Keywords

1 Introductions

In 1998, fuzzy measure range is extended to the fuzzy real number field by Wu et al. [4] etc., which give the definition of Sugeno integral base on fuzzy number fuzzy measure, Guo et al. [5] etc. Also give the definition of (G) integral on fuzzy measure of fuzzy valued functions, which will be generalized the Sugeno integral to fuzzy sets [6]. In 1989 Buckley [7] proposed the concepts of fuzzy complex number, including people need to consider the measure and integration problems of fuzzy complex numbers, introduction fuzzy distance by Zhang [8–12], which discussed the fuzzy real valued measure problem of fuzzy sets, and give the fuzzy real valued fuzzy integral; in 1996, fuzzy measure and measurable function concept is extended to fuzzy complex sets by Qiu et al. [13], which given the concept of complex fuzzy measure, complex fuzzy measurable function and complex fuzzy integral, Wang and Li [14] etc. in 1999 based on the concepts of fuzzy number of Buckley, gives the concept of fuzzy complex valued measures and fuzzy complex valued integral, obtain some important results. The [15–17] study measurable function and its integral of complex fuzzy number set, especially Sugeno and Choquet type fuzzy complex numerical integral and its properties, and its application in classification technique. In this paper, base on the research work on the basis of [1–3], gives the convergence theorem of complex fuzzy set-valued complex fuzzy integral, lays a foundation for the complex fuzzy set-valued complex fuzzy integral theory.

2 Complex Fuzzy Set-Valued Complex Fuzzy Measure

Definition 1

([18]) Suppose (X, F) is a classic measurable space, \(E\rightarrow F\), mapping \(f:X \rightarrow [- \infty , + \infty ]\), f is called real valued measurable function of (X, F) on \(\tilde{E}\), if and only if \(\forall \alpha \in ( - \infty , + \infty )\), \(\tilde{E} \cap {\chi _{{F_\alpha }}} \in \mathcal{F}\), \(\widetilde{E} \cap x_{{F_\alpha }}^c \in \mathcal{F}\), where \({F_\alpha } = \{ x:f(x) \ge \alpha \} \), mapping \(\tilde{f}:X \rightarrow {F^ * }(R)\), f is called real valued measurable function of \((X,\mathcal{F})\) on \(\tilde{E}\), if and only if \(\forall \lambda \in (0,1]\), \({f_\lambda }^ - (x),{f_\lambda }^ + (x)\) is a real valued measurable function, where

$$\tilde{f}(x) = \bigcup _{\lambda \in [0,1]} {\lambda [(f(x))_{_\lambda }^ - } ,(f(x))_{_\lambda }^ + ] \buildrel \varDelta \over = \bigcup _{\lambda \in [0,1]} {\lambda [f_{_\lambda }^ - (x)} ,f_{_\lambda }^ + (x)].$$

Definition 2

([1]) Suppose Z is a non-empty complex numbers set, F(Z) is set kinds on Z that consisting of by all complex fuzzy set \(\tilde{\rho }\) is fuzzy complex valued distance that defined in F(Z), set function

$$\tilde{\mu }: F(Z) \rightarrow F_ + ^ * (K) = \{ \tilde{A} + i\tilde{B}: \tilde{A},\tilde{B} \in F_ + ^ * (R), i = \sqrt{ - 1} \}, $$
$$\tilde{A} + i\tilde{B} \mapsto \tilde{\mu }(\tilde{A} + i\tilde{B}) \in F_ + ^ * (K)$$

called complex fuzzy set-valued complex fuzzy measure on (Z, F(Z)) if and only if

  1. 1.

    \(\tilde{\mu }(\phi ) = \tilde{0}\), \(\tilde{0} = (\tilde{0},\tilde{0})\), \(\tilde{0} \in {F^ * }(K),\)

  2. 2.

    \(\forall \tilde{A},\tilde{B} \in F(Z),\tilde{A} \subset \tilde{B} \Rightarrow \tilde{\mu }(\tilde{A}) \le \tilde{\mu }(\tilde{B})\), where \({{\mathrm{{Re}}}} \tilde{\mu }(\tilde{A}) \le {{\mathrm{{Re}}}} \tilde{\mu }(\tilde{B}),{{\mathrm{{Im}}}} \tilde{\mu } (\tilde{A}) \le {{\mathrm{{Im}}}} \tilde{\mu }(\tilde{B}),\)

  3. 3.

    \(\{ {\tilde{A}_n}\} \subset F\left( Z \right) {\tilde{A}_n} \subset {\tilde{A}_{n + 1}}\left( {n = 1, 2 \ldots } \right) \Rightarrow \tilde{\rho }\mathop {\lim }\limits _{n \rightarrow \infty } \tilde{\mu }{\tilde{A}_n} = \tilde{\mu }\left( {\mathop {\bigcup }\limits _{n = 1}^\infty {{{\tilde{A}}_n}} } \right) ,\)

  4. 4.

    \(\left\{ {{{\tilde{A}}_n}} \right\} \subset F\left( Z \right) {\tilde{A}_n} \supset {\tilde{A}_{n + 1}}\left( {n = 1, 2 \ldots } \right) ,\)

note as \(\tilde{\mu }= {{\mathrm{{Re}}}} \tilde{\mu }+ i{{\mathrm{{Im}}}} \tilde{\mu }\buildrel \varDelta \over = {\tilde{\mu }_R} + i{\tilde{\mu }_I},(i = \sqrt{ - 1} ).\)

Definition 3

([2]) Suppose \(Z \subset K\) is a non-empty set of complex numbers, \((Z,\mathcal{F}(Z))\) is a classical complex measurable space, \(\tilde{E} \in \mathcal{F}(Z)\), mapping \(f: Z \rightarrow K\), called f is a complex valued measurable function on \(\tilde{E}\) about \((Z,\mathcal{F}(Z))\), if and only if \(\forall a + ib \in K\), \(\tilde{E} \cap {\chi _{{F_{a,b}}}} \in \mathcal{F}(Z)\), and \(\tilde{E} \cap {\chi ^c}_{{F_{a,b}}} \in \mathcal{F}(Z)\), where

$$ {F_{a,b}} = \left\{ {z \in K\left| {{{\mathrm{{Re}}}} \left[ {f(z)} \right] \ge a,{{\mathrm{{Im}}}} \left[ {f(z)} \right] \ge b} \right. } \right\} . $$

