Some properties and convergence theorems for fuzzy-valued Kluvánek–Lewis integrals

Abstract

In this paper, we first introduce a new fuzzy-valued integral of scalar-valued functions with respect to a fuzzy-valued measure with some natural properties. Then we prove Vitali type convergence theorem and dominated convergence theorem for this kind of integral.

1 Introduction

Motivated by the applications in several areas of applied science, such as mathematical economics, fuzzy optimal, process control and decision theory, much effort has been devoted to the generalization of classical measure and integral results to the case when outcomes of a random experiment are represented by sets or fuzzy sets, such as the concepts of set-valued measures and integrals (see, e.g., Hiai 1978; Kandilakis 1992; Papageorgiou 1985b; Stojaković 2012; Wu et al. 2001; Zhou and Shi 2015a) and fuzzy valued measures and integrals (see, e.g., Iosif and Gavrilu 2017; Malinowski 2013, 2015; Park 2010; Stojaković 1995, 2011; Stojaković and Stojaković 2007; Xiaoping et al. 1996; Zhou and Shi 2015b, c). Among them, as an extension of the integrals of scalar-valued functions with respect to vector measures, the integral of scalar-valued functions with respect to a set-valued measure was first launched by Papageorgiou (1985b) who considered the bilinear integral of Dinculeanu (1967). After that, Kandilakis (1992) introduced an integral of a bounded real-valued measurable function with respect to a set-valued measure using the set of Kluvánek–Knowles type integrals Kluvánek and Knowles (1975). Wu et al. (2001) introduced the set-valued Bartle integral which is a set of Bartle–Dunford–Schwartz type integrals Bartle et al. (1955) of a scalar-valued function with respect to measure selections of the given set-valued measure. This type of definition has also been considered by Zhang et al. (2007). Recently, Zhou and Shi (2015a) introduced set-valued Kluvánek–Lewis type integral which is a Pettis type weak integral of scalar-valued functions with respect to a set-valued measure in Banach spaces. When it comes to integrals of scalar-valued functions with respect to fuzzy-valued measures, only some approaches can be distinguished. The integral of scalar-valued functions with respect to a fuzzy number measure in $\mathbb R$ was first introduced by Stojaković (1995). Since then, based on Papageorgiou’s set-valued integral Papageorgiou (1985b), the integral of scalar-valued functions with respect to a fuzzy number measure was introduced by Stojaković and Stojaković (2007), where the fuzzy number measure takes on values in the family of fuzzy sets with compact convex $\alpha$-levels in finite-dimensional Banach spaces. Based on the set-valued Bartle integral, Wu et al. (2001) and Park (2010) introduced generalized fuzzy number-valued Bartle integral which is the integration of scalar-valued functions with respect to a generalized fuzzy number measure, where the generalized fuzzy number measure takes on values in the family of fuzzy sets with weakly compact and convex $\alpha$-levels in an infinite-dimensional Banach space. Zhou and Shi (2015c) introduced an integral of scalar-valued functions with respect to a generalized fuzzy number measure which is different from that of Park and main results are extensions of Zhang, Li, Wang and Gao’s results Zhang et al. (2007).

In the present paper, we introduce and study a new fuzzy-valued integral of scalar-valued functions with respect to a generalized fuzzy number measure. The result is an extension of set-valued Kluvánek–Lewis type integrals Zhou and Shi (2015a). We investigate some properties and establish convergence theorems for this kind of integral. The paper is structured as follows. In Sect. 2, we state some basic concepts and preliminary results which will be used in the sequel. In Sect. 3, we first introduce the new integral with some natural properties. Then we prove Vitali type convergence theorem and dominated convergence theorem for this kind of integral.

2 Preliminaries

Throughout this paper, let $(\Omega , \mathscr {A},\mu )$ be a complete finite measure space where $\Omega$ is a nonempty set, $\mathscr {A}$ is a $\sigma$-algebra of subsets of $\Omega$ and $\mu$ is a measure. Let $(\mathcal {X},\Vert \cdot \Vert )$ be a real separable Banach space with its dual space $\mathcal {X}^*$. Let

$\mathscr {P}_0(\mathcal {X})=\{A\subset \mathcal {X}: A\,\, \text{ is } \text{ a } \text{ nonempty } \text{ subset } \text{ of }\,\, \mathcal {X}\},$

$\mathscr {P}_{b(f)(c)}(\mathcal {X})=\{A\in \mathscr {P}_0(\mathcal {X}): A\,\, \text{ is } \text{ bounded } \text{(closed) } \text{(convex) }\},$

$\mathscr {P}_{wkc}(\mathcal {X})=\{A\in \mathscr {P}_0(\mathcal {X}): A\,\, \text{ is } \text{ weakly } \text{ compact } \text{ and } \text{ convex }\}.$

For $A,B\in \mathscr {P}_f(\mathcal {X})$, the Hausdorff metric $d_\mathrm{{H}}$ of A and B is defined by

$$\begin{aligned} d_\mathrm{{H}}(A,B)=\max \left\{ \sup _{a\in A}d(a, B),\sup _{b\in B}d(b, A)\right\} , \end{aligned}$$

where $d(a, B)=\inf _{b\in B}\Vert a-b\Vert$. For $A\subset \mathcal {X}$, the number |A| is defined by $|A|=d_\mathrm{{H}}(A,\{0\})=\sup _{x\in A}\Vert x\Vert .$ Note that $(\mathscr {P}_{wkc}(\mathcal {X}), d_\mathrm{{H}})$ is a complete metric space. We shall denote by $\sigma (\cdot , A)$ the support function of the set $A\subset \mathcal {X}$ defined by

$$\begin{aligned} \sigma (x^*, A)=\sup _{x\in A} x^*(x), \,\,x^*\in \mathcal {X}^*. \end{aligned}$$

The support function satisfies the properties: $\sigma (x^*, A+B)=\sigma (x^*, A)+\sigma (x^*, B)$ and $\sigma (x^*, \lambda A)=\lambda \sigma (x^*, A)$ for all $A,B \in \mathscr {P}_0(\mathcal {X}),$ $x^*\in \mathcal {X}^*$ and $\lambda \ge 0$. In particular,

$$\begin{aligned} d_\mathrm{{H}}(A,B)=\sup _{\Vert x^*\Vert \le 1}\left| \sigma (x^*, A)-\sigma (x^*, B)\right| \end{aligned}$$

whenever A and B are two convex sets.

Let $\tilde{u}:\mathcal {X}\rightarrow [0,1]$. We denote the $\alpha$-level of $\tilde{u}$ by $\tilde{u}_\alpha =\{x\in \mathcal {X}: \tilde{u}(x)\ge \alpha \}$ for any $\alpha \in (0,1].$ $\tilde{u}$ is called a generalized fuzzy number if for each $\alpha \in (0,1],$ $\tilde{u}_\alpha \in \mathscr {P}_{wkc}(\mathcal {X})$ (cf. Xiaoping et al. 1996). Let $\mathscr {F}_{wkc}(\mathcal {X})$ denote the set of all generalized fuzzy numbers on $\mathcal {X}$.

