\{\int {\varphi }_{m}d\mu (x,{\tau }_n)|{\varphi }_m\uparrow f\right\},\hfill \\ \hfill =& \underset{m⟶\infty }{lim}\int {\varphi }_{m}d\mu (x,{\tau }_n).\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}N(f,{\tau }_n)=& \underset{{\tau }_n⟶{\tau }_0}{lim}\underset{m⟶\infty }{lim}\int {\varphi }_{m}d\mu (x,{\tau }_n),\hfill \\ \hfill =& \underset{{\tau }_n⟶{\tau }_0}{lim}\underset{m⟶\infty }{lim}\int \sum _{i=1}^k{a}_i^m{\chi }_{E_i^m}d\mu (x,{\tau }_n),\hfill \\ \hfill =& \underset{{\tau }_n⟶{\tau }_0}{lim}\underset{m⟶\infty }{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{{\tau }_n}{a_i^m}}\right).\hfill \end{array}$$

According to Definition 5 (v), we get

$$\begin{array}{cc}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}N(f,{\tau }_n)=& \underset{{\tau }_n⟶{\tau }_0}{lim}\underset{m⟶\infty }{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{{\tau }_n}{a_i^m}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}\underset{{\tau }_n⟶{\tau }_0}{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{{\tau }_n}{a_i^m}}\right),\hfill \end{array}$$

and by Definition 5 (iii), we get

$$\begin{array}{cc}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}N(f,{\tau }_n)=& \underset{m⟶\infty }{lim}\underset{{\tau }_n⟶{\tau }_0}{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{{\tau }_n}{a_i^m}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}{\ast }_{i=1}^k\underset{{\tau }_n⟶{\tau }_0}{lim}\mu \left(E_i^m,{\displaystyle \frac{{\tau }_n}{a_i^m}}\right).\hfill \end{array}$$

Using Definition 5 (i), we get

$$\begin{array}{cc}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}N(f,{\tau }_n)=& \underset{m⟶\infty }{lim}{\ast }_{i=1}^k\underset{{\tau }_n⟶{\tau }_0}{lim}\mu \left(E_i^m,{\displaystyle \frac{{\tau }_n}{a_i^m}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{{\tau }_0}{a_i^m}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}\int {\varphi }_{m}d\mu (x,{\tau }_0),\hfill \\ \hfill =& inf\left\{\int {\varphi }_{m}d\mu (x,{\tau }_0)\right\},\hfill \\ \hfill =& \int fd\mu (x,{\tau }_0),\hfill \\ \hfill =& N(f,{\tau }_0).\hfill \end{array}$$

(FN6)

Let $f={\displaystyle \sum _{i=1}^k}{a}_i{\chi }_{E_i}$, then

$$\begin{array}{cc}\hfill N(f,\tau )=& \int fd\mu (x,\tau ),\hfill \\ \hfill =& \int \sum _{i=1}^n{a}_i{\chi }_{E_i}d\mu (x,\tau ),\hfill \\ \hfill =& {\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{\tau }{a_i}}\right).\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{\tau ⟶{\tau }_0}{lim}N(f,\tau )=\underset{\tau ⟶{\tau }_0}{lim}{\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{\tau }{a_i}}\right).\end{array}$$

According to Definition 5 (iii), we have

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}N(f,\tau )=& \underset{\tau ⟶0}{lim}{\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{\tau }{a_i}}\right),\hfill \\ \hfill =& {\ast }_{i=1}^k\underset{\tau ⟶0}{lim}\mu \left(E_i,{\displaystyle \frac{\tau }{a_i}}\right),\hfill \end{array}$$

and by Definition 5 (iv),

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}N(f,\tau )=& {\ast }_{i=1}^k\underset{\tau ⟶0}{lim}\mu \left(E_i,{\displaystyle \frac{\tau }{a_i}}\right),\hfill \\ \hfill =& {\ast }_{i=1}^{k}0,\hfill \\ \hfill =& 0.\hfill \end{array}$$

Now let $f\in L^{+}$, so

$$\begin{array}{cc}\hfill N(f,\tau )=& \int fd\mu (x,\tau )=inf\left\{\int {\varphi }_{m}d\mu (x,\tau )\right\},\hfill \\ \hfill =& \underset{m⟶\infty }{lim}\left\{\int {\varphi }_{m}d\mu (x,\tau )\right\},\hfill \\ \hfill =& \underset{m⟶\infty }{lim}\left\{N({\varphi }_m,\tau )\right\}.\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}N(f,\tau )=& \underset{\tau ⟶0}{lim}\underset{m⟶\infty }{lim}\left\{N\right({\varphi }_m,\tau \left)\right\},\hfill \\ \hfill =& \underset{\tau ⟶0}{lim}\underset{m⟶\infty }{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{\tau }{a_i^m}}\right).\hfill \end{array}$$

According to Definition 5 (v), we get

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}N(f,\tau )=& \underset{\tau ⟶0}{lim}\underset{m⟶\infty }{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{\tau }{a_i^m}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}\underset{\tau ⟶0}{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{\tau }{a_i^m}}\right),\hfill \end{array}$$

and from Definition 5 (iii), we get

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}N(f,\tau )=& \underset{m⟶\infty }{lim}\underset{\tau ⟶0}{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{\tau }{a_i^m}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}{\ast }_{i=1}^k\underset{\tau ⟶0}{lim}\mu \left(E_i^m,{\displaystyle \frac{\tau }{a_i^m}}\right).\hfill \end{array}$$