Definition 4

([2]) Suppose Z is a non-empty complex numbers set, \(\tilde{E} \in \mathcal{F}(Z)\), mapping \(\tilde{f}: Z \rightarrow {F_0}\left( K \right) \), \(z \mapsto \tilde{f}\left( z \right) = {{\mathrm{{Re}}}} \tilde{f}\left( z \right) + i{{\mathrm{{Im}}}} \tilde{f}\left( z \right) \in {F_0}\left( K \right) \), \(i = \sqrt{ - 1} \),

$$\tilde{f}\left( z \right) = {\bigcup _{\lambda \in [0,1]} {\lambda \left[ {\left( {{{\mathrm{{Re}}}} \tilde{f}(z} \right) } \right] } _\lambda } + i{\bigcup _{\lambda \in [0,1]} {\lambda \left[ {{{\mathrm{{Im}}}} \tilde{f}\left( z \right) } \right] } _\lambda } $$
$$\buildrel \varDelta \over = \bigcup _{\lambda \in [0,1]} {\lambda {{\mathrm{{Re}}}} {{\tilde{f}}_\lambda }\left( z \right) + } i\bigcup _{\lambda \in [0,1]} {\lambda {{\mathrm{{Im}}}} {{\tilde{f}}_\lambda }\left( z \right) } $$
$$\buildrel \varDelta \over = \bigcup _{\lambda \in [0,1]} {\lambda \left[ {{{\mathrm{{Re}}}} {{\tilde{f}}_\lambda }^ - \left( z \right) ,{{\mathrm{{Re}}}} {{\tilde{f}}_\lambda }^ + \left( z \right) } \right] + } i\bigcup _{\lambda \in [0,1]} {\lambda \left[ {{{\mathrm{{Im}}}} {{\tilde{f}}^ - }_\lambda \left( z \right) ,{{\mathrm{{Im}}}} {{\tilde{f}}^ + }_\lambda \left( z \right) } \right] },$$

then \(\left( {Z,\mathcal{F}\left( Z \right) , \tilde{\mu }} \right) \) is complex fuzzy valued fuzzy measure space, called \(\tilde{f}\) is complex fuzzy valued complex fuzzy measurable function on \(\tilde{E}\) about \((Z,\mathcal{F}\left( Z \right) ,\tilde{\mu })\) if and only if \(\forall \lambda \in \left[ {0,1} \right] \), \({{\mathrm{{Re}}}} {\tilde{f}_\lambda }(z)\), \({{\mathrm{{Im}}}} {\tilde{f}_\lambda }(z)\), which are complex valued measurable function on \(\tilde{E}\) about \((Z,\mathcal{F}(Z))\)

record

$${F_{(\alpha ,\beta ),\lambda }} \buildrel \varDelta \over = \left\{ {z = \alpha + i\beta :{{\mathrm{{Re}}}} f_\lambda ^ \pm \left( z \right) \ge \alpha ,{{\mathrm{{Im}}}} f_\lambda ^ \pm \left( z \right) \ge \beta } \right\} ,$$

where

$${{\mathrm{{Re}}}} f_\lambda ^ \pm \left( z \right) \ge \alpha ,$$

express

$${{\mathrm{{Re}}}} f_\lambda ^ + \left( z \right) \ge \alpha \, {\mathrm{and}} \, {{\mathrm{{Re}}}} f_\lambda ^ - \left( z \right) \ge \alpha , {{\mathrm{{Im}}}} f_\lambda ^ \pm \left( z \right) \ge \beta ,$$

express

$${{\mathrm{{Im}}}} f_\lambda ^ + \left( z \right) \ge \beta \, {\mathrm{and}} \, {{\mathrm{{Im}}}} f_\lambda ^ - \left( z \right) \ge \beta ,\forall \alpha ,\beta \in \left[ {0,\infty } \right) ,$$

thus, \(\tilde{f}\) is complex fuzzy valued complex fuzzy measurable function on \(\tilde{E}\) about \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \) if and only if \(\forall \lambda \in \left[ {0,1} \right] ,\) \(\tilde{E} \cap {\chi _{{F_{\left( {\alpha ,\beta } \right) )\lambda }}}} \in \mathcal{F}\left( Z \right) ,\) \(\tilde{E} \cap {\chi ^c}_{{F_{\left( {\alpha ,\beta } \right) ,\lambda }}} \in \mathcal{F}\left( Z \right) \) \(\tilde{M}\left( {\tilde{E}} \right) \) express all of the complex fuzzy set-valued measurable function on \(\tilde{E}\).

3 Complex Fuzzy Set-valued Complex Fuzzy Integral and Its Properties

Definition 5

([3]) Suppose \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \) is complex fuzzy set-valued fuzzy measure space, \(\tilde{E} \in \mathcal{F}\left( Z \right) \), \(\tilde{f}: Z \rightarrow {F_0}\left( K \right) \), define \(\tilde{f}\) is complex fuzzy set-valued complex fuzzy integral on \(\tilde{E}\) about \(\tilde{\mu }\),

$$\int _{\tilde{E}} {\tilde{f}} d\tilde{\mu }\buildrel \varDelta \over = {\left( {\int _{\tilde{E}} {{{\mathrm{{Re}}}} \tilde{f}} d{{\tilde{\mu }}_R},\int _{\tilde{E}} {{{\mathrm{{Im}}}} \tilde{f}} d\tilde{\mu }} \right) _I},$$