For $\tilde{u}, \tilde{v}\in \mathscr {F}_{wkc}(\mathcal {X})$ and $\lambda \in \mathbb R,$ we define $\tilde{u}+\tilde{v}$ and $\lambda \tilde{u}$ as follows:

$$\begin{aligned} (\tilde{u}+\tilde{v})(x)= & {} \sup _{x=y+z}\min \{\tilde{u}(y), \tilde{v}(z)\},\\ (\lambda \tilde{u})(x)= & {} \left\{ \begin{array}{ll} \tilde{u}\left( \frac{\displaystyle 1}{\displaystyle \lambda }x\right) , &{} \text{ if } \, \, \lambda \ne 0, \\ 0, &{} \text{ if } \, \, \lambda =0. \end{array} \right. \end{aligned}$$

Obviously, we have $(\tilde{u}+\tilde{v})_\alpha =\tilde{u}_\alpha +\tilde{v}_\alpha$ and $(\lambda \tilde{u})_\alpha =\lambda \tilde{u}_\alpha$ for each $\alpha \in (0,1].$ Hence, $\tilde{u}+\tilde{v}, \lambda \tilde{u}\in \mathscr {F}_{wkc}(\mathcal {X})$ (cf. Park 2010; Xiaoping et al. 1996). In the set $\mathscr {F}_{wkc}(\mathcal {X})$, we can define the metric $d_\mathrm{{H}}^\infty$ by

$$\begin{aligned} d_\mathrm{{H}}^\infty (\tilde{u},\tilde{v})=\sup _{\alpha \in (0,1]}d_\mathrm{{H}}(\tilde{u}_\alpha ,\tilde{v}_\alpha ). \end{aligned}$$

$(\mathscr {F}_{wkc}(\mathcal {X}), d_\mathrm{{H}}^\infty )$ is a metric space [cf. Park 2010]. The norm $\Vert \tilde{u}\Vert$ of $\tilde{u}\in \mathscr {F}_{wkc}(\mathcal {X})$ is defined by

$$\begin{aligned} \Vert \tilde{u}\Vert =d_\mathrm{{H}}^\infty (\tilde{u},\tilde{0} )=\sup _{\alpha \in (0,1]}|\tilde{u}_\alpha |, \end{aligned}$$

where $\tilde{0}$ is indicator function of $\{0\}.$

Theorem 2.1

Wu and Wu (2001) If $\tilde{u}\in \mathscr {F}_{wkc}(\mathcal {X})$, then

• (1) $\tilde{u}_\alpha \in \mathscr {P}_{wkc}(\mathcal {X})$ for all $\alpha \in (0,1];$

• (2) $\tilde{u}_\alpha \supseteq \tilde{u}_\beta$ for $0<\alpha \le \beta \le 1;$

• (3) if $\{\alpha _n\}_{n\in \mathbb N}$ is a nondecreasing sequence in [0, 1] converging to $\alpha \in (0,1],$ then $\tilde{u}_\alpha =\bigcap _{n=1}^\infty \tilde{u}_{\alpha _n}.$ Conversely, if $\{A_\alpha : \alpha \in (0,1]\}\subseteq \mathscr {P}_0(\mathcal {X})$ satisfies (1)–(3) above, then there exists a $\tilde{u}\in \mathscr {F}_{wkc}(\mathcal {X})$ such that $\tilde{u}_\alpha =A_\alpha$ for each $\alpha \in (0,1].$

Theorem 2.2

Xue et al. (1994) Let $A_\alpha \in \mathscr {P}_{wkc}(\mathcal {X}),$ $\{A_{\alpha _n}\}_{n\in \mathbb N}\subset \mathscr {P}_{wkc}(\mathcal {X})$ and $\alpha _n\nearrow \alpha$, $A_{\alpha _n}\supset A_{\alpha _{n+1}}\supset A_{\alpha }$, then $\sigma (x^*, A_{\alpha _n})$ converges to $\sigma (x^*, A_{\alpha })$ for each $x^*\in \mathcal {X}^*$ if and only if $A_{\alpha }=\cap _{n=1}^\infty A_{\alpha _n}.$

Definition 2.3

Hiai (1978) Let $(\Omega , \mathscr {A})$ be a measurable space. The mapping $M : \mathscr {A}\rightarrow \mathscr {P}_0(\mathcal {X})$ is called a set-valued measure if it satisfies the following two conditions:

• (1) $M (\emptyset )=\{0\};$

• (2) if $A_1, A_2, \ldots$ are in $\mathscr {A}$, with $A_i \cap A_j=\emptyset$ for $i\ne j$, then

  $$\begin{aligned} M\left( \bigcup _{i=1}^\infty A_i\right) =\sum _{i=1}^\infty M(A_i), \end{aligned}$$

  where $\sum _{i=1}^\infty M(A_i)=\left\{ x\in \mathcal {X}: x=\sum _{i=1}^\infty x_i\,\, ({ \text{ unconditionally } \text{ convergent }}),\, \, x_i\in M\right.$ $\left. (A_i), i\ge 1\right\} .$

We say that M is $\mu$-continuous if for arbitrary $A\in \mathscr {A}$, $\mu (A)=0$, then $M(A)=\{0\}$. As for single-valued measures, we have the notion of total variation |M| of M. For $A\in \mathscr {A}$ we define $|M|(A)=\sup \sum _{i=1}^n|M(A_i)|,$ where the supremum is taken over all finite measurable partitions $\{A_1,\ldots ,A_n\}$ of A. If $|M|(\Omega )<\infty$, then we say that M is of bounded variation.

Definition 2.4

Xiaoping et al. (1996) Let $(\Omega , \mathscr {A})$ be a measurable space. The mapping $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is called a generalized fuzzy number measure if it satisfies the following two conditions:

• (1) $\tilde{M}(\emptyset )=\tilde{0};$

• (2) if $A_1, A_2, \ldots$ are in $\mathscr {A}$, with $A_i \cap A_j=\emptyset$ for $i\ne j$, then

  $$\begin{aligned} \tilde{M}\left( \bigcup _{i=1}^\infty A_i\right) =\sum _{i=1}^\infty \tilde{M}(A_i), \end{aligned}$$

  where $(\sum _{i=1}^\infty \tilde{M}(A_i) )(x)=\sup \{\bigwedge _{i=1}^{\infty }\tilde{M}(A_i)(x_i): x=\sum _{i=1}^{\infty }$ $x_i\,\,$ $({\text{ unconditionally } \text{ convergent }})\, \}.$

