Distribution Function and Nonincreasing Rearrangement of ${\mathbb {B}}{\mathbb {C}}$-Valued Functions with ${\mathbb {B}} {\mathbb {C}}$-Measure

Abstract

This paper investigates the distribution function and nonincreasing rearrangement of $\mathbb{B}\mathbb{C}$-valued functions equipped with the hyperbolic norm. It begins by introducing the concept of the distribution function for $\mathbb{B}\mathbb{C}$-valued functions, which characterizes valuable insights into the behavior and structure of $\mathbb{B}\mathbb{C}$-valued functions, allowing to analyze their properties and establish connections with other mathematical concepts. Next, the nonincreasing rearrangement of $\mathbb{B}\mathbb{C}$-valued functions with the hyperbolic norm are studied. By exploring the nonincreasing rearrangement of $\mathbb{B}\mathbb{C}$-valued functions, it is aimed to determine how the hyperbolic norm influences the rearrangement process and its impact on the function’s behavior and properties.

1 Introduction

$\mathbb{B}\mathbb{C}$-valued functions arise naturally in various mathematical fields, including probability theory, mathematical analysis, and functional analysis, and understanding their properties is crucial for advancing these areas of study. Indeed, the study of modules with bicomplex scalars in the context of functional analysis has gained significant attention in recent years. One influential work that has contributed to this area is the book by Alpay et al., referenced as [1]. The book presents groundbreaking results and insights related to this topic. Functional analysis traditionally deals with vector spaces over a field, such as the complex numbers or the real numbers. However, by considering modules with bicomplex scalars, where the scalars are elements of the bicomplex numbers, a broader framework is introduced. This extension allows for the exploration of new mathematical structures and the investigation of properties beyond the classical setting. The book by Alpay et al. is a valuable resource for researchers and enthusiasts interested in this area. It presents notable results, techniques, and applications pertaining to the study of modules with bicomplex scalars in the context of functional analysis. These results encompass various aspects of functional analysis, such as operator theory, function spaces, and spectral theory, among others. The remarkable results mentioned in the context of the book contribute to the understanding and development of this field. They shed light on the behavior of modules with bicomplex scalars, reveal connections to other areas of mathematics, and potentially find possible applications in physics, engineering, or other disciplines.

The series of articles mentioned in the references highlight the systematic study of topological bicomplex modules and various fundamental theorems related to them. Here is a breakdown of the articles and their contributions:

Kumar and Saini in [8]: This article focused on the study of topological bicomplex modules, exploring their topological properties and investigating concepts such as convergence, continuity, and compactness in this context.

Kumar et al. in [10]: This work presented fundamental theorems, including the principle of uniform boundedness, open mapping theorem, interior mapping theorem and closed graph theorem for bicomplex modules. These theorems establish important results regarding the continuity and invertibility of linear operators between topological bicomplex modules.

Saini et al. in [15]: This article, in collaboration with Kumar et al. in [10], extended the study of fundamental theorems to the setting of topological bicomplex modules. The main aim is generalizing classical results from functional analysis to the bicomplex module framework, providing a deeper understanding of their properties. Also the authors delve further into the study of topological hyperbolic modules, topological bicomplex modules, exploring the properties of linear operators, continuity, and related topological concepts specific to these settings.

Luna-Elizarrarás et al. in [12]: This work investigated the Hahn-Banach theorem for bicomplex modules and hyperbolic modules. This extensions to hyperbolic and bicomplex modules explore the uniqueness and extensions of such functionals on a normed vector space.

Kumar et al. in [11]: This work focused on bicomplex C*-algebras, a generalization of C*-algebras to the bicomplex setting. It covers topics such as bicomplex operator algebras, spectral theory, and other aspects related to the algebraic and topological properties of C*-algebras defined on bicomplex vector space.

Kumar and Singh in [9]: This article investigated bicomplex linear operators on $\mathbb{B}\mathbb{C}$ Hilbert spaces. It explores properties of these operators, functional analysis techniques in the bicomplex setting, and touches upon topics like Littlewood’s subordination theorem, which relates to the order-preserving mapping between functions.

Luna-Elizarrarás et al. in [13]: The book authored by them provide an in-depth exploration of bicomplex analysis and geometry. It covers a wide range of topics, including holomorphic functions, integration, differential equations, and geometric properties specific to the bicomplex domain.

Colombo et al. in [4]: This article focused on $\mathbb{B}\mathbb{C}$ bounded linear operators and bicomplex functional calculus. It provides a detailed study of operators acting on bicomplex modules and explores the construction and properties of functional calculi specific to the bicomplex framework.

Charak et al. in [3]: This article presented the bicomplex spectral decomposition theorem when infinite-dimension, which provides a generalization of the spectral decomposition theorem to bicomplex Hilbert spaces.