where

$$\int _{\tilde{E}} {{{\mathrm{{Re}}}} \tilde{f}} d{\tilde{\mu }_R} = \bigcup _{\lambda \in \left[ {0,1} \right] } {\lambda \left[ {\int _{\tilde{E}} {{\mathrm{{Re}}} {f_\lambda }^ - d{{\tilde{\mu }}_R}} ,\int _{\tilde{E}} {{{\mathrm{{Re}}}} {f_\lambda }^ + d{{\tilde{\mu }}_R}} } \right] }$$
$$= \bigcup _{\lambda \in \left[ {0,1} \right] } {\lambda \left[ { \mathop {\sup }\limits _{\alpha \in [0,\infty )} \alpha \wedge {{\mathrm{{Re}}}} {{\tilde{\mu }}_\lambda }^ - \left( {\tilde{A} \cap {\chi _{{F_{\lambda ,\alpha ,1}}^ - }}} \right) , \mathop {\sup }\limits _{\alpha \in [0,\infty )} \alpha \wedge {{\mathrm{{Re}}}} {{\tilde{\mu }}^ + }_\lambda \left( {\tilde{A} \cap {\chi _{{F_{\lambda ,\alpha ,1}}^ + }}} \right) } \right] } $$
$$\int _{\tilde{E}} {{{\mathrm{{Im}}}} \tilde{f}} d{\tilde{\mu }_I} = \bigcup _{\lambda \in \left[ {0,1} \right] } {\lambda \left[ {\int _{\tilde{E}} {{{\mathrm{{Im}}}} {f_\lambda }^ - d{{\tilde{\mu }}_I}} ,\int _{\tilde{E}} {{{\mathrm{{Im}}}} {f_\lambda }^ + d{{\tilde{\mu }}_I}} } \right] }$$
$$=\bigcup _{\lambda \in \left[ {0,1} \right] } {\lambda \left[ { \mathop {\sup }\limits _{\alpha \in [0,\infty )} \alpha \wedge {{\mathrm{{Im}}}} {{\tilde{\mu }}_\lambda }^ - (\tilde{A} \cap {\chi _{{F_{\lambda ,\alpha ,2}}^ - }}, \mathop {\sup }\limits _{\alpha \in [0,\infty )} \alpha \wedge {{\mathrm{{Im}}}} {{\tilde{\mu }}^ + }_\lambda (\tilde{A} \cap {\chi _{{F_{\lambda ,\alpha ,2}}^ + }}} \right] },$$

where

$${F_{\lambda ,\alpha ,1}}^ - = \left\{ {z|{{\mathrm{{Re}}}} {f_\lambda }^ - \left( z \right) \ge \alpha } \right\} ,$$
$${F_{\lambda ,\alpha ,1}}^ + = \left\{ {z|{{\mathrm{{Re}}}} {f_\lambda }^ + \left( z \right) \ge \alpha } \right\} ,$$
$${F_{\lambda ,\alpha ,2}}^ - = \left\{ {z|{{\mathrm{{Im}}}} {f_\lambda }^ - \left( z \right) \ge \alpha } \right\} ,$$
$${F_{\lambda ,\alpha ,2}}^ + = \left\{ {z|{{\mathrm{{Im}}}} {f_\lambda }^ + \left( z \right) \ge \alpha } \right\} ,$$

now called \(\tilde{f}\) complex fuzzy set-value complex fuzzy integrable in \(\tilde{E}\) about \(\tilde{\mu }\).

Complex fuzzy set-valued complex fuzzy integral has the following important properties:

Theorem 1

([3]) Suppose \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \) is complex fuzzy set-valued fuzzy measure space:

  1. 1.

    \(\forall \tilde{E} \in \mathcal{F}\left( Z \right) \), \(\tilde{f} \in \tilde{M}\left( {\tilde{E}} \right) \), then

    $$\int _{\tilde{E}} {\tilde{f}} d\tilde{\mu }\in {F^ * }\left( K \right) ,$$
  2. 2.

    Suppose \(\tilde{E} \in \mathcal{F}\left( Z \right) \), if \(\tilde{f} \in \tilde{M}\left( {\tilde{E}} \right) \), \({\chi _{\tilde{E}}}\) is the characteristic function of \(\tilde{E}\), then \(\int _{\tilde{E}} {\tilde{f}} d\tilde{\mu }= \int {\tilde{f}} {\chi _{\tilde{E}}}d\tilde{\mu }, \)

  3. 3.

    Let \(\tilde{E} \in \mathcal{F}\left( Z \right) \). If \(\tilde{f} \in \tilde{M}\left( {\tilde{E}} \right) \), if \(\tilde{\mu }\left( {\tilde{E}} \right) = \tilde{0}\), then

    $$\int _{\tilde{E}} {\tilde{f}} d\tilde{\mu }= \tilde{0},$$
  4. 4.

    Let \(\tilde{A},\tilde{B} \in \mathcal{F}\left( Z \right) \). If \(\tilde{f} \in \tilde{M}\left( {\tilde{E}} \right) \), if \(\tilde{A} \subset \tilde{B}\), then

    $$\int _{\tilde{A}} {\tilde{f}} d\tilde{\mu }\subseteq \int _{\tilde{B}} {\tilde{f}} d\tilde{\mu }, $$
  5. 5.

    Let \(\tilde{A} \in \mathcal{F}\left( Z \right) \). If \({\tilde{f}_1},{\tilde{f}_2} \in \tilde{M}\left( {\tilde{E}} \right) \), if \({\tilde{f}_1} \subseteq {\tilde{f}_2}\) in \(\tilde{A}\), then

    $$\int _{\tilde{A}} {{{\tilde{f}}_1}} d\tilde{\mu }\subseteq \int _{\tilde{A}} {{{\tilde{f}}_2}} d\tilde{\mu }, $$

    these properties are demonstrated in [18]. Here ignore.

4 Complex Fuzzy Set-Value Complex Fuzzy Integral and Its Convergence Theorem

Theorem 2

Suppose \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \) is complex fuzzy set-valued fuzzy measure space, \(\left\{ {{{\tilde{f}}_n}} \right\} \) is non-negative complex fuzzy set-valued complex fuzzy integrable function sequence in \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \), \(A \in \mathcal{F}\left( Z \right) \), if in A, \(\left\{ {{{\tilde{f}}_n}} \right\} \) monotone convergence in \(\tilde{f}\) incrementing, then \( \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}d} \tilde{\mu }= \int _A {\tilde{f}d} \tilde{\mu }. \)

Proof

Suppose \(A = X\), because \({\tilde{f}_n} \le \tilde{f}\), \(\left( {n = 1,2,\ldots } \right) \), so by the generalized complex fuzzy set-valued complex fuzzy integral properties have the following result

$$ \mathop {\lim }\limits _{n \rightarrow \infty } \int _X {{{\tilde{f}}_n}d} \tilde{\mu }\le \int _X {\tilde{f}d} \tilde{\mu }, $$

now to proof the opposite inequality.