From Definition 5 (iv), we get

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}N(f,\tau )=& \underset{m⟶\infty }{lim}{\ast }_{i=1}^k\underset{\tau ⟶0}{lim}\mu \left(E_i^m,{\displaystyle \frac{\tau }{a_i^m}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}{\ast }_{i=1}^{k}0,\hfill \\ \hfill =& 0.\hfill \end{array}$$

Similarly,

$$\begin{array}{c}\hfill \underset{\tau ⟶\infty }{lim}N(f,\tau )=1.\end{array}$$

 □

We have proved $(L^{+},N,\ast )$ is a ∗-fuzzy normed space. Define $M:L^{+}\times L^{+}\times (0,\infty )⟶(0,1]$ by

$$\begin{array}{c}\hfill M(f,g,\tau )=N(|f-g|,\tau )=\int |f-g|d\mu (x,\tau ),\end{array}$$

then M is a fuzzy metric on $L^{+}$ and $(L^{+},M,\ast )$ is called the ∗-fuzzy metric induced by the ∗-fuzzy normed space $(L^{+},N,\ast )$.

Theorem 5

([3]). If $f\in L^{+}$ and $\epsilon >0$, there is an integrable fuzzy simple function $\varphi ={\displaystyle \sum _{j=1}^n}{a}_j{\chi }_{E_J}$ such that $\int |f-\varphi |d\mu (x,\tau )>1-\epsilon$ for each $\tau >0$ (that is, the integrable simple functions are dense in $L^{+}$).

Now, we show $L^{+}$ is a complete space.

Theorem 6.

$L^{+}$ is a ∗-fuzzy Banach space.

Proof. 

Let $\left\{f_n\right\}\subseteq L^{+}$ is a Cauchy sequence, then $\left\{f_n\left(x\right)\right\}\subset {\mathbb{R}}^{+}$ is a Cauchy sequence for every $x\in X$ and $\mathbb{R}$ is complete so there exist $y\in \mathbb{R}$ such that $f_n\left(x\right)⟶y$. We get $f:X⟶\mathbb{R},f\left(x\right)=y$ according to corollary 3.16 [3], f is fuzzy measurable so $f\in L_{+}$ and according to Theorem (3), $f\in L^{+}$ so, $\underset{n⟶\infty }{lim}{f}_n\left(x\right)=f\left(x\right)$ almost everywhere or $\underset{n⟶\infty }{lim}{f}_n=f$. □

4. ∗-Fuzzy ${\left({\mathit{L}}^{+}\right)}^{\mathit{p}}$ 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 $(X,\mathcal{C},\ast )$ be a ∗-fuzzy measure space. We define

$$\begin{array}{ccc}& & {\left(L^{+}\right)}^p\hfill \\ & =& \left\{f:X⟶{\mathbb{R}}^{+}\mathit{in}\mathit{which}f\mathit{is}\mathit{fuzzy}\mathit{measurable}\mathit{function}\mathit{and}\int f^{p}d\mu (x,\tau )>0,p\ge 1\right\}.\hfill \end{array}$$

There is an order on $({\left(L^{+}\right)}^p,\le )$ such that $f,g\in {\left(L^{+}\right)}^p$ we have $f\le g$ if and only if $f\left(x\right)\le g\left(x\right)$. Furthermore, if $f,g\in {\left(L^{+}\right)}^p$ then $|f-g|\in {\left(L^{+}\right)}^p$, and ${|f-g|}^p\le f^p$ or $g^p$ hence ${\int |f-g|}^{p}d\mu (x,\tau )\ge max[\int f^{p}d\mu (x,\tau ),\int g^{p}d\mu (x,\tau )]$.

In the next theorem we prove ∗-fuzzy ${\left(L^{+}\right)}^p$ is a ∗- fuzzy normed space.

Theorem 7.

Define $N_p:{\left(L^{+}\right)}^p\times (0,\infty )⟶(0,1]$ by $N_p(f,\tau )=\int f^{p}d\mu (x,\tau )$ then $({\left(L^{+}\right)}^p,N_p,\ast )$ is a ∗- fuzzy normed space.

Proof. 

(FN1)

$N_p(f,\tau )=\int f^{p}d\mu (x,\tau )>0$.

(FN2)

By theorem 4.5 of [3] we have,

$N_p(f,\tau )=1⟺\int f^{p}d\mu (x,\tau )=1⟺f^p=0⟺f=0$, almost everywhere.