These references collectively represent significant contributions to the study of bicomplex modules, functional analysis on $\mathbb{B}\mathbb{C}$, and related areas. They showcase the exploration of properties, the development of new theorems, and the application of functional analysis techniques in the context of bicomplex numbers. Researchers and readers interested in these topics can refer to these articles and the books for detailed insights into the respective areas of study.

2 Preliminaries on $\mathbb{B}\mathbb{C}$

Now, we will give a summary of bicomplex numbers with some basic properties .The set bicomplex numbers $\mathbb{B}\mathbb{C}$ which is a two-dimensional extension of the complex numbers is defined as

$$\begin{aligned} {\mathbb {B}}{\mathbb {C}}:=\{W=w_{1}+jw_{2}|~~w_{1},w_{2}\in {\mathbb {C}} (i)\} \end{aligned}$$

where i and j are imaginary units satisfying $ij=ji$, $i^{2}=j^{2}=-1$. Here ${\mathbb {C}} (i)$ is the field of complex numbers with the imaginary unit i. According to ring structure, for any $Z=z_{1}+jz_{2},~W=w_{1}+jw_{2}$ in $\mathbb{B}\mathbb{C}$ usual addition and multiplication are defined as

$$\begin{aligned} Z+W= & {} \left( z_{1}+w_{1}\right) +j\left( z_{2}+w_{2}\right) \\ ZW= & {} \left( z_{1}w_{1}-z_{2}w_{2}\right) +j\left( z_{2}w_{1}+z_{1}w_{2}\right) . \end{aligned}$$

The set $\mathbb{B}\mathbb{C}$ forms a commutative ring under the usual addition and multiplication of bicomplex numbers. The bicomplex numbers have a unit element denoted as $1_{{\mathbb {B}}{\mathbb {C}}}:=1$ and it acts as the identity for multiplication, such that for any bicomplex number W, $1W=W1=W$. In the sense of module structure, the set $\mathbb{B}\mathbb{C}$ is a module over itself. This means that $\mathbb{B}\mathbb{C}$ satisfies the properties of a module, including scalar multiplication and distributivity. The product of the imaginary units i and j bring out a hyperbolic unit k, such that $k^{2}=1$. This implies that k is a square root of 1 and is distinct from i and j. The product operation of all units i, j,  and k in the bicomplex numbers is commutative. Specifically, the following relations hold:

$$\begin{aligned} ij=k,jk=-i \quad and \quad ik=-j. \end{aligned}$$

These properties summarize the basic characteristics of bicomplex numbers and their algebraic structure.

Hyperbolic numbers ${\mathbb {D}}$ are a two-dimensional extension of the real numbers that form a number system known as the hyperbolic plane or hyperbolic plane algebra. They can be represented in the form $\beta =\beta _{1}+k\beta _{2}$, where $\beta _{1}$ and $\beta _{2}$ are real numbers, and k is the hyperbolic unit. In the hyperbolic number system, for any two hyperbolic numbers $\beta =\beta _{1}+k\beta _{2}$ and $\gamma =\delta _{1}+k\delta _{2}$, addition and multiplication are defined as follows:

$$\begin{aligned} \beta +\gamma= & {} \left( \beta _{1}+\delta _{1}\right) +k\left( \beta _{2}+\delta _{2}\right) \\ \beta \gamma= & {} \left( \beta _{1}\delta _{1}+\beta _{2}\delta _{2}\right) +k\left( \beta _{1}\delta _{2}+\beta _{2}\delta _{1}\right) . \end{aligned}$$

The hyperbolic numbers form a ring, which means that addition and multiplication are closed operations, associative, and distributive and a module over itself. However, unlike the complex numbers, the hyperbolic numbers do not have a multiplicative inverse for all nonzero elements. The nonzero hyperbolic numbers that have multiplicative inverses are called units. The hyperbolic numbers can also be considered a significant subset of the bicomplex numbers $\mathbb{B}\mathbb{C}$.

Let $W=w_{1}+jw_{2}\in {\mathbb {B}}{\mathbb {C}}$ where $w_{1},w_{2}\in {\mathbb {C}} (i)$. By the notation of W with imaginary units i and j, the conjugations are formed for bicomplex numbers in [1, 13] as follows:

(i):

The first conjugation ${\overline{W}}=\overline{w_{1}}+j \overline{w_{2}}$ (the bar-conjugation)

(ii):

The second conjugation $W^{\dagger }=w_{1}-jw_{2}$ (the $\dagger$–conjugation)

(iii):

The third one $W^{*}=\overline{w_{1}}-j\overline{ w_{2}}$ (the $*$–conjugation)

where $\overline{w_{1}}$ and $\overline{w_{2}}$ are the usual complex conjugates of $w_{1},~w_{2}\in {\mathbb {C}}\left( i\right)$, respectively.

In summary, ${\overline{W}}$ and $W^{\dagger }$ are individual conjugations obtained by taking the complex conjugate of the components and combining them with the imaginary unit j. The third conjugation, $W^{*}$, is obtained by composing ${\overline{W}}$ and $W^{\dagger }$, resulting in a new expression involving both real and imaginary parts.