Let \(I = \int _X {\tilde{f}d} \tilde{\mu }\). Then

  1. 1.

    if \(I = 0\), conclusion obvious,

  2. 2.

    if \(0 < I < \infty + i\infty \) then

    $$I = \mathop {\sup }\limits _{{{\mathrm{{Re}}}} \alpha \in [0,\infty )} S\left( {{{\mathrm{{Re}}}} \alpha ,{{\mathrm{{Re}}}} \tilde{\mu }{{\left( {\tilde{f}} \right) }_\alpha }} \right) + i \mathop {\sup }\limits _{{\mathrm{{Im}}} \alpha \in [0,\infty )} S\left( {{\mathrm{{Im}}} \alpha ,{{\mathrm{{Im}}}} \tilde{\mu }\left( {{{\tilde{f}}_\alpha }} \right) } \right) $$

    to know the exist \({\alpha _k} > 0\) makes

    $$\begin{aligned}&S\left( {{{\mathrm{{Re}}}} {\alpha _k},{\mathrm{{Re}}} \tilde{\mu }\left( {{{\tilde{f}}_{{\alpha _k}}}} \right) } \right) \mathrm{{ > }}ReI\mathrm{{ - }}\frac{1}{{2k}}, S\left( {{{\mathrm{{Im}}}} {\alpha _k},{{\mathrm{{Im}}}} \tilde{\mu }\left( {{{\tilde{f}}_{{\alpha _k}}}} \right) } \right) \mathrm{{ > }}ImI \mathrm{{ - }} \frac{1}{{2k}},\\&\left( {k = 1,2,\ldots } \right) , \quad another \quad {\tilde{f}_n} \quad \uparrow \tilde{\quad }f \quad have \end{aligned}$$
    $${\left( {{{\tilde{f}}_n}} \right) _{{\alpha _k}}} \uparrow {\left( {\tilde{f}} \right) _{{\alpha _k}}},$$

    then using the properties of generalized triangle norm, know exist \({n_k}\), such that,

    when \(n \ge {n_k}\),

    $$S\left( {{{\mathrm{{Re}}}} {\alpha _k},{{\mathrm{{Re}}}} \tilde{\mu }\left( {{{\tilde{f}}_{{\alpha _k}}}} \right) } \right) $$
    $$\mathrm{{ > }}ReI\mathrm{{ - }}\frac{1}{{2k}},S\left( {{{\mathrm{{Im}}}} {\alpha _k},{{\mathrm{{Im}}}} \tilde{\mu }\left( {{{\tilde{f}}_{{\alpha _k}}}} \right) } \right) , \mathrm{{ > }}ImI\mathrm{{ - }}\frac{1}{{2k}},\left( {k = 1,2,\ldots } \right) ,$$

    when \(n \ge {n_k}\),

    $$\int _X {{{\tilde{f}}_n}} d\tilde{\mu }> I - \frac{1}{k}, \left( {k = 1,2,\ldots } \right) ,$$

    by the arbitrariness of k have

    $$\int _X {\tilde{f}} d\tilde{\mu }\le \mathop {\lim }\limits _{n \rightarrow \infty } \int _X {{{\tilde{f}}_n}} d\tilde{\mu }. $$
  3. 3.

    if \(I = \infty + i\infty \), then \({\alpha _k} > 0\) makes

    $$S\left( {{{\mathrm{{Re}}}} {\alpha _k},{{\mathrm{{Re}}}} \tilde{\mu }\left( {\left( {{{\tilde{f}}_{{\alpha _k}}}} \right) } \right) } \right) \mathrm{{ > k}},$$
    $$S({{\mathrm{{Im}}}} {\alpha _k}, {{\mathrm{{Im}}}} \tilde{\mu }({\tilde{f}_{{\alpha _k}}}))\mathrm{{ > k}}, \left( {k = 1,2,\ldots } \right) ,$$

    exit \({n_k}\), such that when \(n \ge {n_k}\),

    $$S\left( {{{\mathrm{{Re}}}} {\alpha _k},{{\mathrm{{Re}}}} \tilde{\mu }\left( {{{\tilde{f}}_{{\alpha _k}}}} \right) } \right) \mathrm{{ > k}},$$
    $$S\left( {{{\mathrm{{Im}}}} {\alpha _k},{{\mathrm{{Im}}}} \tilde{\mu }\left( {{{\tilde{f}}_{{\alpha _k}}}} \right) } \right) \mathrm{{ > k}},$$

    then

    $$ \mathop {\lim }\limits _{n \rightarrow \infty } \int _X {{{\tilde{f}}_n}} d\tilde{\mu }\ge \int _X {\tilde{f}} d\tilde{\mu }\ge S\left( {{{\mathrm{{Re}}}} {\alpha _k},{{\mathrm{{Re}}}} \tilde{\mu }\left( {{{\tilde{f}}_{{\alpha _k}}}} \right) } \right) $$
    $$ {+} iS\left( {{{\mathrm{{Im}}}} {\alpha _k},{{\mathrm{{Im}}}} \tilde{\mu }\left( {{{\tilde{f}}_{{\alpha _k}}}} \right) } \right) > k + ik\left( {k = 1,2,\ldots } \right) ,$$

that is \( \mathop {\lim }\limits _{n \rightarrow \infty } \int _X {{{\tilde{f}}_n}} d\tilde{\mu }\ge \int _X {\tilde{f}} d\tilde{\mu }. \)

Theorem 3

Suppose \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \) is complex fuzzy set-valued fuzzy measure space, \(\left\{ {{{\tilde{f}}_n}} \right\} \) is complex fuzzy set value complex fuzzy integrable function sequence in \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \), \(A \in \mathcal{F}\left( Z \right) \), if in A, \(\left\{ {{{\tilde{f}}_n}} \right\} \) decrease monotonically converges to \(\tilde{f}\), and for arbitrarily \({\varepsilon _i} > 0,\left( {i = 1,2} \right) \), where \(\varepsilon = {\varepsilon _1} + i{\varepsilon _2}\) exit \({n_0}\) makes

$$\tilde{\mu }\left( {\left\{ {x|{{\tilde{f}}_{{n_0}}} > \int _A {\tilde{f}d\tilde{\mu }+ \varepsilon } } \right\} \cap A} \right) < \infty + i\infty , $$

then

$$ \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}d} \tilde{\mu }= \int _A {\tilde{f}d} \tilde{\mu }. $$