We say that $\tilde{M}$ is $\mu$-continuous if for arbitrary $A\in \mathscr {A}$, $\mu (A)=0$, then $\tilde{M}(A)=\tilde{0}.$ For $A\in \mathscr {A}$, we define $|\tilde{M}|(A)=\sup \sum _{i=1}^n\Vert \tilde{M}(A_i)\Vert ,$ where the supremum is taken over all finite measurable partitions $\{A_1,\ldots ,A_n\}$ of A. If $|\tilde{M}|(\Omega )<\infty$, then we say that $\tilde{M}$ is of bounded variation. Note that for each generalized fuzzy number measure $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ and $A\in \mathscr {A}$, we have $|\tilde{M}|(A)=\sup _{\alpha \in (0,1]} |\tilde{M}_\alpha (A)|$. Therefore, $|\tilde{M}|$ is a real-valued measure. If $\tilde{M}$ is of bounded variation, then for each $\alpha \in (0,1],$ $\tilde{M}_\alpha$ is of bounded variation. For a real-valued function $f:\Omega \rightarrow \mathbb R$, if f is $|\tilde{M}|$-integrable, then f is $|\tilde{M}_\alpha |$-integrable for each $\alpha \in (0,1]$ (cf. Park 2010). In the following, let $L^1(\Omega , \mathbb R, |\tilde{{M}}|)$ be the space of all functions $f: \Omega \rightarrow \mathbb R$ which are $\mathscr {A}$-measurable and $|\tilde{{M}}|$-integrable and $L^1(\Omega , \mathbb R^+, |\tilde{M}|)$ the space of all functions $f: \Omega \rightarrow \mathbb R^+$, where $\mathbb R^+$ denotes the set of all nonnegative real numbers, which are $\mathscr {A}$-measurable and $|\tilde{M}|$-integrable.

Theorem 2.5

Xiaoping et al. (1996) The mapping $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is a generalized fuzzy number measure if and only if there exists a family of set-valued measures $M_\alpha : \mathscr {A}\rightarrow \mathscr {P}_{wkc}(\mathcal {X}), \alpha \in (0,1]$ satisfying the following three conditions:

• (1) for arbitrary $\alpha ,\beta \in (0,1]$ and $A\in \mathscr {A}$, if $\alpha \le \beta ,$ then $M_\alpha (A)\supseteq M_\beta (A)$;

• (2) for arbitrary $\{\alpha _n\}_{n\in \mathbb N}\subseteq (0,1]$ and $A\in \mathscr {A},$ if $\alpha _n\nearrow \alpha ,$ then $M_\alpha (A)=\bigcap _{n=1}^\infty M_{\alpha _n}(A);$

• (3) for arbitrary $A\in \mathscr {A}$, we have

  $$\begin{aligned} \tilde{M}(A)(x) = \left\{ \begin{array}{ll} \sup \{\alpha \in (0,1]: x\in M_{\alpha }(A)\}, &{} \quad \mathrm{{if}} \, \, \{\alpha \in (0,1]: x\in M_{\alpha }(A)\}\ne \emptyset ;\\ 0, &{} \quad \mathrm{{if}} \, \, \{\alpha \in (0,1]: x\in M_{\alpha }(A)\}=\emptyset . \end{array} \right. \end{aligned}$$

Note that for a generalized fuzzy number measure $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$, the set-valued measure $M_\alpha : \mathscr {A}\rightarrow \mathscr {P}_{wkc}(\mathcal {X})$ is determined by

$$\begin{aligned} M_\alpha (A)=\{x\in \mathcal {X}:\tilde{M}(A)(x)\ge \alpha \}, \end{aligned}$$

i.e., $M_\alpha (A)=[\tilde{M}(A)]_\alpha$ for each $A\in \mathscr {A}$ and $\alpha \in (0,1].$

The integral of measurable set-valued function $X: \Omega \rightarrow \mathscr {P}_0(\mathcal {X})$ with respect to a measure $\mu$ is defined by $\int _\Omega X (\omega )\,\mathrm {d}\mu (\omega )= \{\int _\Omega f (\omega )\,\mathrm {d}\mu (\omega ): f\in S_X \},$ where $S_X$ denotes the set of all measurable selectors of $X(\omega )$ and $\int _\Omega f (\omega )\,\mathrm {d}\mu (\omega )$ is defined in the sense of Bochner. A measurable set-valued function $X: \Omega \rightarrow \mathscr {P}_0(\mathcal {X})$ is integrably bounded if there exists an integrable function $h: \Omega \rightarrow \mathbb R^+$ such that $|X(\omega )|=\sup _{x\in X(\omega )}\Vert x\Vert \le h(\omega )$ $\mu$-a.e.

A Banach space $\mathcal {X}$ has the Radon–Nikodým property (RNP) if for each finite measure space $(\Omega , \mathscr {A}, \mu )$ and each $\mu$-continuous $\mathcal {X}$-valued measure $m : \mathscr {A}\rightarrow \mathcal {X}$ of bounded variation, there exists a Bochner integrable function $f : \Omega \rightarrow \mathcal {X}$ such that $m(A) =\int _A f(\omega )\,\mathrm {d}\mu (\omega )$ for all $A\in \mathscr {A}.$ A measurable set-valued function $X(\omega )$ is said to be a Radon–Nikodým derivative of the set-valued measure M with respect to $\mu$ if $M(A)=\int _A X (\omega )\,\mathrm {d}\mu (\omega )$ for all $A\in \mathscr {A}$ and we write $\mathrm {d}M=X \mathrm {d}\mu$.

Let $L^1(\Omega , \mathbb R, |M|)$ be the space of all functions $f: \Omega \rightarrow \mathbb R$ which are $\mathscr {A}$-measurable and |M|-integrable.

Definition 2.6

Zhou and Shi (2015a) Let $M : \mathscr {A}\rightarrow \mathscr {P}_{wkc}(\mathcal {X})$ be a set-valued measure and $f: \Omega \rightarrow \mathbb R$ be an element of $L^1(\Omega , \mathbb R, |M|)$. f is said to be Kluvánek–Lewis integrable with respect to M (for shortly, (KL) M-integrable) if

• (1) f is $\sigma (x^*, M(\cdot ))$-integrable for each $x^*\in \mathcal {X}^*$;

• (2) for each $A\in \mathscr {A}$, there exists a $W_A\in \mathscr {P}_{wkc}(\mathcal {X})$ such that

  $$\begin{aligned} \sigma (x^*, W_A)=\int _A f(\omega )\,\mathrm {d}\sigma (x^*,M(\omega )) \end{aligned}$$

  for each $x^*\in \mathcal {X}^*.$ In the case, we write $W_A=({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}M(\omega )$ for each $A\in \mathscr {A}$ and call it set-valued Kluvánek–Lewis integral of f with respect to M on A.