For any bicomplex number W, they also wrote the following three moduli in [1, 13]:

(i):

$|W|_{i}^{2}=W\cdot W^{\dagger }=w_{1}^{2}+w_{2}^{2}\in {\mathbb {C}}\left( i\right)$,

(ii):

$|W|_{j}^{2}=W\cdot {\overline{W}} =(|w_{1}|^{2}-|w_{2}|^{2})+j\left( 2\text {Re}\left( w_{1}\overline{w_{2}} \right) \right) \in {\mathbb {C}}\left( j\right) ,$

(iii):

$|W|_{k}^{2}=W\cdot W^{*}=(|w_{1}|^{2}+|w_{2}|^{2})+k\left( -2\text {Im}\left( w_{1}\overline{w_{2}} \right) \right) \in {\mathbb {D}}.$

Furthermore, ${\mathbb {B}}{\mathbb {C}}$ is a normed space with the norm $\left\| W\right\| _{{\mathbb {B}}{\mathbb {C}}} =\sqrt{|w_{1}|^{2}+|w_{2}|^{2} }$ for any $W=w_{1}+jw_{2}$ in ${\mathbb {B}}{\mathbb {C}}$. According to this,

$$\begin{aligned} \left\| W_{1}W_{2}\right\| _{{\mathbb {B}}{\mathbb {C}}}\le \sqrt{2} \left\| W_{1}\right\| _{{\mathbb {B}}{\mathbb {C}}}\left\| W_{2}\right\| _{{\mathbb {B}}{\mathbb {C}}} \end{aligned}$$

for every $W_{1},W_{2}\in {\mathbb {B}}{\mathbb {C}}$, and finally ${\mathbb {B}} {\mathbb {C}}$ is a modified Banach algebra [14].

If the hyperbolic numbers $e_{1}$ and $e_{2}$ defined as

$$\begin{aligned} e_{1}=\frac{1+k}{2} \quad and \quad e_{2}=\frac{1-k}{2}\text{, } \end{aligned}$$

then it is easy to see that

$$\begin{aligned} e_{1}^{2}=e_{1},~~~e_{2}^{2}=e_{2},~~~{e_{1}^{*}}=e_{1},~~~{e_{2}^{*}}=e_{2},~~~e_{1}+e_{2}=1,~~~e_{1}\cdot e_{2}=0 \end{aligned}$$

are satisfied with $\left\| e_{1}\right\| _{{\mathbb {B}}{\mathbb {C}} }=\left\| e_{2}\right\| _{{\mathbb {B}}{\mathbb {C}}}=\frac{\sqrt{2}}{2}$. By using this linearly independent set $\left\{ e_{1},e_{2}\right\}$, any $W=w_{1}+jw_{2}\in \mathbb{B}\mathbb{C}$ can be written as a linear combination of $e_{1}$ and $e_{2}$ uniquely. That is, $W=w_{1}+jw_{2}$ can be written as

$$\begin{aligned} W=w_{1}+jw_{2}=e_{1}z_{1}+e_{2}z_{2} \end{aligned}$$

(2.1)

where $z_{1}=w_{1}-iw_{2}$ and $z_{2}=w_{1}+iw_{2}$ [1]. Here $z_{1}$ and $z_{2}$ are elements of ${\mathbb {C}}\left( i\right)$ and the formula in (2.1) is called the idempotent representation of the bicomplex number W.

Besides the Euclidean-type norm $\left\| \cdot \right\| _{{\mathbb {B}} {\mathbb {C}}}$, another norm named with (${\mathbb {D}}$-valued) hyperbolic-valued norm $\left| W\right| _{k}$ of any bicomplex number $W=e_{1}z_{1}+e_{2}z_{2}$ is defined as

$$\begin{aligned} \left| W\right| _{k}=e_{1}\left| z_{1}\right| +e_{2}\left| z_{2}\right| . \end{aligned}$$

For any hyperbolic number $\alpha =\beta _{1}+k\beta _{2}\in {\mathbb {D}}$, an idempotent representation can also be written as $\mathbb {D\subset BC}$. Thus $\alpha =\beta _{1}+k\beta _{2}\in {\mathbb {D}}$ can be written as

$$\begin{aligned} \alpha =e_{1}\alpha _{1}+e_{2}\alpha _{2} \end{aligned}$$

where $\alpha _{1}=\beta _{1}+\beta _{2}$ and $\alpha _{2}=\beta _{1}-\beta _{2}$ are real numbers. If $\alpha _{1}>0$ and $\alpha _{2}>0$ for any $\alpha =\beta _{1}+k\beta _{2}\in {\mathbb {D}}$, then we say that $\alpha$ is a positive hyperbolic number. Thus, the set of non-negative hyperbolic numbers ${\mathbb {D}}^{+}\cup \left\{ 0\right\}$ is defined by

$$\begin{aligned} {\mathbb {D}}^{+}\cup \left\{ 0\right\}= & {} \left\{ \alpha =\beta _{1}+k\beta _{2}:\beta _{1}^{2}-\beta _{2}^{2}\ge 0,~\beta _{1}\ge 0\right\} \\= & {} \left\{ \alpha =e_{1}\alpha _{1}+e_{2}\alpha _{2}:\alpha _{1}\ge 0,~\alpha _{2}\ge 0~\right\} . \end{aligned}$$