(FN3)

Let $f=\varphi ={\displaystyle \sum _{i=1}^n}{a}_i{\chi }_{E_i}$ then,

$$\begin{array}{cc}\hfill N_p(c\varphi ,\tau )=& \int {\left(c\varphi \right)}^{p}d\mu ,\hfill \\ \hfill =& \int {\left(\sum _{i=1}^{n}c{a}_i{\chi }_{E_i}\right)}^{p}d\mu ,\hfill \\ \hfill =& {\ast }_{i=1}^n\mu \left(E_i,{\displaystyle \frac{\tau }{c^p{a}_i^p}}\right).\hfill \end{array}$$

(14)

On the other hand,

$$\begin{array}{cc}\hfill N_p(\varphi ,{\displaystyle \frac{\tau }{c^p}})=& \int {\varphi }^{p}d\mu (x,{\displaystyle \frac{\tau }{c^p}}),\hfill \\ \hfill =& \int {\left(\sum _{i=1}^n{a}_i{\chi }_{E_i}\right)}^{p}d\mu (x,{\displaystyle \frac{\tau }{c^p}}),\hfill \\ \hfill =& \int \sum _{i=1}^n{a}_i^p{\chi }_{E_i}d\mu (x,{\displaystyle \frac{\tau }{c^p}}),\hfill \\ \hfill =& {\ast }_{i=1}^n\mu \left(E_i,{\displaystyle \frac{\tau }{c^p{a}_i^p}}\right).\hfill \end{array}$$

(15)

From (14) and (15) we conclude that

$$\begin{array}{c}\hfill N_p(cf,\tau )=N_p\left(f,{\displaystyle \frac{\tau }{c}}\right).\end{array}$$

Now let $f\in {\left(L^{+}\right)}^p$, then we have

$$\begin{array}{c}\hfill N_p(cf,\tau )=\int {\left(cf\right)}^{p}d\mu (x,\tau )=inf\left\{\int {\left(c{\varphi }_n\right)}^{p}d\mu (x,\tau ):{\left(c{\varphi }_n\right)}^p\uparrow {\left(cf\right)}^p\right\}.\end{array}$$

(16)

On the other hand,

$$\begin{array}{ccc}\hfill & & N_p(f,\frac{\tau }{c})=\int f^{p}d\mu (x,\frac{\tau }{c})\hfill \\ \hfill & =& inf\left\{\int {\varphi }_n^{p}d\mu (x,\frac{\tau }{c}):{\varphi }_n^p\uparrow f_n^p\right\}.\hfill \end{array}$$

(17)

From (14) and (15) we get

$$\begin{array}{c}\hfill \int {\left(c{\varphi }_n\right)}^{p}d\mu (x,\tau )=N_p(c{\varphi }_n,\tau )=N_p({\varphi }_n,{\displaystyle \frac{\tau }{c}})=\int {\varphi }_n^{p}d\mu (x,{\displaystyle \frac{\tau }{c}}).\end{array}$$

Using (16) and (17) we get

$$\begin{array}{c}\hfill N_p(cf,\tau )=N_p(f,{\displaystyle \frac{\tau }{c}}).\end{array}$$

(FN4)

Let $f=\varphi$ and $g=\psi$ be simple functions. Then,

$$\begin{array}{cc}\hfill N_p\left(\varphi +\psi ,s+\tau \right)=& N_p\left(\sum _{i=1}^n{a}_i{\chi }_{E_i}+\sum _{j=1}^m{b}_j{\chi }_{F_j},s+\tau \right),\hfill \\ \hfill =& N_p\left(\sum _{i,j}(a_i+b_j){\chi }_{E_i\cap F_j},s+\tau \right),\hfill \\ \hfill =& \int {\left(\sum _{i,j}(a_i+b_j){\chi }_{E_i\cap F_j}\right)}^{p}d\mu (x,s+\tau ),\hfill \\ \hfill =& \int \sum _{i,j}{(a_i+b_j)}^p{\chi }_{E_i\cap F_j}d\mu (x,s+\tau ),\hfill \\ \hfill =& {\ast }_{i,j}\mu \left(E_i\cap F_j,{\displaystyle \frac{s+\tau }{{(a_i+b_j)}^p}}\right).\hfill \end{array}$$

(18)

On the other hand,

$$\begin{array}{cc}\hfill N_p(\varphi ,s)\ast N_p(\psi ,\tau )=& \left(\int {\varphi }^{p}d\mu (x,s)\right)\ast \left(\int {\psi }^{p}d\mu (x,\tau )\right),\hfill \\ \hfill =& \left(\int {\left(\sum _{i=1}^n{a}_i{\chi }_{E_i\cap F_j}\right)}^{p}d\mu (x,s)\right)\ast \left(\int {\left(\sum _{j=1}^m{b}_j{\chi }_{E_i\cap F_j}\right)}^{p}d\mu (x,\tau )\right),\hfill \\ \hfill =& \left(\int \sum _{i=}^n{a}_i^p{\chi }_{E_i\cap F_j}d\mu (x,s)\right)\ast \left(\int \sum _{j=1}^m{b}_j^p{\chi }_{E_i\cap F_j}d\mu (x,\tau )\right),\hfill \\ \hfill =& \left({\ast }_{i,j}\mu \left(E_i\cap F_j,{\displaystyle \frac{s}{a_i^p}}\right)\right)\ast \left({\ast }_{i,j}\mu \left(E_i\cap F_j,{\displaystyle \frac{\tau }{b_j^p}}\right)\right),\hfill \\ \hfill =& {\ast }_{i,j}\left(\mu \left(E_i\cap F_j,{\displaystyle \frac{s}{a_i^p}}\right)\ast \mu \left(E_i\cap F_j,{\displaystyle \frac{\tau }{b_j^p}}\right)\right),\hfill \\ \hfill \le & {\ast }_{i,j}\left(\mu \left(E_i\cap F_j,min\left\{{\displaystyle \frac{s}{a_i^p}},{\displaystyle \frac{\tau }{b_j^p}}\right\}\right)\right)\hfill \\ \hfill \le & {\ast }_{i,j}\mu \left(E_i\cap F_j,{\displaystyle \frac{s+\tau }{{(a_i+b_j)}^p}}\right).\hfill \end{array}$$