Theorem 2.7

Zhou and Shi (2015a) Suppose that $\mathcal {X}$ has the RNP, $\mathcal {X}^*$ is separable, ${M} : \mathscr {A}\rightarrow \mathscr {P}_{wkc}(\mathcal {X})$ is a $\mu$-continuous set-valued measure of bounded variation and $f :\Omega \rightarrow \mathbb R^+$ is (KL) M-integrable. Then there exists an integrably bounded set-valued function $X: \Omega \rightarrow \mathscr {P}_{wkc}(\mathcal {X})$ such that

$$\begin{aligned} ({\mathrm {KL}})\int _\Omega f(\omega ) \,\mathrm {d}M(\omega )=\int _\Omega f(\omega )X(\omega ) \,\mathrm {d}\mu (\omega ). \end{aligned}$$

3 Main results

In the sequel, let $L^1_{KL}(\Omega , \mathbb R, |\tilde{{M}}|)$ (respectively, $L^1_{KL}(\Omega , \mathbb R^+, |\tilde{{M}}|)$ be the space of all functions $f\in L^1(\Omega , \mathbb R, |\tilde{{M}}|)$ (respectively, $f\in L^1(\Omega , \mathbb R^+, |\tilde{{M}}|)$) which are (KL) $\tilde{{M}}_\alpha$-integrable for all $\alpha \in (0,1]$. Then we generalize the set-valued Kluvánek–Lewis integral to generalized fuzzy number measures as follows:

Definition 3.1

Let $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ be a generalized fuzzy number measure on the measurable space $(\Omega , \mathscr {A})$ and $f\in L^1_{KL}(\Omega , \mathbb R, |\tilde{M}|)$. For each $A\in \mathscr {A}$, the mapping $\tilde{M}^\prime (A): \mathcal {X}\rightarrow [0,1]$ is defined by

$$\begin{aligned} \tilde{M}^\prime (A)(x)= \sup \{\alpha \in (0,1]: x\in \tilde{M}^\prime _{\alpha }(A)\}, \quad \forall x\in \mathcal {X}, \end{aligned}$$

where $\tilde{M}^\prime _{\alpha }(A) =({\mathrm {KL}}) \int _A f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )$. We write

$$\begin{aligned} \tilde{M}^\prime (A)=({\mathrm {KL}}) \int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega ) \end{aligned}$$

and call it fuzzy-valued Kluvánek–Lewis integral of f with respect to $\tilde{M}$ on A.

In the following, we show that fuzzy-valued Kluvánek–Lewis integral is, in fact, a new generalized fuzzy number measure if $f\in L^1_{KL}(\Omega , \mathbb R^+, |\tilde{M}|)$.

Theorem 3.2

Suppose that $\mathcal {X}$ has the RNP, $\mathcal {X}^*$ is separable, $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is a $\mu$-continuous generalized fuzzy number measure of bounded variation. Then for each $f\in L^1_{KL}(\Omega , \mathbb R^+, |\tilde{M}|)$ and $A\in \mathscr {A}$, we have

$$\begin{aligned} ({\mathrm {KL}}) \int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega )\in \mathscr {F}_{wkc}(\mathcal {X}). \end{aligned}$$

Proof

Since $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is a $\mu$-continuous generalized fuzzy number measure of bounded variation, by Theorem 2.5, $\tilde{M}_\alpha (A)=[\tilde{M}(A)]_\alpha$ is a $\mu$-continuous set-valued measure of bounded variation for each $\alpha \in (0,1].$ Therefore, by Theorem 8 Zhou and Shi (2015a), there exists an integrably bounded set-valued function $X_\alpha : \Omega \rightarrow \mathscr {P}_{wkc}(\mathcal {X})$ such that

$$\begin{aligned} \tilde{M}^\prime _{\alpha }(\Omega )= ({\mathrm {KL}})\int _\Omega f(\omega ) \,\mathrm {d}\tilde{M}_\alpha (\omega )=\int _\Omega f(\omega )X_\alpha (\omega ) \,\mathrm {d}\mu (\omega ). \end{aligned}$$

To end the proof, we show that $\{\tilde{M}^\prime _\alpha (\Omega )\}_{\alpha \in (0,1]}$ satisfies all conditions of Theorem 2.1. By the above equality and Corollary of Proposition 3.1 in Papageorgiou (1985a), we have $\tilde{M}^\prime _\alpha (\Omega )\in \mathscr {P}_{wkc}(\mathcal {X})$ for all $\alpha \in (0,1].$ Since $0<\alpha \le \beta \le 1$ implies $\tilde{M}_\alpha (A)\supseteq \tilde{M}_\beta (A)$ for any $A\in \mathscr {A}$, we have $\sigma (x^*,\tilde{M}_\alpha (A))\ge \sigma (x^*,\tilde{M}_\beta (A))$ for each $x^*\in \mathcal {X}$ and $A\in \mathscr {A}$. This implies that

$$\begin{aligned} \int _\Omega f(\omega )\,\mathrm {d}\sigma (x^*,\tilde{M}_\alpha (\omega ))\ge \int _\Omega f(\omega )\,\mathrm {d}\sigma (x^*,\tilde{M}_\beta (\omega )) \end{aligned}$$

for each $x^*\in \mathcal {X}$, i.e., $\sigma (x^*,\tilde{M}^\prime _\alpha (\Omega ))\ge \sigma (x^*,\tilde{M}^\prime _\beta (\Omega ))$ for each $x^*\in \mathcal {X}$, which implies that $\tilde{M}^\prime _\alpha (\Omega )\supseteq \tilde{M}^\prime _\beta (\Omega ).$ Now let $\{\alpha _n\}_{n\in \mathbb N}$ be a nondecreasing sequence in [0, 1] converging to $\alpha \in (0,1].$ We use Theorem 2.2 to prove that $\tilde{M}^\prime _\alpha (\Omega )=\cap _{n=1}^\infty \tilde{M}^\prime _{\alpha _n}(\Omega )$. Note that Hausdorff convergence implies weak convergence for arbitrary sequence in $\mathscr {P}_{bfc}(\mathcal {X})$. To prove that $\sigma (x^*, \tilde{M}^\prime _{\alpha _n}(\Omega ))$ converges to $\sigma (x^*, \tilde{M}^\prime _{\alpha }(\Omega ))$ for each $x^*\in \mathcal {X}^*$, we first show that $\tilde{M}^\prime _{\alpha _n}(\Omega )$ Hausdorff converges to $\tilde{M}^\prime _{\alpha }(\Omega )$. By properties of the support function, we have