Now, let $\alpha$ and $\gamma$ be any two elements of ${\mathbb {D}}$. In [1, 12] and [13], a relation $\preceq$ is defined on ${\mathbb {D}}$ by

$$\begin{aligned} \alpha \preceq \gamma \Leftrightarrow \gamma -\alpha \in {\mathbb {D}}^{+}\cup \left\{ 0\right\} . \end{aligned}$$

It is showed in [1] that this relation “$\preceq$” has reflexive, anti-symmetric and transitive properties. Therefore “$\preceq$” defines a partial order on ${\mathbb {D}}$. If idempotent representations of the hyperbolic numbers $\alpha$ and $\gamma$ are written as $\alpha =e_{1}\alpha _{1}+e_{2}\alpha _{2}$ and $\gamma =e_{1}\gamma _{1}+e_{2}\gamma _{2}$, then $\alpha \preceq \gamma$ implies that $\alpha _{1}\le \gamma _{1}$ and $\alpha _{2}\le \gamma _{2}$. By $\alpha \prec \gamma$, we mean $\alpha _{1}<\gamma _{1}$ and $\alpha _{2}<\gamma _{2}$. Any function f defined on ${\mathbb {D}}$ is called

• ${\mathbb {D}}$-increasing if $f\left( \alpha \right) \prec f\left( \gamma \right)$

• ${\mathbb {D}}$-decreasing if $f\left( \alpha \right) \succ f\left( \gamma \right)$

• ${\mathbb {D}}$-nonincreasing if $f\left( \alpha \right) \succeq f\left( \gamma \right)$

• ${\mathbb {D}}$-nondecreasing if $f\left( \alpha \right) \preceq f\left( \gamma \right)$

whenever $\alpha \prec \gamma$. For more details on hyperbolic numbers ${\mathbb {D}}$ and partial order “$\preceq$”, one can refer to [1, Section 1.5] and [13].

Definition 1

Let A be a subset of ${\mathbb {D}}$. A is called a ${\mathbb {D}}$-bounded above set if there is a hyperbolic number $\delta$ such that $\delta \succeq \alpha$ for all $\alpha \in A$. If $A\subset$ ${\mathbb {D}}$ is ${\mathbb {D}}$-bounded from above, then the ${\mathbb {D}}$-supremum of A is defined as the smallest member of the set of all upper bounds of A. In other words, a hyperbolic number $\lambda$ is an upper bound of the set A , can be described by the following two properties:

(i):

$\alpha \preceq \lambda$, for each $\alpha \in A$

(ii):

For any $\varepsilon \succ 0$, there exists $\theta \in A$ such that $\theta \succ \lambda -\varepsilon$ [15].

In other words, the hyperbolic number $\lambda =e_{1}\lambda _{1}+e_{2}\lambda _{2}$, where $\lambda _{1}$ and $\lambda _{2}$ are real numbers, is the ${\mathbb {D}}$-supremum of A if

(1):

$e_{1}\alpha _{1}+e_{2}\alpha _{2}\preceq e_{1}\lambda _{1}+e_{2}\lambda _{2}$ for each $\alpha =e_{1}\alpha _{1}+e_{2}\alpha _{2}\in A$

(2):

For any $\varepsilon =e_{1}\varepsilon _{1}+e_{2}\varepsilon _{2}\succ 0$, there exists $\theta =e_{1}\theta _{1}+e_{2}\theta _{2}\in A$ such that $e_{1}\theta _{1}+e_{2}\theta _{2}\succ e_{1}\left( \lambda _{1}-\varepsilon _{1}\right) +e_{2}\left( \lambda _{2}-\varepsilon _{2}\right)$ are satisfied.

Remark 1

Let A be a ${\mathbb {D}}$-bounded above subset of ${\mathbb {D}}$ and $A_{1}:=\left\{ \lambda _{1}:e_{1}\lambda _{1}\right.$ $\left. +e_{2}\lambda _{2}\in A\right\}$, $A_{2}:=\left\{ \lambda _{2}:e_{1}\lambda _{1}+e_{2}\lambda _{2}\in A\right\}$. Then the sup$_{{\mathbb {D}}}A$ is given by

$$\begin{aligned} {sup}_{{\mathbb {D}}}A:=e_{1}{sup} A_{1}+e_{2}{sup} A_{2}. \end{aligned}$$

Similarly, for any ${\mathbb {D}}$-bounded below set A, ${\mathbb {D}}$-infimum of A is defined as

$$\begin{aligned} {inf}_{{\mathbb {D}}}A=e_{1}{inf} A_{1}+e_{2}{inf} A_{2} \end{aligned}$$

where $A_{1}$ and $A_{2}$ are as above [1, Remark 1.5.2].