Proof

\({\tilde{f}_1} \ge {\tilde{f}_2} \ge \cdots ,\) so \(\int _A {{{\tilde{f}}_1}} d\tilde{\mu }\ge \int _A {{{\tilde{f}}_2}} d\tilde{\mu }\ge \cdots ,\) then

$$ \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}} d\tilde{\mu }= \mathop {\wedge }\limits _{n = 1}^\infty \int _A {{{\tilde{f}}_n}} d\tilde{\mu },$$

there

$$ \mathop {\wedge }\limits _{n = 1}^\infty \int _A {{{\tilde{f}}_n}} d\tilde{\mu }= \mathop {\wedge }\limits _{n = 1}^\infty {\mathrm{{Re}}} \int _A {{{\tilde{f}}_n}} d\tilde{\mu }+ i \mathop {\wedge }\limits _{n = 1}^\infty {{\mathrm{{Im}}}} \int _A {{{\tilde{f}}_n}} d\tilde{\mu }, $$

due to \(\forall n,{\tilde{f}_n} \ge \tilde{f}\), so

$$\int _A {{{\tilde{f}}_n}} d\tilde{\mu }\ge \int _A {\tilde{f}} d\tilde{\mu }$$

have

$$ \mathop {\wedge }\limits _{n = 1}^\infty \int _A {{{\tilde{f}}_n}} d\tilde{\mu }\ge \int _A {\tilde{f}} d\tilde{\mu }$$

if \( \mathop {\wedge }\limits _{n = 1}^\infty \int _A {{{\tilde{f}}_n}} d\tilde{\mu }> \int _A {\tilde{f}} d\tilde{\mu }, \) then \(\int _A {\tilde{f}} d\tilde{\mu }= \lambda < \infty + i\infty \) where \(\lambda = {\lambda _1} + i{\lambda _2}\), and there is \({\gamma _i} \in (0,\infty )\) where \(\gamma = {\gamma _1} + i{\gamma _2}\), \(\left( {i = 1,2} \right) \) makes

$$ \mathop {\wedge }\limits _{n = 1}^\infty \int _A {{{\tilde{f}}_n}} d\tilde{\mu }> \gamma > \lambda ,$$
$$\Rightarrow \forall n, \mathop {\sup }\limits _{{{\mathrm{{Re}}}} \alpha \in \left[ {0,\infty } \right) } S\left( {{{\mathrm{{Re}}}} \alpha ,{\mathrm{{Re}}} \tilde{\mu }[{{\left( {{{\tilde{f}}_n}} \right) }_\alpha } \cap A]} \right) $$
$$+ i \mathop {\sup }\limits _{{\mathrm{{Im}}} \alpha \in \left[ {0,\infty } \right) } S\left( {{{\mathrm{{Im}}}} \alpha ,{{\mathrm{{Im}}}} \tilde{\mu }\left( {{{\left( {{{\tilde{f}}_n}} \right) }_\alpha } \cap A]} \right) } \right) > \gamma ,$$
$$\Rightarrow \forall n,S\left( {\gamma ,\tilde{\mu }\left( {{{\left( {{{\tilde{f}}_n}} \right) }_\gamma } \cap A} \right) } \right) = S\left( {{{\mathrm{{Re}}}} \gamma ,{{\mathrm{{Re}}}} \tilde{\mu }\left( {{{\left( {{{\tilde{f}}_n}} \right) }_\gamma } \cap A} \right) } \right) $$
$${+} iS\left( {{{\mathrm{{Im}}}} \gamma ,{{\mathrm{{Im}}}} \tilde{\mu }\left( {{{\left( {{{\tilde{f}}_n}} \right) }_\gamma } \cap A} \right) } \right) > \gamma ,$$

take \({\varepsilon _l} = \frac{{{\gamma _l} + {\lambda _l}}}{2}\) \(\left( {l = 1,2} \right) \) exit \({n_0}\),

$$\tilde{\mu }\left( {\left\{ {x|{{\tilde{f}}_{{n_0}}} > \int _A {\tilde{f}d\tilde{\mu }} + \varepsilon } \right\} \cap A} \right) < \infty + i\infty ,{\gamma _l} = {\lambda _l} + 2{\varepsilon _l} > {\lambda _l} + {\varepsilon _l}$$
$$\Rightarrow \left\{ {x|{{\tilde{f}}_{{n_0}}} \ge \gamma } \right\} \subseteq \left\{ {x|{{\tilde{f}}_{{n_0}}} > \lambda + \varepsilon } \right\} ,$$
$$\Rightarrow \tilde{\mu }\left( {A \cap {{\left( {{{\tilde{f}}_{{n_0}}}} \right) }_\gamma }} \right) \le \tilde{\mu }\left( {\left\{ {x|{{\tilde{f}}_{{n_0}}} > \lambda + \varepsilon } \right\} \cap A} \right) < \infty + i\infty .$$

\({\textit{By}} \; {\textit{the}} \; {\textit{continuity}} \; {\textit{of}} \;\tilde{\mu }\), \(A \cap {\left( {{{\tilde{f}}_{{n_1}}}} \right) _\gamma } \supseteq A \cap {\left( {{{\tilde{f}}_{{n_2}}}} \right) _\gamma } \supseteq \ldots \),

$$\tilde{\mu }\left( {A \cap {{\left( {\tilde{f}} \right) }_\gamma }} \right) $$
$$= \tilde{\mu }\left( { \mathop {\cap }\limits _{n = 1}^\infty \left[ {A \cap {{\left( {{{\tilde{f}}_n}} \right) }_\gamma }} \right] } \right) = \mathop {\lim }\limits _{n \rightarrow \infty } \tilde{\mu }\left( {A \cap {{\left( {{{\tilde{f}}_n}} \right) }_\gamma }} \right) \ge \gamma \ge \lambda $$
$$\int _A {\tilde{f}} d\tilde{\mu }\buildrel \varDelta \over = \mathop {\sup }\limits _{{{\mathrm{{Re}}}} \alpha \in \left[ {0,\infty } \right) } S\left( {{{\mathrm{{Re}}}} \alpha ,{{\mathrm{{Re}}}} \tilde{\mu }\left[ {{{\tilde{f}}_\alpha } \cap A} \right] } \right) $$
$${+\,i}\ \mathop {\sup }\limits _{{{\mathrm{{Im}}}} \alpha \in \left[ {0,\infty } \right) } S\left( {{{\mathrm{{Im}}}} \alpha ,{{\mathrm{{Im}}}} \tilde{\mu }\left[ {{{\tilde{f}}_\alpha } \cap A} \right] } \right) $$
$$\ge S\left( {{{\mathrm{{Re}}}} \gamma ,{{\mathrm{{Re}}}} \tilde{\mu }\left[ {{{\tilde{f}}_\gamma } \cap A} \right] } \right) + iS\left( {{{\mathrm{{Im}}}} \gamma ,{{\mathrm{{Im}}}} \tilde{\mu }\left[ {{{\tilde{f}}_\gamma } \cap A} \right] } \right) > \lambda = {\lambda _1} + i{\lambda _2},$$