$$\begin{aligned}&d_\mathrm{{H}}\left( \tilde{M}^\prime _{\alpha _n}(\Omega ),\tilde{M}^\prime _{\alpha }(\Omega )\right) \\&\quad =d_\mathrm{{H}}\left( \int _\Omega f(\omega )X_{\alpha _n}(\omega )\,\mathrm {d}\mu (\omega ), \int _\Omega f(\omega )X_\alpha (\omega )\,\mathrm {d}\mu (\omega )\right) \\&\quad =\sup _{\Vert x^*\Vert \le 1}\left| \sigma \left( x^*, \int _\Omega f(\omega )X_{\alpha _n}(\omega )\,\mathrm {d}\mu (\omega )\right) -\sigma \left( x^*, \int _\Omega f(\omega )X_\alpha (\omega )\,\mathrm {d}\mu (\omega )\right) \right| \\&\quad =\sup _{\Vert x^*\Vert \le 1}\left| \int _\Omega \sigma \left( x^*, f(\omega )X_{\alpha _n}(\omega )\right) \,\mathrm {d}\mu (\omega ) -\int _\Omega \sigma \left( x^*, f(\omega )X_\alpha (\omega )\right) \,\mathrm {d}\mu (\omega )\right| \\&\quad \le \int _\Omega \sup _{\Vert x^*\Vert \le 1}\left| \sigma \left( x^*, f(\omega )X_{\alpha _n}(\omega )\right) -\sigma \left( x^*, f(\omega )X_\alpha (\omega )\right) \right| \,\mathrm {d}\mu (\omega )\\&\quad =\int _\Omega d_\mathrm{{H}}\left( f(\omega ) X_{\alpha _n}(\omega ), f(\omega ) X_\alpha (\omega ) \right) \,\mathrm {d}\mu (\omega )\\&\quad {=}{\int _\Omega \left| f(\omega )\right| d_\mathrm{{H}}\left( X_{\alpha _n}(\omega ), X_\alpha (\omega ) \right) \,\mathrm {d}\mu (\omega )}\\&\quad {\le }{\int _\Omega |f(\omega )| \left( | X_{\alpha _n}(\omega )|+|X_\alpha (\omega )|\right) \,\mathrm {d}\mu (\omega )}\\&\quad {=}{\int _\Omega f(\omega ) \left( | X_{\alpha _n}(\omega )|+|X_\alpha (\omega )|\right) \,\mathrm {d}\mu (\omega ).} \end{aligned}$$

Since $X_{\alpha _1}(\omega )$ is integrably bounded, there exists a nonnegative $\mu$-integrable function $\varphi (\omega )$ such that $|X_{\alpha _1}(\omega )|\le \varphi (\omega )$ $\mu$-a.e. In fact, $X_{\alpha }(\omega )$ and $X_{\alpha _n}(\omega )$ are Radon–Nikodým derivatives of $\tilde{M}_{\alpha }$ and $\tilde{M}_{\alpha _n}$ with respect to $\mu$, respectively. Hence, by Theorem 6.4.6 Zhang et al. (2007), we have $X_{\alpha }(\omega )\subseteq \cdots \subseteq X_{\alpha _n}(\omega )\cdots \subseteq X_{\alpha _1}(\omega )$, which implies that $|X_{\alpha }(\omega )|\le \varphi (\omega )$ and $|X_{\alpha _n}(\omega )|\le \varphi (\omega )$ $\mu$-a.e. for all $n\in \mathbb N$. Then

$$\begin{aligned} { \int _\Omega f(\omega ) \left( |X_{\alpha _n}(\omega )|+|X_\alpha (\omega )|\right) \,\mathrm {d}\mu (\omega ) \le \int _\Omega 2 f(\omega ) \varphi (\omega )\,\mathrm {d}\mu (\omega )<\infty } \end{aligned}$$

enables the application of Lebesgue’s dominated convergence theorem

$$\begin{aligned} \lim _{n\rightarrow \infty }d_\mathrm{{H}}\left( \tilde{M}^\prime _{\alpha _n}(\Omega ),\tilde{M}^\prime _{\alpha }(\Omega )\right)= & {} \lim _{n\rightarrow \infty }d_\mathrm{{H}}\left( \int _\Omega f(\omega )X_{\alpha _n}(\omega )\,\mathrm {d}\mu (\omega ), \int _\Omega f(\omega )X_\alpha (\omega )\,\mathrm {d}\mu (\omega )\right) \\\le & {} \lim _{n\rightarrow \infty }\int _\Omega { f(\omega )}d_\mathrm{{H}}\left( X_{\alpha _n}(\omega ), X_\alpha (\omega ) \right) \,\mathrm {d}\mu (\omega )\\= & {} \int _\Omega {f(\omega )}\lim _{n\rightarrow \infty }d_\mathrm{{H}}\left( X_{\alpha _n}(\omega ), X_\alpha (\omega ) \right) \,\mathrm {d}\mu (\omega ). \end{aligned}$$

In the same manner as in the proof of Theorem  3.1 Zhou and Shi (2015c), we have

$$\begin{aligned} \lim _{n\rightarrow \infty }d_\mathrm{{H}}\left( X_{\alpha _n}(\omega ), X_\alpha (\omega ) \right) =0, \end{aligned}$$

which follows that

$$\begin{aligned} \lim _{n\rightarrow \infty }d_\mathrm{{H}}\left( \tilde{M}^\prime _{\alpha _n}(\Omega ),\tilde{M}^\prime _{\alpha }(\Omega )\right)\le & {} \int _\Omega {f(\omega )}\lim _{n\rightarrow \infty }d_\mathrm{{H}}\left( X_{\alpha _n}(\omega ), X_\alpha (\omega ) \right) \,\mathrm {d}\mu (\omega )\\= & {} 0, \end{aligned}$$

i.e., $\tilde{M}^\prime _{\alpha _n}(\Omega ){\mathop {\rightarrow }\limits ^{\mathrm {d_\mathrm{{H}}}}}\tilde{M}^\prime _{\alpha }(\Omega )$, which implies that $\lim _{n\rightarrow \infty }\sigma (x^*, \tilde{M}^\prime _{\alpha _n}(\Omega ) ) =\sigma (x^*, \tilde{M}^\prime _{\alpha }(\Omega ) )$ for each $x^*\in \mathcal {X}^*.$ Then, by Theorem 2.2, we obtain that $\tilde{M}^\prime _\alpha (\Omega )=\cap _{n=1}^\infty \tilde{M}^\prime _{\alpha _n}(\Omega ).$

Up to now, $\{\tilde{M}^\prime _\alpha (\Omega )\}_{\alpha \in (0,1]}$ satisfies all conditions of Theorem 2.1. Hence, the family $\{\tilde{M}^\prime _\alpha (\Omega )\}_{\alpha \in (0,1]}$ generates the generalized fuzzy number $\tilde{M}^\prime (\Omega ): \mathcal {X}\rightarrow [0,1]$ defined by

$$\begin{aligned} \tilde{M}^\prime (\Omega )(x)=\sup \{\alpha \in (0,1]: x\in \tilde{M}^\prime _{\alpha }(\Omega )\} \end{aligned}$$

for each $x\in \mathcal {X}.$ To prove that for each $A\in \mathscr {A}$, $\{\tilde{M}^\prime _\alpha (A)\}_{\alpha \in (0,1]}$ defines a generalized fuzzy number, we repeat the preceding proof for any $A\in \mathscr {A}$ instead of $\Omega .$ This completes the proof. $\square$

Theorem 3.3

Suppose that $\mathcal {X}$ has the RNP, $\mathcal {X}^*$ is separable and $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is a $\mu$-continuous generalized fuzzy number measure of bounded variation. Then for each $f\in L^1_{KL}(\Omega , \mathbb R^+, |\tilde{M}|),$ $\tilde{M}^\prime : \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ defined by

$$\begin{aligned} \tilde{M}^\prime (A)=({\mathrm {KL}}) \int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega ), \quad \forall A\in \mathscr {A}, \end{aligned}$$

is a $\mu$-continuous generalized fuzzy number measure.