Remark 2

A ${\mathbb {B}}{\mathbb {C}}$-module space or ${\mathbb {D}}$-module space Y can be decomposed as

$$\begin{aligned} Y=e_{1}Y_{1}+e_{2}Y_{2} \end{aligned}$$

(2.2)

where $Y_{1}=e_{1}Y$ and $Y_{2}=e_{2}Y$ are ${\mathbb {R}}$-vector or ${\mathbb {C}} \left( i\right) -$vector spaces. The spelling in (2.2) is called as the idempotent decomposition of the space Y [1, 15].

Definition 2

Let ${\mathfrak {M}}$ be a $\sigma$-algebra on a set $\Omega$. A bicomplex-valued function $\mu =\mu _{1}e_{1}+\mu _{2}e_{2}$ defined on $\Omega$ is called a ${\mathbb {B}}{\mathbb {C}}$-measure on ${\mathfrak {M}}$ if $\mu _{1},~\mu _{2}$ are complex measures on ${\mathfrak {M}}$. In particular if $\mu _{1},~\mu _{2}$ are positive measures on ${\mathfrak {M}}$ i.e range of both $\mu _{1},~\mu _{2}$ are $\left[ 0,\infty \right]$ then $\mu$ is called a ${\mathbb {D}}$-measure on ${\mathfrak {M}}$ and if $\mu _{1},~\mu _{2}$ are real measures on ${\mathfrak {M}}$ i.e range of both $\mu _{1},~\mu _{2}$ are $\left[ 0,\infty \right)$ then $\mu$ is called a ${\mathbb {D}}^{+}$ -measure on ${\mathfrak {M}}$ [6].

Assume that $\Omega =\left( \Omega ,{\mathfrak {M}},\mu \right)$ is a $\sigma$ -finite complete measure space and $f_{1},~f_{2}$ are complex-valued (real-valued) measurable functions on $\Omega$. The function having idempotent decomposition $f=f_{1}e_{1}+~f_{2}e_{2}$ is called as a $\mathbb { BC}$-measurable function and $\left| f\right| _{k}=\left| f_{1}\right| e_{1}+~\left| f_{2}\right| e_{2}$ is called a ${\mathbb {D}}$-valued measurable function on $\Omega$ [5]. Thus for any given complex valued function space $\left( F\left( \Omega \right) ,\left\| \cdot \right\| _{\Omega }\right)$, one can create a $\mathbb { BC}$-valued function space $\left( F\left( \Omega ,{\mathbb {B}}{\mathbb {C}} \right) ,\left\| \cdot \right\| _{{\mathbb {B}}{\mathbb {C}}}\right)$ by combining all $f_{1}$, $f_{2}$ and bringing out functions of the type $f=f_{1}e_{1}+~f_{2}e_{2}$ where $f_{1}$ and $f_{2}$ are in $\left( F\left( \Omega \right) ,\left\| \cdot \right\| _{\Omega }\right)$ with $\left\| f\right\| _{{\mathbb {B}}{\mathbb {C}}}^{2}=\frac{1}{2}\left( \left\| f_{1}\right\| _{\Omega }^{2}+\left\| f_{2}\right\| _{\Omega }^{2}\right)$. Similar definition can be given for any hyperbolic measurable function.

For any $\mathbb{B}\mathbb{C}$-valued measurable function $f=f_{1}e_{1}+~f_{2}e_{2}$, it is easy to see that $\left| f\right| _{k}=\left| f_{1}\right| e_{1}+~\left| f_{2}\right| e_{2}$ is ${\mathbb {D}}$ -valued measurable. Because if $f=f_{1}e_{1}+~f_{2}e_{2}$ is a ${\mathbb {B}} {\mathbb {C}}$-valued measurable function, then $f_{1},~f_{2}$ are ${\mathbb {C}}$-measurable functions. Therefore real and imaginary parts of $f_{1}$ and $~f_{2}$ are ${\mathbb {R}}$-valued measurable and so does $\left| f_{1}\right|$ and $~\left| f_{2}\right|$. As a result, $\left| f\right| _{k}$ is ${\mathbb {D}}$-measurable. Also for any two ${\mathbb {B}}{\mathbb {C}}$-valued measurable functions f and g,  it can be easily seen that their sum and multiplication functions are also ${\mathbb {B}}{\mathbb {C}}$-measurable functions [5, 6]. More results on ${\mathbb {D}}$-topology such as ${\mathbb {D}}$-limit, ${\mathbb {D}}$-continuity, ${\mathbb {D}}$-Cauchy and $\mathbb { D}$-convergence etc. can be found in [6] and the references therein.