\(conflicting \; with \; \int _A {\tilde{f}} d\tilde{\mu }= \lambda \), so \( \mathop {\wedge }\limits _{n = 1}^\infty \int _A {{{\tilde{f}}_n}d} \tilde{\mu }= \int _A {\tilde{f}d} \tilde{\mu }\), that is

$$ \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}d} \tilde{\mu }= \int _A {\tilde{f}d} \tilde{\mu }. $$

Theorem 4

Suppose \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \) is complex fuzzy set-valued fuzzy measure space, \(\left\{ {{{\tilde{f}}_n}} \right\} \) is complex fuzzy set value complex fuzzy integrable function sequence in \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \) \(A \in \mathcal{F}\left( Z \right) \), if in A, \(\left\{ {{{\tilde{f}}_n}} \right\} \) convergence in \(\tilde{f}\), and for arbitrary \({\varepsilon _k} > 0,\left( {k = 1,2} \right) \), where, \(\varepsilon = {\varepsilon _1} + i{\varepsilon _2}\) exit \({n_0}\) makes

$$\tilde{\mu }\left( {\left\{ {x| \mathop {\sup }\limits _{n \ge {n_0}} {{\tilde{f}}_n} > \int _A {\tilde{f}d\tilde{\mu }+ \varepsilon } } \right\} \cap A} \right) < \infty + i\infty .$$

then \(\mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}d} \tilde{\mu }= \int _A {\tilde{f}d} \tilde{\mu }\).

Proof

Let \({\tilde{h}_n} = \mathop {\vee }\limits _{k = 1}^n {\tilde{f}_k},{\tilde{g}_k} = \mathop {\wedge }\limits _{k = 1}^n {\tilde{f}_k}\). Then \(\left\{ {{{\tilde{h}}_n}} \right\} \downarrow \tilde{f}\) and \(\left\{ {{{\tilde{g}}_n}} \right\} \uparrow \tilde{f}\), \({\tilde{g}_n} \le {\tilde{f}_n} \le {\tilde{h}_n}\), so

$$\int _A {{{\tilde{g}}_n}d\tilde{\mu }} \le \int _A {{{\tilde{f}}_n}d\tilde{\mu }} \le \int _A {{{\tilde{h}}_n}d\tilde{\mu }}$$

by theory 3,

$$ \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{g}}_n}d\tilde{\mu }} = \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}d\tilde{\mu }} $$
$$= \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{h}}_n}d\tilde{\mu }} = \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {\tilde{f}d\tilde{\mu }}. $$

Definition 6

([3]) Given a fuzzy complex value fuzzy measure space \(\left( {C,\tilde{F},\tilde{\mu }} \right) \), let \({\tilde{f}_n}\left( {n = 1,2, \ldots } \right) \) and \(\tilde{f}:C \rightarrow {F^ * }\left( C \right) \) are fuzzy complex value fuzzy measurable function, \(\tilde{A} \in \tilde{F}\) then

(1) \(\left\{ {{{\tilde{f}}_n}} \right\} \) almost everywhere converges to \(\tilde{f}\) on \(\tilde{A}\), if \(\tilde{\mu }\left( {\tilde{E}} \right) = \tilde{0}\) for \(\tilde{E} \in \tilde{F}\)

and makes \(\left\{ {{{\tilde{f}}_n}} \right\} \) converges to \(\tilde{f}\) point by point on \(\tilde{A} - \tilde{E}\), note that \({\tilde{f}_n} \mathop {\rightarrow }\limits ^{a.e} \tilde{f}\);

(2) \(\left\{ {{{\tilde{f}}_n}} \right\} \) almost uniform converges to \(\tilde{f}\) on \(\tilde{A}\), if \(\exists \tilde{E} \in \tilde{F}\) for \(\varepsilon > 0\), \(\left| {{{\tilde{\mu }}_\alpha }\left( {\tilde{E}} \right) } \right| < \varepsilon \) and makes \(\left\{ {{{\tilde{f}}_n}} \right\} \) uniform converges to \(\tilde{f}\) point by point, note that \({\tilde{f}_n} \mathop {\rightarrow }\limits ^{a.u} \tilde{f};\)

(3) \(\left\{ {{{\tilde{f}}_n}} \right\} \) pseudo almost everywhere converges to \(\tilde{f}\), if \(\hat{\mu }\left( {\tilde{A} - \tilde{E}} \right) = \tilde{\mu }\left( {\tilde{E}} \right) \) for \(\tilde{E} \in \tilde{F}\) and makes \(\left\{ {{{\tilde{f}}_n}} \right\} \) converges to \(\tilde{f}\) point by point on \(\tilde{A} - \tilde{E}\), note that \({\tilde{f}_n} \mathop {\rightarrow }\limits ^{p.a.e} \tilde{f};\)

(4) \(\left\{ {{{\tilde{f}}_n}} \right\} \) pseudo almost uniform converges to \(\tilde{f}\), if \( \mathop {\lim }\limits _{n \rightarrow \infty } \tilde{\mu }\left( {\tilde{A} - {{\tilde{E}}_k}} \right) = \tilde{\mu }\left( {\tilde{A}} \right) \) for \(\left\{ {{{\tilde{E}}_k}} \right\} \subset \tilde{F}\) and makes \(\left\{ {{{\tilde{f}}_n}} \right\} \) uniform converges to \(\tilde{f}\) point by point on \(\tilde{A} - \tilde{E}\) for any fixed point, \(k = 1,2,3 \ldots \) note that \({\tilde{f}_n} \mathop {\rightarrow }\limits ^{p.a.u} \tilde{f};\)

(5) \(\left\{ {{{\tilde{f}}_n}} \right\} \) converges in measure to \(\tilde{f}\), if

$$ \mathop {\lim }\limits _{n \rightarrow \infty } \tilde{\mu }\left( {\left\{ {x|\left\{ {{{\tilde{f}}_n}\left( x \right) - \tilde{f}\left( x \right) } \right\} > \varepsilon } \right\} \cap \tilde{A}} \right) = 0$$

for any \(\varepsilon > 0\), note that \({\tilde{f}_n} \mathop {\rightarrow }\limits ^{\tilde{\mu }} \tilde{f};\)