Proof

To end the proof, we use Theorem 2.5. First, by Theorem 7 Zhou and Shi (2015a), we have that $\tilde{M}^\prime _\alpha : \mathscr {A}\rightarrow \mathscr {P}_{wkc}(\mathcal {X})$ defined by

$$\begin{aligned} \tilde{M}^\prime _\alpha (A)=({\mathrm {KL}}) \int _A f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega ), \quad \forall A\in \mathscr {A}, \end{aligned}$$

is a $\mu$-continuous set-valued measure for each $\alpha \in (0,1].$ It is obvious that $\tilde{M}^\prime _\alpha$ satisfies conditions (1) and (2) of Theorem 2.5. For arbitrary $A\in \mathscr {A}$, if $\tilde{M}^\prime (A)(x)>0$, then

$$\begin{aligned} \tilde{M}^\prime (A)(x) =\sup \{\alpha \in (0,1]: \tilde{M}^\prime (A)(x)\ge \alpha \} =\sup \{\alpha \in (0,1]: x\in \tilde{M}^\prime _{\alpha }(A)\}. \end{aligned}$$

If $\tilde{M}^\prime (A)(x)=0$, then for all $\alpha \in (0,1]$, $x\notin \tilde{M}^\prime _\alpha (A),$ i.e., $\{\alpha \in (0,1]: x\in \tilde{M}^\prime _{\alpha }(A)\}=\emptyset .$ Hence, for arbitrary $A\in \mathscr {A}$, $\tilde{M}^\prime (A)$ satisfies

$$\begin{aligned} \tilde{M}^\prime (A)(x)= \left\{ \begin{array}{ll} \sup \{\alpha \in (0,1]: x\in \tilde{M}^\prime _{\alpha }(A)\},\,\, \text{ if } \, \, \{\alpha \in (0,1]: x\in \tilde{M}^\prime _{\alpha }(A)\}\ne \emptyset ; \\ 0, \,\, \text{ if } \, \, \{\alpha \in (0,1]: x\in \tilde{M}^\prime _{\alpha }(A)\}=\emptyset . \end{array} \right. \end{aligned}$$

$\{\tilde{M}^\prime _\alpha (\cdot )\}_{\alpha \in (0,1]}$ satisfies all conditions of Theorem 2.5. It follows that $\tilde{M}^\prime : \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is a generalized fuzzy number measure. It is easy to see that $\tilde{M}^\prime$ is absolutely continuous with respect to $\tilde{M}$, which implies that $\tilde{M}^\prime$ is $\mu$-continuous.

This completes the proof. $\square$

In what follows, some properties of the new integral will be given.

Theorem 3.4

Suppose that $\mathcal {X}$ has the RNP, $\mathcal {X}^*$ is separable and $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is a $\mu$-continuous generalized fuzzy number measure of bounded variation. If $f,g\in L^1_{KL}(\Omega , \mathbb R^+, |\tilde{M}|)$, then

$$\begin{aligned} ({\mathrm {KL}}) \int _\Omega \left( f(\omega )+g(\omega )\right) \,\mathrm {d}\tilde{M}(\omega ) =({\mathrm {KL}}) \int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )+({\mathrm {KL}}) \int _\Omega g(\omega )\,\mathrm {d}\tilde{M}(\omega ). \end{aligned}$$

Proof

From Theorem 3.2, we have $({\mathrm {KL}}) \int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega ), ({\mathrm {KL}}) \int _\Omega g(\omega )\,\mathrm {d}\tilde{M}(\omega )\in \mathscr {F}_{wkc}(\mathcal {X})$ and

$$\begin{aligned} \left[ ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha= & {} ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega ),\\ \left[ ({\mathrm {KL}})\int _\Omega g(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha= & {} ({\mathrm {KL}})\int _\Omega g(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega ) \end{aligned}$$

for each $\alpha \in (0,1]$. It follows that

$$\begin{aligned}&\left[ ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )+ ({\mathrm {KL}})\int _\Omega g(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha \\&\quad =\left[ ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha +\left[ ({\mathrm {KL}})\int _\Omega g(\omega )\,\mathrm {d}\tilde{M}(\omega ) \right] _\alpha \\&\quad = ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )+ ({\mathrm {KL}})\int _\Omega g(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )\\&\quad =\int _\Omega f(\omega )X_\alpha (\omega )\,\mathrm {d}\mu (\omega ) +\int _\Omega g(\omega )X_\alpha (\omega )\,\mathrm {d}\mu (\omega )\\&\quad =\int _\Omega (f(\omega ) + g(\omega ))X_\alpha (\omega )\,\mathrm {d}\mu (\omega )\\&\quad = ({\mathrm {KL}})\int _\Omega \left( f(\omega ) + g(\omega )\right) \,\mathrm {d}\tilde{M}_\alpha (\omega )\\&\quad ={\left[ ({\mathrm {KL}})\int _\Omega (f(\omega ) + g(\omega ))\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha } \end{aligned}$$

for each $\alpha \in (0,1].$ Therefore, we have

$$\begin{aligned}&\left( ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )+({\mathrm {KL}})\int _\Omega g(\omega )\,\mathrm {d}\tilde{M}(\omega )\right) (x) \\&\quad =\sup \left\{ \alpha \in (0,1]: x\in \left[ ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega ) +({\mathrm {KL}})\int _\Omega g(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha \right\} \\&\quad =\sup \left\{ \alpha \in (0,1]: x\in {\left[ ({\mathrm {KL}})\int _\Omega (f(\omega ) + g(\omega ))\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha }\right\} \\&\quad =\left( ({\mathrm {KL}})\int _\Omega \left( f(\omega )+g(\omega )\right) \,\mathrm {d}\tilde{M}(\omega )\right) (x) \end{aligned}$$

for each $x\in \mathcal {X}$, which implies that

$$\begin{aligned} ({\mathrm {KL}})\int _\Omega \left( f(\omega )+g(\omega )\right) \,\mathrm {d}\tilde{M}(\omega ) =({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )+({\mathrm {KL}})\int _\Omega g(\omega )\,\mathrm {d}\tilde{M}(\omega ). \end{aligned}$$

This completes the proof. $\square$

Theorem 3.5

Suppose that $\mathcal {X}$ has the RNP, $\mathcal {X}^*$ is separable and $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is a $\mu$-continuous generalized fuzzy number measure of bounded variation. If $f\in L^1_{KL}(\Omega , \mathbb R^+, |\tilde{M}|)$ and $\lambda \ge 0$, then

$$\begin{aligned} ({\mathrm {KL}})\int _\Omega \lambda f(\omega )\,\mathrm {d}\tilde{M}(\omega )=\lambda ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega ). \end{aligned}$$