Theorem 1

Let $u=u_{1}e_{1}+~u_{2}e_{2},v=v_{1}e_{1}+~v_{2}e_{2}$ and $u_{n}=u_{1}^{n}e_{1}+~u_{2}^{n}e_{2}$ be ${\mathbb {B}}{\mathbb {C}}$-measurable functions. Assume that $\lambda \in {\mathbb {B}}{\mathbb {C}}$ and $\psi$ is a continuous map on an open set $U\subset {\mathbb {B}}{\mathbb {C}}$. Then:

(a):

Real and imaginary parts of the functions $u_{1}$,$u_{2}$ ,$v_{1}$ and $v_{2}$ are ${\mathbb {R}}$-valued measurable

(b):

$u_{1},~u_{2},~v_{1}$ and $v_{2}$ are ${\mathbb {C}}$-valued measurable

(c):

$u+v$,$u\cdot v$ and $\lambda u$ are ${\mathbb {B}}\mathbb {C }$-valued measurable

(d):

${sup}_{{\mathbb {D}}}\left| u_{n}\right| _{k}$, ${inf}_{{\mathbb {D}}}\left| u_{n}\right| _{k}$, ${limsup}_{\mathbb {D }}\left| u_{n}\right| _{k}$, ${liminf}_{{\mathbb {D}}}\left| u_{n}\right| _{k}$, ${lim} \left| u_{n}\right| _{k}$ are $\mathbb { D}$-valued measurable

(e):

$\psi \circ u$ is ${\mathbb {B}}{\mathbb {C}}$-valued measurable

where they are defined in ${\mathfrak {M}}$.

Proof

The proofs of each item can be done by using the definition of measurable function and the similar techniques used in [2, Appendix A]. $\square$

Definition 3

Let $\left( \Omega ,{\mathfrak {M}},\vartheta \right)$ be a measure space, ${\mathfrak {F}}\left( \Omega ,{\mathfrak {M}}\right)$ indicate the set of all ${\mathfrak {M}}$-measurable functions on $\Omega$ and $u\in {\mathfrak {F}}\left( \Omega ,{\mathfrak {M}}\right)$ be a ${\mathbb {B}}{\mathbb {C}}$-valued function. Let $E_{M}=\left\{ x\in \Omega :\left| u\left( x\right) \right| _{k}\succ M\right\}$ for any $M\succeq 0$. Since u is a ${\mathfrak {M}}$ -measurable function, we can say that $\left| u\right| _{k}=\left| u_{1}\right| e_{1}+~\left| u_{2}\right| e_{2}$ is ${\mathbb {D}}$-valued measurable, i.e. $E_{M}\in {\mathfrak {M}}$ for any $M\succeq 0$. If the set A is defined as $A=\left\{ M\succ 0:\vartheta \left( E_{M}\right) =0\right\} =\left\{ M\in {\mathbb {D}}^{+}:\left| u\left( x\right) \right| _{k}\preceq M~~\vartheta -a.e.\right\}$, then $\mathbb {D-}$essential supremum of u, denoted by essup$_{{\mathbb {D}}}u$ or $\left\| u\right\| _{\infty }^{{\mathbb {D}}}$ is defined by $\left\| u\right\| _{\infty }^{{\mathbb {D}}}=$essup$_{{\mathbb {D}}}u =$inf$_{{\mathbb {D}} }\left( A\right)$.

3 $\mathbb {D-}$Distribution Function

Now suppose that $\left( \Omega ,{\mathfrak {M}},\vartheta \right)$ is a $\sigma$-finite complete $\mathbb{B}\mathbb{C}$-measurable space and ${\mathfrak {F}} \left( \Omega ,{\mathfrak {M}}\right)$ is the set of all measurable $\mathbb {BC }$-valued functions on $\Omega$.

Definition 4

Let $u=u_{1}e_{1}+~u_{2}e_{2}$ be an element of ${\mathfrak {F}}\left( \Omega , {\mathfrak {M}}\right)$, $\lambda =\lambda _{1}e_{1}+~\lambda _{2}e_{2}$ be in ${\mathbb {D}}^{+}\cup \left\{ 0\right\}$ and $\vartheta =\vartheta _{1}e_{1}+~\vartheta _{2}e_{2}$ be a ${\mathbb {B}}{\mathbb {C}}$-measure. Then $\mathbb{B}\mathbb{C}$-distribution function $D_{u}^{\mathbb{B}\mathbb{C}}:{\mathbb {D}}^{+}\cup \left\{ 0\right\} \rightarrow$ ${\mathbb {D}}^{+}\cup \left\{ 0\right\}$ of u is given by

$$\begin{aligned} D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right)= & {} D_{u_{1}}\left( \lambda _{1}\right) e_{1}+D_{u_{2}}\left( \lambda _{2}\right) e_{2} \nonumber \\= & {} \vartheta _{1}\left\{ x\in \Omega :\left| u_{1}\left( x\right) \right|>\lambda _{1}\right\} e_{1}+\vartheta _{2}\left\{ x\in \Omega :\left| u_{2}\left( x\right) \right| >\lambda _{2}\right\} e_{2} \end{aligned}$$

(3.1)

where $\lambda \succeq 0$.