(19)

(FN5)

Let $f={\displaystyle \sum _{i=1}^k}{a}_i{\chi }_{E_i}$, then

$$\begin{array}{cc}\hfill N_p(f,{\tau }_n)=& \int {\left(\sum _{i=1}^k{a}_i{\chi }_{E_i}\right)}^{p}d\mu (x,{\tau }_n),\hfill \\ \hfill =& {\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{{\tau }_n}{{\left(a_i\right)}^p}}\right),\hfill \end{array}$$

and so

$$\begin{array}{c}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}{N}_p(f,{\tau }_n)=lim{\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{{\tau }_n}{{\left(a_i\right)}^p}}\right).\end{array}$$

Using Definition 5 (iii), we get

$$\begin{array}{cc}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}{N}_p(f,{\tau }_n)=& \underset{{\tau }_n⟶{\tau }_0}{lim}{\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{{\tau }_n}{{\left(a_i\right)}^p}}\right)\hfill \\ \hfill =& {\ast }_{i=1}^k\underset{{\tau }_n⟶{\tau }_0}{lim}\mu \left(E_i,{\displaystyle \frac{{\tau }_n}{{\left(a_i\right)}^p}}\right),\hfill \end{array}$$

and according to Definition 5 (i),

$$\begin{array}{cc}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}{N}_p(f,{\tau }_n)=& {\ast }_{i=1}^k\underset{{\tau }_n⟶{\tau }_0}{lim}\mu \left(E_i,{\displaystyle \frac{{\tau }_n}{{\left(a_i\right)}^p}}\right)\hfill \\ \hfill =& {\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{{\tau }_0}{{\left(a_i\right)}^p}}\right)\hfill \\ \hfill =& \int f^{p}d\mu (x,{\tau }_0),\hfill \\ \hfill =& N_p(f,{\tau }_0).\hfill \end{array}$$

Now let $f\in {\left(L^{+}\right)}^p$, we have

$$\begin{array}{cc}\hfill N_p(f,{\tau }_n)=& \int f^{p}d\mu (x,{\tau }_n)\hfill \\ \hfill =& inf\left\{\int {\left({\varphi }_m\right)}^{p}d\mu (x,{\tau }_n)|{\varphi }_m\uparrow f\right\}\hfill \\ \hfill =& \underset{m⟶\infty }{lim}\int {\left({\varphi }_m\right)}^{p}d\mu (x,{\tau }_n).\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill =\underset{{\tau }_n⟶{\tau }_0}{lim}{N}_p(f,{\tau }_n)=& \underset{{\tau }_n⟶{\tau }_0}{lim}\underset{m⟶\infty }{lim}\int {\left({\varphi }_m\right)}^{p}d\mu (x,{\tau }_n),\hfill \\ \hfill =& \underset{{\tau }_n⟶{\tau }_0}{lim}\underset{m⟶\infty }{lim}\int \left(\sum _{i=1}^k{\left(a_i^m{\chi }_{E_i^m}\right)}^{p}d\mu (x,{\tau }_n)\right)\hfill \\ \hfill =& \underset{{\tau }_n⟶{\tau }_0}{lim}\underset{m⟶\infty }{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{{\tau }_n}{{\left(a_i^m\right)}^p}}\right).\hfill \end{array}$$

Using Definition 5 (v), we get

$$\begin{array}{cc}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}{N}_p(f,{\tau }_n)=& \underset{{\tau }_n⟶{\tau }_0}{lim}\underset{m⟶\infty }{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{{\tau }_n}{{\left(a_i^m\right)}^p}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}\underset{{\tau }_n⟶{\tau }_0}{lim}{\ast }_{i=1}^k\mu {\left(E_i^m,{\displaystyle \frac{{\tau }_n}{{\left(a_i^m\right)}^p}}\right)}^{\prime }\hfill \end{array}$$

and according to Definition 5 (iii)

$$\begin{array}{cc}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}{N}_p(f,{\tau }_n)=& \underset{m⟶\infty }{lim}\underset{{\tau }_n⟶{\tau }_0}{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{{\tau }_n}{{\left(a_i^m\right)}^p}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}{\ast }_{i=1}^k\underset{{\tau }_n⟶{\tau }_0}{lim}\mu \left(E_i^m,{\displaystyle \frac{{\tau }_n}{{\left(a_i^m\right)}^p}}\right).\hfill \end{array}$$