Proof

If $f\in L^1(\Omega , \mathbb R^+, |\tilde{M}|)$ and $\lambda \ge 0$, then $\lambda f\in L^1(\Omega , \mathbb R^+, |\tilde{M}|).$ According to Theorem 3.2, we have $({\mathrm {KL}}) \int _\Omega \lambda f(\omega )\,\mathrm {d}\tilde{M}(\omega )\in \mathscr {F}_{wkc}(\mathcal {X})$ and

$$\begin{aligned} \left[ ({\mathrm {KL}}) \int _\Omega \lambda f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha =({\mathrm {KL}}) \int _\Omega \lambda f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega ) \end{aligned}$$

for each $\alpha \in (0,1].$ From properties of the support function and Theorem 2.7, we have

$$\begin{aligned}&d_\mathrm{{H}}\left( ({\mathrm {KL}}) \int _\Omega \lambda f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega ), \lambda ({\mathrm {KL}}) \int _\Omega f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )\right) \\&\quad = d_\mathrm{{H}}\left( \int _\Omega \lambda f(\omega )X_{\alpha }(\omega )\,\mathrm {d}\mu (\omega ), \lambda \int _\Omega f(\omega )X_\alpha (\omega )\,\mathrm {d}\mu (\omega )\right) \\&\quad =\sup _{\Vert x^*\Vert \le 1}\left| \sigma \left( \int _\Omega \lambda f(\omega )X_{\alpha }(\omega )\,\mathrm {d}\mu (\omega ),x^*\right) -\sigma \left( \lambda \int _\Omega f(\omega )X_\alpha (\omega )\,\mathrm {d}\mu (\omega ), x^*\right) \right| \\&\quad =\sup _{\Vert x^*\Vert \le 1}\left| \int _\Omega \sigma \left( \lambda f(\omega )X_{\alpha }(\omega ),x^*\right) \,\mathrm {d}\mu (\omega ) -\lambda \int _\Omega \sigma \left( f(\omega )X_\alpha (\omega ),x^*\right) \,\mathrm {d}\mu (\omega )\right| \\&\quad =\sup _{\Vert x^*\Vert \le 1}\left| \int _\Omega \sigma \left( \lambda f(\omega )X_{\alpha }(\omega ),x^*\right) \,\mathrm {d}\mu (\omega ) -\int _\Omega \sigma \left( \lambda f(\omega )X_\alpha (\omega ),x^*\right) \,\mathrm {d}\mu (\omega )\right| \\&\quad =0 \end{aligned}$$

for each $\alpha \in (0,1].$ Since $(\mathscr {P}_{wkc}(\mathcal {X}), d_\mathrm{{H}})$ is a complete metric space, we have

$$\begin{aligned} ({\mathrm {KL}}) \int _\Omega \lambda f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega ) = \lambda ({\mathrm {KL}}) \int _\Omega f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega ) \end{aligned}$$

for each $\alpha \in (0,1].$ This follows that

$$\begin{aligned} ({\mathrm {KL}}) \int _\Omega \lambda f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )= & {} \lambda ({\mathrm {KL}}) \int _\Omega f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )\\= & {} \lambda \left[ ({\mathrm {KL}}) \int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha \\= & {} \left[ \lambda ({\mathrm {KL}}) \int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha \end{aligned}$$

for each $\alpha \in (0,1].$ Then we have

$$\begin{aligned} \left( ({\mathrm {KL}}) \int _\Omega \lambda f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right) (x)= & {} \sup \left\{ \alpha \in (0,1]: x\in ({\mathrm {KL}}) \int _\Omega \lambda f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )\right\} \\= & {} \sup \left\{ \alpha \in (0,1]: x\in \left[ \lambda ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha \right\} \\= & {} \left( \lambda ({\mathrm {KL}}) \int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right) (x) \end{aligned}$$

for each $x\in \mathcal {X}$, which implies that

$$\begin{aligned} ({\mathrm {KL}})\int _\Omega \lambda f(\omega )\,\mathrm {d}\tilde{M}(\omega )=\lambda ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega ). \end{aligned}$$

This completes the proof. $\square$

Theorem 3.6

Suppose that $\mathcal {X}$ has the RNP, $\mathcal {X}^*$ is separable and $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is a $\mu$-continuous generalized fuzzy number measure of bounded variation. If $f\in L^1_{KL}(\Omega , \mathbb R^+, |\tilde{M}|)$, then we have

$$\begin{aligned} \left\| ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right\| \le \int _A {f(\omega )}\,\mathrm {d}|\tilde{M}|(\omega ) \end{aligned}$$

for each $A\in \mathscr {A}$.

Proof

By Theorem 3.2 and Theorem 9 Zhou and Shi (2015a), we have

$$\begin{aligned} \left| ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )\right| \le \int _A{f(\omega )}\,\mathrm {d}|\tilde{M}_\alpha |(\omega ) \end{aligned}$$

for each $A\in \mathscr {A}$ and $\alpha \in (0,1]$. Since $|\tilde{M}_\alpha |(A) \le |\tilde{M}|(A)$ for each $A\in \mathscr {A}$, we have

$$\begin{aligned} \left| ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )\right| \le \int _A{f(\omega )}\,\mathrm {d}|\tilde{M}|(\omega ) \end{aligned}$$

for each $A\in \mathscr {A}$ and $\alpha \in (0,1]$. It follows that

$$\begin{aligned} \left\| ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right\|= & {} \sup _{\alpha \in (0,1]} \left| \left[ ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha \right| \\= & {} \sup _{\alpha \in (0,1]} \left| ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )\right| \\\le & {} \int _A{f(\omega )}\,\mathrm {d}|\tilde{M}|(\omega ). \end{aligned}$$

This completes the proof. $\square$

Corollary 3.7

Suppose that $\mathcal {X}$ has the RNP, $\mathcal {X}^*$ is separable, $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is a $\mu$-continuous generalized fuzzy number measure of bounded variation. If $f\in L^1_{KL}(\Omega , \mathbb R^+, |\tilde{M}|)$, then we have

• (1) $\lim _{A\rightarrow \emptyset } \Vert ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega ) \Vert =0;$

• (2) if $A=\emptyset$, then $(\mathrm {KL})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega )=\tilde{0}$;

• (3) if $f=0$, then $(\mathrm {KL})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega )=\tilde{0}$ for each $A\in \mathscr {A}.$

Proof

• (1) By Theorem 3.6, we have

  $$\begin{aligned} \left\| ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right\|\le & {} \int _A {f(\omega )}\,\mathrm {d} | \tilde{M} |(\omega ), \\ \end{aligned}$$

  which implies that

  $$\begin{aligned} \lim _{A\rightarrow \emptyset } \left\| ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right\|\le & {} \lim _{A\rightarrow \emptyset } \int _A {f(\omega )}\,\mathrm {d}|\tilde{M}|(\omega )=0.\\ \end{aligned}$$

  Thus, $\lim _{A\rightarrow \emptyset } \Vert ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega ) \Vert =0.$

• (2) If $A=\emptyset$, then $\int _A {f(\omega )}\,\mathrm {d}| \tilde{M}|(\omega )=0$. This follows that

  $$\begin{aligned} d_\mathrm{{H}}^\infty \left( ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega ),\tilde{0}\right)= & {} \left\| ({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right\| \\\le & {} \int _A {f(\omega )}\,\mathrm {d} | \tilde{M} |(\omega )=0,\\ \end{aligned}$$

  which implies that $d_\mathrm{{H}}^\infty (({\mathrm {KL}})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega ),\tilde{0} )=0$. Since $(\mathscr {F}_{wkc}(\mathcal {X}), d_\mathrm{{H}}^\infty )$ is a metric space, $(\mathrm {KL})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega )=\tilde{0}$.