Theorem 2

Let u and v be two $\mathbb{B}\mathbb{C}$-valued functions in ${\mathfrak {F}}\left( \Omega ,{\mathfrak {M}}\right)$ with $u=u_{1}e_{1}+u_{2}e_{2}$ and $v=v_{1}e_{1}+v_{2}e_{2}$. Then for any $\lambda ,~\alpha ,~\gamma \in {\mathbb {D}}^{+}\cup \left\{ 0\right\}$:

(a):

$D_{u}^{\mathbb{B}\mathbb{C}}\left( \cdot \right)$ is decreasing in the sense of ${\mathbb {D}}$;

(b):

Being $\left| v\right| _{k}\preceq \left| u\right| _{k}$ $\vartheta -$a.e. says that $D_{v}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) \preceq D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right)$;

(c):

$D_{du}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) =D_{u}^{ \mathbb{B}\mathbb{C}}\left( \frac{\lambda }{\left| d\right| _{k}}\right)$ for all $d\in$ ${\mathbb {B}} {\mathbb {C}}$ with non-zero components;

(d):

$D_{u+v}^{\mathbb{B}\mathbb{C}}\left( \alpha +\gamma \right) \preceq D_{u}^{\mathbb{B}\mathbb{C}}\left( \alpha \right) +D_{v}^{\mathbb{B}\mathbb{C}}\left( \gamma \right)$;

(e):

$D_{uv}^{\mathbb{B}\mathbb{C}}\left( \alpha \gamma \right) \preceq D_{u}^{\mathbb{B}\mathbb{C}}\left( \alpha \right) +D_{v}^{\mathbb{B}\mathbb{C}}\left( \gamma \right)$;

(f):

If $\left| u\right| _{k}\preceq$liminf$_{{\mathbb {D}}}\left| u_{n}\right| _{k}$ $\vartheta -$a.e., then $D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) \preceq$liminf$_{\mathbb { D}}D_{u_{n}}^{\mathbb{B}\mathbb{C}}\left( \lambda \right)$;

(g):

If $\left| u_{n}\right| _{k}$ is a ${\mathbb {D}}$ -increasing sequence and $\mathbb {D-}$convergent to $\left| u\right| _{k}$, then lim$_{{\mathbb {D}}}D_{u_{n}}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) =D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right)$.

Proof

(a) Let $\alpha ,~\gamma \succeq 0$ be any two non-negative hyperbolic numbers with $\alpha \preceq \gamma$. Then $\gamma _{1}\ge \alpha _{1}$ and $\gamma _{2}\ge \alpha _{2}$ for $\alpha =\alpha _{1}e_{1}+\alpha _{2}e_{2}$ and $\gamma =\gamma _{1}e_{1}+\gamma _{2}e_{2}$. Therefore, we have

$$\begin{aligned} \left\{ x\in \Omega :\left| u_{j}\left( x\right) \right|>\gamma _{j}\right\} \subset \left\{ x\in \Omega :\left| u_{j}\left( x\right) \right| >\alpha _{j}\right\} \end{aligned}$$

and $D_{u_{j}}\left( \gamma _{j}\right) \le D_{u_{j}}\left( \alpha _{j}\right)$ by the monotonicity of the measure for all $j=1,2.$ Hence

$$\begin{aligned} D_{u}^{\mathbb{B}\mathbb{C}}\left( \gamma \right)= & {} D_{u_{1}}\left( \gamma _{1}\right) e_{1}+D_{u_{2}}\left( \gamma _{2}\right) e_{2} \\= & {} \vartheta _{1}\left\{ x\in \Omega :\left| u_{1}\left( x\right) \right|>\gamma _{1}\right\} e_{1}+\vartheta _{2}\left\{ x\in \Omega :\left| u_{2}\left( x\right) \right|>\gamma _{2}\right\} e_{2} \\\preceq & {} \vartheta _{1}\left\{ x\in \Omega :\left| u_{1}\left( x\right) \right|>\alpha _{1}\right\} e_{1}+\vartheta _{2}\left\{ x\in \Omega :\left| u_{2}\left( x\right) \right| >\alpha _{2}\right\} e_{2} \\= & {} D_{u_{1}}\left( \alpha _{1}\right) e_{1}+D_{u_{2}}\left( \alpha _{2}\right) e_{2}=D_{u}^{\mathbb{B}\mathbb{C}}\left( \alpha \right) \end{aligned}$$

and so $D_{u}^{\mathbb{B}\mathbb{C}}\left( \cdot \right)$ is ${\mathbb {D}}$-decreasing.