By Definition 5 (i), we have

$$\begin{array}{cc}\hfill \underset{{\tau }_n⟶{\tau }_0}{lim}{N}_p(f,{\tau }_n)=& \underset{m⟶\infty }{lim}{\ast }_{i=1}^k\underset{{\tau }_n⟶{\tau }_0}{lim}\mu \left(E_i^m,{\displaystyle \frac{{\tau }_n}{{\left(a_i^m\right)}^p}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{{\tau }_0}{{\left(a_i^m\right)}^p}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}\int {\left({\varphi }_m\right)}^{p}d\mu (x,{\tau }_0),\hfill \\ \hfill =& inf\left\{\int {\left({\varphi }_m\right)}^{p}d\mu (x,{\tau }_0)\right\},\hfill \\ \hfill =& \int f^{p}d\mu (x,{\tau }_0),\hfill \\ \hfill =& N_p(f,{\tau }_0).\hfill \end{array}$$

(FN6)

Let $f={\displaystyle \sum _{i=1}^k}{a}_i{\chi }_{E_i}$, then

$$\begin{array}{cc}\hfill N_p(f,\tau )=& \int f^{p}d\mu (x,\tau ),\hfill \\ \hfill =& \int {\left(\sum _{i=1}^k{a}_i{\chi }_{E_i}\right)}^{p}d\mu (x,\tau ),\hfill \\ \hfill =& {\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{\tau }{{\left(a_i\right)}^p}}\right),\hfill \end{array}$$

and so

$$\begin{array}{c}\hfill \underset{\tau ⟶{\tau }_0}{lim}{N}_p(f,\tau )=\underset{\tau ⟶{\tau }_0}{lim}{\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{\tau }{{\left(a_i\right)}^p}}\right).\end{array}$$

Using Definition 5 (iii),

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}{N}_p(f,\tau )=& \underset{\tau ⟶0}{lim}{\ast }_{i=1}^k\mu \left(E_i,{\displaystyle \frac{\tau }{{\left(a_i\right)}^p}}\right),\hfill \\ \hfill =& {\ast }_{i=1}^k\underset{\tau ⟶0}{lim}\mu \left(E_i,{\displaystyle \frac{\tau }{{\left(a_i\right)}^p}}\right)\hfill \end{array}$$

and by Definition 5 (iv), we have

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}{N}_p(f,\tau )=& {\ast }_{i=1}^k\underset{\tau ⟶0}{lim}\mu \left(E_i,{\displaystyle \frac{\tau }{{\left(a_i\right)}^p}}\right),\hfill \\ \hfill =& {\ast }_{i=1}^{k}0,\hfill \\ \hfill =& 0.\hfill \end{array}$$

Now, let $f\in {\left(L^{+}\right)}^p$, then

$$\begin{array}{cc}\hfill N_P(f,\tau )=& \int f^{p}d\mu (x,\tau )=inf\left\{\int {\left({\varphi }_m\right)}^{p}d\mu (x,\tau ):{\varphi }_m\uparrow f\right\},\hfill \\ \hfill =& \underset{m⟶\infty }{lim}\left\{\int {\left({\varphi }_m\right)}^{p}d\mu (x,\tau )\right\},\hfill \end{array}$$

and so

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}{N}_p(f,\tau )=& =\underset{\tau ⟶0}{lim}\underset{m⟶\infty }{lim}\left\{N_p({\varphi }_m,\tau )\right\},\hfill \\ \hfill =& \underset{\tau ⟶0}{lim}\underset{m⟶\infty }{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{\tau }{{\left(a_i^m\right)}^p}}\right).\hfill \end{array}$$

Using Definition 5 (v), we get

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}{N}_p(f,\tau )=& \underset{\tau ⟶0}{lim}\underset{m⟶\infty }{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{\tau }{{\left(a_i^m\right)}^p}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}\underset{\tau ⟶0}{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{\tau }{{\left(a_i^m\right)}^p}}\right),\hfill \end{array}$$

and by Definition 5 (iii), we have

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}{N}_p(f,\tau )=& \underset{m⟶\infty }{lim}\underset{\tau ⟶0}{lim}{\ast }_{i=1}^k\mu \left(E_i^m,{\displaystyle \frac{\tau }{{\left(a_i^m\right)}^p}}\right),\hfill \\ \hfill =& \underset{m⟶\infty }{lim}{\ast }_{i=1}^k\underset{\tau ⟶0}{lim}\mu \left(E_i^m,{\displaystyle \frac{\tau }{{\left(a_i^m\right)}^p}}\right).\hfill \end{array}$$

from Definition 5 (iv), we get

$$\begin{array}{cc}\hfill \underset{\tau ⟶0}{lim}{N}_p(f,\tau )=& \underset{\tau ⟶0}{lim}{\ast }_{i=1}^{k}0,\hfill \\ \hfill =& 0.\hfill \end{array}$$

 □

We proved $({\left(L^{+}\right)}^p,N_p,\ast )$ is a ∗-fuzzy normed space. Now, define the fuzzy set $M:{\left(L^{+}\right)}^p\times {\left(L^{+}\right)}^p\times (0,\infty )⟶(0,1]$ by

$$\begin{array}{c}\hfill M(f,g,\tau )=N_p(|f-g|,\tau )=\int {|f-g|}^{p}d\mu (x,\tau ).\end{array}$$

Then, M is a fuzzy metric on ∗-fuzzy ${\left(L^{+}\right)}^p$ and $({\left(L^{+}\right)}^p,M,\ast )$ is called the ∗-fuzzy metric space induced by the ∗-fuzzy normed space $({\left(L^{+}\right)}^p,N_p,\ast )$. Now, we study further properties of ∗-fuzzy ${\left(L^{+}\right)}^p$.