(6) \(\left\{ {{{\tilde{f}}_n}} \right\} \) pseudo converges in measure to \(\tilde{f}\), if

$$ \mathop {\lim }\limits _{n \rightarrow \infty } \tilde{\mu }\left( {\left\{ {x|\left\{ {{{\tilde{f}}_n}\left( x \right) - \tilde{f}\left( x \right) } \right\} > \varepsilon } \right\} \cap \tilde{A}} \right) = \tilde{\mu }\left( {\tilde{A}} \right) ,$$

note that \({\tilde{f}_n} \mathop {\rightarrow }\limits ^{p.\tilde{\mu }} \tilde{f}.\)

Theorem 5

Suppose \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \) is Complex fuzzy set-valued fuzzy measure space \(\left\{ {{{\tilde{f}}_n}} \right\} ,\tilde{f}\) is complex fuzzy set-valued complex fuzzy measurable function in \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \), \(A \in \mathcal{F}\left( Z \right) \), if \({\tilde{f}_n} \mathop {\rightarrow }\limits ^{a.e} \tilde{f}\), \(\tilde{\mu }\) is zero-additive, and for arbitrarily \({\varepsilon _k} > 0,\left( {k = 1,2} \right) \), where \(\varepsilon = {\varepsilon _1} + i{\varepsilon _2}\) exit \({n_0}\) makes

$$\tilde{\mu }\left( {\left\{ {x| \mathop {\sup }\limits _{n \ge {n_0}} {{\tilde{f}}_n} \rangle } \right\} \int _A {\tilde{f}d\tilde{\mu }+ \varepsilon \cap A} } \right) < \infty + i\infty , $$

then \( \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}d} \tilde{\mu }= \int _A {\tilde{f}d} \tilde{\mu }.\)

Proof

Because \({\tilde{f}_n} \mathop {\rightarrow }\limits ^{a.e} \tilde{f}\) in A, then exit \(B \in \mathcal{F}\left( Z \right) \), \(\tilde{\mu }\left( B \right) = 0\), \(\tilde{\mu }\) is zero-additive, \(\int _{A\backslash B} {\tilde{f}d} \tilde{\mu }\buildrel \varDelta \over = \)

\( \mathop {\sup }\limits _{{\mathrm{Re}} \alpha \in \left[ {0,\infty } \right) } S\left( {{{\mathrm{{Re}}}} \alpha ,{{\mathrm{{Re}}}} \tilde{\mu }\left[ {{{\tilde{f}}_\alpha } \cap \left( {A\backslash B} \right) } \right] } \right) \) \( + i \mathop {\sup }\limits _{{\mathrm{Im}} \alpha \in \left[ {0,\infty } \right) } S\left( {{{\mathrm{{Im}}}} \alpha ,{{\mathrm{{Im}}}} \tilde{\mu }\left[ {{{\tilde{f}}_\alpha } \cap \left( {A\backslash B} \right) } \right] } \right) \)

\( = \mathop {\sup }\limits _{{\mathrm{Re}} \alpha \in \left[ {0,\infty } \right) } S\left( {{{\mathrm{{Re}}}} \alpha ,{{\mathrm{{Re}}}} \tilde{\mu }\left[ {\left( {A \cap {{\tilde{f}}_\alpha }} \right) \backslash B} \right] } \right) \) \(+ i \mathop {\sup }\limits _{{{\mathrm{{Im}}}} \alpha \in \left[ {0,\infty } \right) } S\left( {{\mathrm{Im}}\alpha ,{\mathrm{Im}}\tilde{\mu }\left[ {\left( {A \cap {{\tilde{f}}_\alpha }} \right) \backslash B} \right] } \right) \)

\( = \mathop {\sup }\limits _{{{\mathrm{{Re}}}} \alpha \in \left[ {0,\infty } \right) } S\left( {{{\mathrm{{Re}}}} \alpha ,{{\mathrm{{Re}}}} \tilde{\mu }\left( {A \cap {{\tilde{f}}_\alpha }} \right) } \right) + i \mathop {\sup }\limits _{{{\mathrm{{Re}}}} \alpha \in \left[ {0,\infty } \right) } S\left( {{{\mathrm{{Im}}}} \alpha ,{\mathrm{Im}} \tilde{\mu }\left( {A \cap {{\tilde{f}}_\alpha }} \right) } \right) \)

\( = \int _A {\tilde{f}d} \tilde{\mu }.\)

Similarly

$$\int _A {{{\tilde{f}}_n}d} \tilde{\mu }= \int _{A\backslash B} {{{\tilde{f}}_n}d} \tilde{\mu },$$

and because

$$\tilde{\mu }\left( {\left\{ {x| \mathop {\sup }\limits _{n \ge {n_0}} {{\tilde{f}}_n} > \int _A {\tilde{f}d\tilde{\mu }} + \varepsilon } \right\} \cap A} \right) < \infty + i\infty , $$

and from the Theorem 2, obtained \( \mathop {\lim }\limits _{n \rightarrow \infty } \int _{A\backslash B} {{{\tilde{f}}_n}d} \tilde{\mu }= \int _{A\backslash B} {\tilde{f}d} \tilde{\mu }\), so \( \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}d} \tilde{\mu }= \int _A {\tilde{f}d} \tilde{\mu }.\)

Theorem 6

Suppose \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \) is complex fuzzy set-valued fuzzy measure space \(\left\{ {{{\tilde{f}}_n}} \right\} ,\tilde{f}\) is complex fuzzy set-valued complex fuzzy measurable function in \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \), \(A \in \mathcal{F}\left( Z \right) \), if \(\left\{ {{{\tilde{f}}_n}} \right\} \) uniform convergence in \(\tilde{f}\) in A, then

$$ \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}d} \tilde{\mu }= \int _A {\tilde{f}d} \tilde{\mu }. $$

Proof

(1) If \(\int _A {\tilde{f}d} \tilde{\mu }= \infty + i\infty \), let \({\tilde{g}_n} = \mathop {\wedge }\limits _{k = 1}^n {\tilde{f}_k}\) then

$$ \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}d} \tilde{\mu }\ge \int _A {{{\tilde{g}}_n}d} \tilde{\mu }= \int _A {\tilde{f}d} \tilde{\mu }= \infty + i\infty . $$

(2) If

$$\int _A {\tilde{f}d} \tilde{\mu }= \mathop {\sup }\limits _{{{\mathrm{{Re}}}} \alpha \in \left[ {0,\infty } \right) } S\left( {{{\mathrm{{Re}}}} \alpha ,{{\mathrm{{Re}}}} \tilde{\mu }\left[ {{{\tilde{f}}_\alpha } \cap A} \right] } \right) + i \mathop {\sup }\limits _{{{\mathrm{{Im}}}} \alpha \in \left[ {0,\infty } \right) } S\left( {{{\mathrm{{Im}}}} \alpha , {\mathrm{Im}}\tilde{\mu }\left[ {{{\tilde{f}}_\alpha } \cap A} \right] } \right) $$