• (3) If $f=0$, then for each $x^*\in \mathcal {X}^*$, $A\in \mathscr {A}$ and $\alpha \in (0,1],$ we have

  $$\begin{aligned} \sigma \left( x^*, ({\mathrm {KL}})\int _{A } f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )\right) =\int _{A } f(\omega )\,\mathrm {d} \sigma (x^*, \tilde{M}_\alpha (\omega )) =0, \end{aligned}$$

  which implies that $(\mathrm {KL})\int _A f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )=\{0\}$ for each $A\in \mathscr {A}$ and $\alpha \in (0,1].$ It follows that $(\mathrm {KL})\int _A f(\omega )\,\mathrm {d}\tilde{M}(\omega )=\tilde{0}$ for each $A\in \mathscr {A}.$ This completes the proof.

$\square$

In the following, we give the Vitali type convergence theorem for the fuzzy-valued Kluvánek–Lewis integral.

Theorem 3.8

Suppose that $\mathcal {X}$ has the RNP, $\mathcal {X}^*$ is separable, $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is a $\mu$-continuous generalized fuzzy number measure of bounded variation. Let $f ,f_n \in L^1_{KL}(\Omega , \mathbb R^+, |\tilde{M}|), n\in \mathbb N,$ be such that $\{f_n\}_{n\in \mathbb N}$ is uniformly integrable with respect to $|\tilde{M}|$ and $\lim _{n\rightarrow \infty }f_n=f$ $|\tilde{M}|$-almost everywhere. Then

$$\begin{aligned} \lim _{n\rightarrow \infty }d_\mathrm{{H}}^\infty \left( ({\mathrm {KL}})\int _\Omega f_n(\omega )\,\mathrm {d} \tilde{M} (\omega ), ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d} \tilde{M} (\omega )\right) =0. \end{aligned}$$

Proof

By Theorem 3.2, we have $({\mathrm {KL}})\int _\Omega f_n(\omega )\,\mathrm {d}\tilde{M}(\omega )$, $({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )\in \mathscr {F}_{wkc}(\mathcal {X}),$ $n\in \mathbb N,$ such that

$$\begin{aligned} \left[ ({\mathrm {KL}}) \int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha =({\mathrm {KL}}) \int _\Omega f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega ) \end{aligned}$$

and

$$\begin{aligned} \left[ ({\mathrm {KL}}) \int _\Omega f_n(\omega )\,\mathrm {d}\tilde{M}(\omega )\right] _\alpha =({\mathrm {KL}}) \int _\Omega f_n(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega ) \end{aligned}$$

for each $\alpha \in (0,1]$ and $n\in \mathbb N$. Then, by Theorem 11 Zhou and Shi (2015a) and Theorem 3.5 Park (2010), we have

$$\begin{aligned}&d_\mathrm{{H}}\left( ({\mathrm {KL}})\int _\Omega f_n(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega ), ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )\right) \\&\quad \le \int _\Omega \left| f_n(\omega )-f(\omega )\right| \,\mathrm {d} |\tilde{M}_\alpha |(\omega )\\&\quad \le \int _\Omega \left| f_n(\omega )-f(\omega )\right| \,\mathrm {d} |\tilde{M}|(\omega ) \end{aligned}$$

for each $\alpha \in (0,1].$ It follows that

$$\begin{aligned}&d_\mathrm{{H}}^\infty \left( ({\mathrm {KL}})\int _\Omega f_n(\omega )\,\mathrm {d}\tilde{M}(\omega ), ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right) \\&\quad =\sup _{\alpha \in (0,1]}d_\mathrm{{H}}\left( ({\mathrm {KL}})\int _\Omega f_n(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega ), ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}_\alpha (\omega )\right) \\&\quad \le \int _\Omega \left| f_n(\omega )-f(\omega )\right| \,\mathrm {d} |\tilde{M}|(\omega ). \end{aligned}$$

Therefore, by classical Vitali’s convergence theorem, we can conclude that

$$\begin{aligned}&d_\mathrm{{H}}^\infty \left( ({\mathrm {KL}})\int _\Omega f_n(\omega )\,\mathrm {d}\tilde{M}(\omega ), ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d}\tilde{M}(\omega )\right) \\&\quad \le \int _\Omega \left| f_n(\omega )-f(\omega )\right| \,\mathrm {d} |\tilde{M}|(\omega )\rightarrow 0 \end{aligned}$$

as $n\rightarrow \infty .$ This completes the proof. $\square$

Similarly, we can obtain the dominated convergence theorem for the fuzzy-valued Kluvánek–Lewis integral as follows:

Theorem 3.9

Suppose that $\mathcal {X}$ has the RNP, $\mathcal {X}^*$ is separable, $\tilde{M}: \mathscr {A}\rightarrow \mathscr {F}_{wkc}(\mathcal {X})$ is a $\mu$-continuous generalized fuzzy number measure of bounded variation. Let $f ,f_n \in L^1_{KL}(\Omega , \mathbb R^+, |\tilde{M}|), n\in \mathbb N,$ be such that $\lim _{n\rightarrow \infty }f_n=f$ $|\tilde{M}|$-almost everywhere. If there exists a nonnegative, $|\tilde{M}|$-integrable function $g : \Omega \rightarrow \mathbb R$ such that $|f_n(\omega )|\le g(\omega )$ for all $n\in \mathbb N$ and $\omega \in \Omega$, then we have

$$\begin{aligned} \lim _{n\rightarrow \infty }d_\mathrm{{H}}^\infty \left( ({\mathrm {KL}})\int _\Omega f_n(\omega )\,\mathrm {d} \tilde{M}(\omega ), ({\mathrm {KL}})\int _\Omega f(\omega )\,\mathrm {d} \tilde{M} (\omega )\right) =0. \end{aligned}$$

Proof

By Theorem 12 Zhou and Shi (2015a), Theorem 3.5 Park (2010) and classical Lebesgue-dominated convergence theorem, we can obtain the result. $\square$

4 Conclusions

In the current paper, a new integral of scalar-valued functions with respect to a fuzzy-valued measure is introduced. Some natural properties of this kind of integral are investigated and convergence theorems are established. In all applications which involve measure and integral, when measurement or data are fuzzy, the structure defined in this paper can be applied. There are several issues for further investigation connected with current results: specific properties of the integral, such as Chebyshev type inequalities and Markov type inequalities, application on random case—expectation, conditional expectation, martingales and similar structures, and application in economy.