Theorem 8.

For $1\le p<\infty$, the set of simple functions $g={\displaystyle \sum _{i=1}^n}{a}_i{\chi }_{E_i}$ where $\mu (E_i,\tau )>0$ for all $i\in \{1,2,...,n\}$ and for all $\tau >0$, is dense in ∗-fuzzy ${\left(L^{+}\right)}^p$.

Proof. 

Clearly simple functions $g={\displaystyle \sum _{i=1}^n}{a}_i{\chi }_{E_i}$ are in ∗-fuzzy ${\left(L^{+}\right)}^p$. Let $f\in {\left(L^{+}\right)}^p$, by theorem 3.20 in [3] we can choose a sequence $\left\{f_n\right\}$ of simple functions such that $f_n\uparrow f$ almost everywhere, and so ${(f-f_n)}^p\downarrow 0$.

We assert ${(f-f_n)}^p\in L^{+}$ because

$$\begin{array}{c}\hfill {(f-f_n)}^p\le f^p,\end{array}$$

and so

$$\begin{array}{c}\hfill \int {(f-f_n)}^{p}d\mu (x,\tau )\ge \int f^{p}d\mu (x,\tau )>0,\end{array}$$

then ${(f-f_n)}^p\in L^{+}$ and ${(f-f_n)}^p⟶0$. Using the fundamental convergence Theorem 3, we get

$$\begin{array}{c}\hfill \underset{n⟶\infty }{lim}\int {(f-f_n)}^{p}d\mu (x,\tau )=\int 0d\mu (x,\tau )=1.\end{array}$$

Then, $\underset{n⟶\infty }{lim}{N}_p(f-f_n,\tau )=1$ i.e., $f_n\stackrel{N_p}{\to }f$. □

In the next theorem we prove that ∗-fuzzy ${\left(L^{+}\right)}^p$ spaces are complete.

Theorem 9.

For $1\le p<\infty$, ∗-fuzzy ${\left(L^{+}\right)}^p$ is a ∗-fuzzy Banach space.

Proof. 

Let $\left\{f_n\right\}\subseteq {\left(L^{+}\right)}^p$ be a Cauchy sequence, then for every $x\in X$, $\left\{f_n\left(x\right)\right\}\subseteq \mathbb{R}$ is a Cauchy sequence in $\mathbb{R}$ and since $\mathbb{R}$ is complete, there exist $y\in \mathbb{R}$ such that $f_n\left(x\right)⟶y$, we define $f:X⟶\mathbb{R}$ by $f\left(x\right)=y$. Since $f_n⟶f$ almost everywhere, so ${\left(f_n\right)}^p⟶{\left(f\right)}^p$ almost everywhere, and ${\left(f_n\right)}^p\in L^{+}$ by the fundamental converge Theorem 3 we have ${\left(f\right)}^p\in L^{+}$ and $lim\int {\left(f_n\right)}^{p}d\mu (x,\tau )=\int {\left(f\right)}^{p}d\mu (x,\tau )$, hence $f\in {\left(L^{+}\right)}^p$. □

5. Inequalities on ∗-Fuzzy ${\left({\mathit{L}}^{+}\right)}^{\mathbf{p}}$

In this section, we are ready to prove some important inequalities on ∗-fuzzy ${\left(L^{+}\right)}^p$.

Lemma 1

([16]). If $a\ge 0$, $b\ge 0$, and $0<\lambda <1$, then

$$\begin{array}{c}\hfill a^{\lambda }{b}^{1-\lambda }\le \lambda a+(1-\lambda )b,\end{array}$$

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

Theorem 10

(Hölder’s Inequality). Suppose $1<p<\infty$ and ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$. If f and g are fuzzy measurable functions on X then,

$$\begin{array}{c}\hfill N(fg,\tau )\ge N_p\left(f,{\left(p\right)}^{\frac{1}{p}}\tau \right)\ast N_q\left(g,{\left(q\right)}^{\frac{1}{q}}\tau \right).\end{array}$$

Proof. 

We apply Lemma 1 with ${\left(f\left(x\right)\right)}^p=a$, $b={\left(g\left(x\right)\right)}^q$, and $\lambda ={\displaystyle \frac{1}{p}}$ to obtain

$$\begin{array}{c}\hfill {\left({\left(f\left(x\right)\right)}^p\right)}^{\frac{1}{p}}.{\left({\left(g\left(x\right)\right)}^q\right)}^{1-\frac{1}{p}}\le {\displaystyle \frac{1}{p}}{\left(f\left(x\right)\right)}^p+(1-{\displaystyle \frac{1}{p}}){\left(g\left(x\right)\right)}^q,\end{array}$$

then

$$\begin{array}{c}\hfill f\left(x\right).g\left(x\right)\le {\left({({\displaystyle \frac{1}{p}})}^{\frac{1}{p}}f\left(x\right)\right)}^p+{\left({({\displaystyle \frac{1}{q}})}^{\frac{1}{q}}g\left(x\right)\right)}^q.\end{array}$$