\( = \lambda < \infty + i\infty \), then,

$$\forall \alpha \le \lambda ,\tilde{\mu }\left[ {{{\tilde{f}}_\alpha } \cap A} \right] \ge \lambda ;$$
$$\forall \alpha > \lambda ,\tilde{\mu }\left[ {{{\tilde{f}}_\alpha } \cap A} \right] \le \lambda ;$$

\({\alpha _n}\) monotone decreasing trend to \(\lambda \), then \({\tilde{f}_{{\alpha _1}}} \subseteq {\tilde{f}_{{\alpha _2}}} \subseteq \cdots \) and \( \mathop {\cap }\limits _{n = 1}^\infty {\tilde{f}_{{\alpha _n}}} = {\tilde{f}_{ \mathop {\lambda }\limits _\cdot }}\) by under continuous of \(\tilde{\mu },\tilde{\mu }\left( {{{\tilde{f}}_{ \mathop {\lambda }\limits _{\cdot } }} \cap A} \right) = \mathop {\lim }\limits _{n \rightarrow \infty } \tilde{\mu }\left( {{{\tilde{f}}_{{\alpha _n}}} \cap A} \right) \le \lambda \), uniform convergence in \(\tilde{f}\) in A, arbitrarily \({\varepsilon '_k} > 0,\left( {k = 1,2} \right) \) where \(\varepsilon ' = {\varepsilon '_1} + i{\varepsilon '_2}\) exit \({n_0},\forall x \in A\),

$${\tilde{f}_n}\left( x \right) \le \tilde{f}\left( x \right) + \varepsilon \Rightarrow \mathop {\sup }\limits _{n \ge {n_0}} {\tilde{f}_n}\left( x \right) \le \tilde{f}\left( x \right) + \varepsilon $$
$$\Rightarrow \left\{ {x| \mathop {\sup }\limits _{n \ge {n_0}} {{\tilde{f}}_n}(x) \ge \lambda + \varepsilon } \right\} \cap A \subseteq \left\{ {x|\tilde{f}(x) \ge \lambda } \right\} \cap A = {\tilde{f}_{ \mathop {\lambda }\limits _\cdot }} \cap A$$
$$\Rightarrow \tilde{\mu }\left( {\left\{ {x| \mathop {\sup }\limits _{n \ge {n_0}} {{\tilde{f}}_n}(x) \ge \lambda + \varepsilon } \right\} \cap A} \right) $$

\(~~~~~~~~~~~~~~~~\le \lambda < \infty + i\infty \)

by Theorem 3 has \( \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}d} \tilde{\mu }= \int _A {\tilde{f}d} \tilde{\mu }.\)

Theorem 7

Suppose \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \) is complex fuzzy set-valued fuzzy measure space, \(\left\{ {{{\tilde{f}}_n}} \right\} ,\tilde{f}\) are complex fuzzy set-valued complex fuzzy measurable function on \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \), \(A \in \mathcal{F}\left( Z \right) \), if \({\tilde{f}_n} \mathop {\rightarrow }\limits ^{a.e.u} \tilde{f}\) on A, and \(\tilde{\mu }\) is zero-additive, then \( \mathop {\lim }\limits _{n \rightarrow \infty } \int _A {{{\tilde{f}}_n}d} \tilde{\mu }= \int _A {\tilde{f}d} \tilde{\mu }.\)

Proof

Similar to the proof method of Theorem 6.

Theorem 8

Suppose \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \) is complex fuzzy set-valued fuzzy measure space \(\left\{ {{{\tilde{f}}_n}} \right\} ,\tilde{f}\) are complex fuzzy set-valued complex fuzzy measurable function on \(\left( {Z,\mathcal{F}\left( Z \right) ,\tilde{\mu }} \right) \), \(A \in \mathcal{F}\left( Z \right) \), if \({\tilde{f}_n} \mathop {\rightarrow }\limits ^{a.u} \tilde{f}\) in A, and \(\tilde{\mu }\) is zero-additive, and exit

$$\{ {B_k}\} \subseteq \mathcal{F}(Z),{B_1} \supseteq {B_2} \supseteq \cdots ,\tilde{\mu }({B_k}) \rightarrow 0$$

makes

$$ \mathop {\lim }\limits _{n \rightarrow \infty } \int _{A\backslash {B_k}} {{{\tilde{f}}_n}d} \tilde{\mu }= \int _{A\backslash {B_k}} {\tilde{f}d} \tilde{\mu }.$$

Proof

If \({\tilde{f}_n} \mathop {\rightarrow }\limits ^{a.u} \tilde{f}\) in A, then exit \(\{ {E_k}\} \subseteq \mathcal{F}(Z)\), \(\tilde{\mu }({E_k}) \rightarrow 0\), \({\tilde{f}_n} \mathop {\rightarrow }\limits ^u \tilde{f}\) in \(A\backslash {E_k}\), let

$${B_k} = \mathop {\cap }\limits _{i = 1}^k {E_i} \subseteq {E_k}.$$

Then

$$A\backslash {B_k} = \mathop {\cap }\limits _{i = 1}^k \left( {A\backslash {E_i}} \right) ,$$

\(\forall k\), \({\tilde{f}_n} \mathop {\rightarrow }\limits ^u \tilde{f}\) in \(A\backslash {E_k}\), \({\tilde{f}_n} \mathop {\rightarrow }\limits ^u \tilde{f}\) in \(A\backslash {B_k}\), by Theorem 6 know the conclusion is right.

5 Conclusion

In this paper, the fuzzy measure concepts was extended from general classical set to ordinary complex fuzzy set, fuzzy, research complex fuzzy set-valued complex fuzzy measure and its properties, and measurable function in complex fuzzy set value complex fuzzy measure space and its properties was studied, of the of extension of the scope of classical measure theory, generalization of the corresponding conclusion of classical measure theory; research Integral theory problem of complex fuzzy set-valued function base on complex fuzzy set-valued measure, establish complex fuzzy set-valued complex fuzzy Integral theory, which is important work in fuzzy complex analysis. This work extends the fuzzy measure and fuzzy integral theory, to lay a solid foundation for our future research on complex fuzzy set-valued complex fuzzy integral application problem.