Takeing integral of both sides, we get

$$\begin{array}{cc}\hfill \int f(x).g(x)d\mu (x,\tau )\ge & \int \left[{\left({({\displaystyle \frac{1}{p}})}^{\frac{1}{p}}f\left(x\right)\right)}^p+{\left({({\displaystyle \frac{1}{q}})}^{\frac{1}{q}}g\left(x\right)\right)}^q\right]d\mu (x,\tau ),\hfill \\ \hfill \ge & \left(\int {\left({({\displaystyle \frac{1}{p}})}^{\frac{1}{p}}f\left(x\right)\right)}^{p}d\mu (x,\tau )\right)\ast \left(\int {\left({({\displaystyle \frac{1}{q}})}^{\frac{1}{q}}g\left(x\right)\right)}^{q}d\mu (x,\tau )\right),\hfill \\ \hfill =& N_p\left({({\displaystyle \frac{1}{p}})}^{\frac{1}{p}}f,\tau \right)\ast N_q\left({({\displaystyle \frac{1}{q}})}^{\frac{1}{q}}g,\tau \right),\hfill \\ \hfill =& N_p\left(f,{\left(p\right)}^{\frac{1}{p}}\tau \right)\ast N_q\left(g,{\left(q\right)}^{\frac{1}{q}}\tau \right).\hfill \end{array}$$

Then,

$$\begin{array}{c}\hfill N_1\left(f.g,\tau \right)\ge N_p\left(f,{\left(p\right)}^{\frac{1}{p}}\tau \right)\ast N_q\left(g,{\left(q\right)}^{\frac{1}{q}}\tau \right).\end{array}$$

 □

In the next theorem we compare two ∗-fuzzy ${\left(L^{+}\right)}^p$ spaces.

Theorem 11.

If $0<p<q<r<\infty$, then ${\left(L^{+}\right)}^q\subseteq {\left(L^{+}\right)}^p+{\left(L^{+}\right)}^r$, that is, each $f\in {\left(L^{+}\right)}^q$ is the sum of a function in ∗-fuzzy ${\left(L^{+}\right)}^p$ and a function in ∗-fuzzy ${\left(L^{+}\right)}^r$.

Proof. 

If $f\in {\left(L^{+}\right)}^q$, let $E=\{x:f(x)>1\}$ and set $g=f{\chi }_E$ and $h=f{\chi }_{E^c}$, then

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

However,

$$\begin{array}{c}\hfill g^p={\left(f{\chi }_E\right)}^p=f^p{\chi }_E\le f^q{\chi }_E,\end{array}$$

then,

$$\begin{array}{c}\hfill \int g^{p}d\mu \ge \int f^q{\chi }_{E}d\mu >0,\end{array}$$

then,

$$\begin{array}{c}\hfill g\in {\left(L^{+}\right)}^P.\end{array}$$

On the other hand,

$$\begin{array}{c}\hfill h^r={\left(f{\chi }_{E^c}\right)}^r=f^r{\chi }_{E^c}\le f^q{\chi }_{E^c},\end{array}$$

then,

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

and so

$$\begin{array}{c}\hfill h\in {\left(L^{+}\right)}^r.\end{array}$$

 □

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

Theorem 12.

If $0<p<q<r<\infty$, then $L^p\cap L^r\subseteq L^q$ and

$$\begin{array}{c}\hfill N_q(f,\tau )\ge N_p\left(f,{\left({\displaystyle \frac{p}{\lambda q}}\right)}^{\frac{1}{p}}\tau \right)\ast N_r\left(f,{\left({\displaystyle \frac{r}{(1-\lambda )q}}\right)}^{\frac{1}{r}}\tau \right),\end{array}$$

where $\lambda \in (0,1)$ is defined by $\lambda ={\displaystyle \frac{\frac{1}{q}-\frac{1}{r}}{\frac{1}{p}-\frac{1}{r}}}$.

Proof. 

From $\int f^{q}d\mu (x,\tau )=\int f^{\lambda q}.f^{(1-\lambda )q}d\mu (x,\tau )$ and Hölder’s inequality Theorem 10, we have

$$\begin{array}{cc}\hfill \int f^{q}d\mu (x,\tau )=& \int f^{\lambda q}.f^{q(1-\lambda )}d\mu (x,\tau ),\hfill \\ \hfill \ge & \left(\int {\left({\left({\displaystyle \frac{\lambda q}{p}}\right)}^{\frac{\lambda q}{p}}{f}^{\lambda q}\right)}^{\frac{p}{\lambda q}}d\mu (x,\tau )\right)\ast {\left(\int {\left({\displaystyle \frac{(1-\lambda )q}{r}}\right)}^{\frac{(1-\lambda )q}{r}}{f}^{q(1-\lambda )}d\mu (x,\tau )\right)}^{\frac{r}{(1-\lambda )q}},\hfill \\ \hfill \ge & \left(\int {\displaystyle \frac{\lambda q}{p}}{f}^{p}d\mu (x,\tau )\right)\ast \left(\int \left({\displaystyle \frac{(1-\lambda )q}{r}}\right)f^{r}d\mu (x,\tau )\right),\hfill \\ \hfill =& {\left(\int {\left({\displaystyle \frac{\lambda q}{p}}\right)}^{\frac{1}{p}}f\right)}^{p}d\mu (x,\tau ))\ast \left(\int {\left({\left({\displaystyle \frac{(1-\lambda )q}{r}}\right)}^{\frac{1}{r}}f\right)}^{r}d\mu (x,\tau )\right),\hfill \\ \hfill =& N_p\left({\left({\displaystyle \frac{\lambda q}{p}}\right)}^{\frac{1}{p}}f,\tau \right)\ast N_r\left({\left({\displaystyle \frac{(1-\lambda )q}{r}}\right)}^{\frac{1}{r}}f,\tau \right),\hfill \\ \hfill =& N_p\left(f,{\left({\displaystyle \frac{p}{\lambda q}}\right)}^{\frac{1}{p}}\tau \right)\ast N_r\left(f,{\left({\displaystyle \frac{r}{(1-\lambda )q}}\right)}^{\frac{1}{r}}\tau \right).\hfill \end{array}$$

then,

$$\begin{array}{c}\hfill N_q(f,\tau )\ge N_p\left(f,{\left({\displaystyle \frac{p}{\lambda q}}\right)}^{\frac{1}{p}}\tau \right)\ast N_r\left(f,{\left({\displaystyle \frac{r}{(1-\lambda )q}}\right)}^{\frac{1}{r}}\tau \right).\end{array}$$

 □

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

Theorem 13.

If $\mu (X,\tau )>0$ and $0<p<q<\infty$, then $L^p\left(\mu \right)\supset L^q\left(\mu \right)$ and,

$$\begin{array}{c}\hfill N_p(f,\tau )\ge N_q\left(f,{(\frac{q}{p})}^{\frac{p}{q}}\tau \right)\ast \mu \left(X,{\left(\frac{q}{q-p}\right)}^{\frac{q-p}{q}}\tau \right).\end{array}$$

Proof. 

By Theorem 7 and Hölder’s inequality Theorem 10, we get

$$\begin{array}{cc}\hfill N_p(f,\tau )=& \int f^p.1d\mu (x,\tau ),\hfill \\ \hfill \ge & N_{\frac{q}{p}}\left(f^p,{(\frac{q}{p})}^{\frac{p}{q}}\tau \right)\ast N_{\frac{q}{q-p}}\left(1,{(\frac{q}{q-p})}^{\frac{q-p}{q}}\tau \right),\hfill \\ \hfill =& \int {\left(f^p\right)}^{{\displaystyle \frac{q}{p}}}d\mu \left(x,{(\frac{q}{p})}^{\frac{p}{q}}\tau \right)\ast \int 1d\mu \left(x,{(\frac{q}{q-p})}^{\frac{q-p}{q}}\tau \right),\hfill \\ \hfill =& \int f^{q}d\mu \left(x,{(\frac{q}{p})}^{\frac{p}{q}}\tau \right)\ast \mu \left(X,{(\frac{q}{q-p})}^{\frac{q-p}{q}}\tau \right),\hfill \\ \hfill =& N_q\left(f,{(\frac{q}{p})}^{\frac{p}{q}}\tau \right)\ast \mu \left(X,{(\frac{q}{q-p})}^{\frac{q-p}{q}}\tau \right).\hfill \end{array}$$

 □

Finally, we prove the Chebyshev’s Inequality in ∗-fuzzy ${\left(L^{+}\right)}^p$ spaces.

Theorem 14

(Chebyshev’s Inequality). If $f\in {\left(L^{+}\right)}^p(0<p<\infty )$ then for any $a>0$, $N_p(f,\tau )\le N_p({\chi }_{E_a},\frac{\tau }{a})$ with respect to $E_a=\{x:f\left(x\right)>a\}$.

Proof. 

We have,

$$\begin{array}{c}\hfill f^p>{\left(f{\chi }_{E_a}\right)}^p=f^p{\chi }_{E_a},\end{array}$$

then

$$\begin{array}{c}\hfill \int f^{p}d\mu (x,\tau )\le \int f^{p}d\mu (x,\tau ){\chi }_{E_a}={\int }_{E_a}{f}^{p}d\mu (x,\tau ),\end{array}$$

(20)

and on $E_a$ we have

$$\begin{array}{c}\hfill {\int }_{E_a}{f}^{p}d\mu (x,\tau )\le {\int }_{E_a}{a}^{p}d\mu (x,\tau )=\int a^p{\chi }_{E_a}d\mu (x,\tau ).\end{array}$$

(21)

By (20) and (21) we get

$$\begin{array}{cc}\hfill \int f^{p}d\mu (x,\tau )\le & \int a^p{\chi }_{E_a}d\mu (x,\tau ),\hfill \\ \hfill =& \int {\left(a{\chi }_{E_a}\right)}^{p}d\mu (x,\tau ).\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill N_p(f,\tau )\le & N_p(a{\chi }_{E_a},\tau ),\hfill \\ \hfill =& N_p({\chi }_{E_a},{\displaystyle \frac{\tau }{a}}).\hfill \end{array}$$

 □

6. Conclusions

We have considered an uncertainty measure $\mu$ 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 ∗-fuzzy ${\left(L^{+}\right)}^p$. Moreover, we have got a norm on ∗-fuzzy ${\left(L^{+}\right)}^p$ spaces and proved that ∗-fuzzy ${\left(L^{+}\right)}^p$ spaces are ∗-fuzzy Banach spaces. Finally, we have proved Chebyshev’s Inequality and Hölder’s Inequality.