Now let $\gamma _{0}^{1}e_{1}+\gamma _{0}^{2}e_{2}=\gamma _{0}\succeq 0$. Since the distribution functions $D_{u_{j}}\left( \cdot \right) :\left[ 0,\infty \right] \rightarrow \left[ 0,\infty \right]$ are continuous from right for $j=1,2$ [2, Theorem 4.3], we can write the following

$$\begin{aligned} \underset{h_{1}\rightarrow 0^{+}}{lim }D_{u_{1}}\left( \gamma _{0}^{1}+h_{1}\right) =D_{u_{1}}\left( \gamma _{0}^{1}\right) \text{ and } \underset{h_{2}\rightarrow 0^{+}}{lim }D_{u_{2}}\left( \gamma _{0}^{2}+h_{2}\right) =D_{u_{2}}\left( \gamma _{0}^{2}\right) \end{aligned}$$

by the continuity of the measures. Let $h\overset{{\mathbb {D}}}{\rightarrow } 0^{+}$ means $h_{1}\rightarrow 0^{+}$, $h_{2}\rightarrow 0^{+}$ where $h=h_{1}e_{1}+h_{2}e_{2}\in {\mathbb {D}}^{+}$. Then

$$\begin{aligned} \underset{h\overset{{\mathbb {D}}}{\rightarrow }0^{+}}{ {lim}_{ {\mathbb {D}}}}D_{u}^{\mathbb {B}\mathbb {C}}\left( \gamma _{0}+h\right) ={} & {} {} \underset{h \overset{{\mathbb {D}}}{\rightarrow }0^{+}}{ {lim}_{{\mathbb {D}}}} \left\{ D_{u_{1}}\left( \gamma _{0}^{1}+h_{1}\right) e_{1}+D_{u_{2}}\left( \gamma _{0}^{2}+h_{2}\right) e_{2}\right\} \\={} & {} {} \left\{ \underset{h_{1}\rightarrow 0^{+}}{lim} D_{u_{1}}\left( \gamma _{0}^{1}+h_{1}\right) \right\} e_{1}+\left\{ \underset{h_{2}\rightarrow 0^{+}}{lim}D_{u_{2}}\left( \gamma _{0}^{2}+h_{2}\right) \right\} e_{2} \\={} & {} {} D_{u_{1}}\left( \gamma _{0}^{1}\right) e_{1}+D_{u_{2}}\left( \gamma _{0}^{2}\right) e_{2} \\={} & {} {} D_{u}^{\mathbb {B}\mathbb {C}}\left( \gamma _{0}\right) . \end{aligned}$$

(b) Let $\left| u\right| _{k}=\left| u_{1}\right| e_{1}+\left| u_{2}\right| e_{2}$, $\left| v\right| _{k}=\left| v_{1}\right| e_{1}+\left| v_{2}\right| e_{2}$ and $\left| v\right| _{k}\preceq \left| u\right| _{k}$ $\vartheta -$a.e. Then $\left| v_{1}\left( x\right) \right| \le \left| u_{1}\left( x\right) \right|$ and $\left| v_{2}\left( x\right) \right| \le \left| u_{2}\left( x\right) \right|$ $\vartheta -$a.e. Therefore, the inclusion

$$\begin{aligned} \left\{ x\in \Omega :\left| v_{j}\left( x\right) \right|>\lambda _{j}\right\} \subset \left\{ x\in \Omega :\left| u_{j}\left( x\right) \right| >\lambda _{j}\right\} \end{aligned}$$

exists for all $j\in 1,2$ and for any $\lambda \in {\mathbb {D}}^{+}\cup \left\{ 0\right\}$. Since the measure is monotone, one can write for $j\in 1,2$ that

$$\begin{aligned} \vartheta _{j}\left\{ x\in \Omega :\left| v_{j}\left( x\right) \right|>\lambda _{j}\right\} \le \vartheta _{j}\left\{ x\in \Omega :\left| u_{j}\left( x\right) \right| >\lambda _{j}\right\} \end{aligned}$$

and

$$\begin{aligned} D_{v}^{\mathbb{B}\mathbb{C}}\left( \lambda \right)= & {} \vartheta _{1}\left\{ x\in \Omega :\left| v_{1}\left( x\right) \right|>\lambda _{1}\right\} e_{1}+\vartheta _{2}\left\{ x\in \Omega :\left| v_{2}\left( x\right) \right|>\lambda _{2}\right\} e_{2} \\\preceq & {} \vartheta _{1}\left\{ x\in \Omega :\left| u_{1}\left( x\right) \right|>\lambda _{1}\right\} e_{1}+\vartheta _{2}\left\{ x\in \Omega :\left| u_{2}\left( x\right) \right| >\lambda _{2}\right\} e_{2} \\= & {} D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) . \end{aligned}$$

(c) Let $d=d_{1}e_{1}+d_{2}e_{2}$ be any ${\mathbb {B}} {\mathbb {C}}$-number with $d_{1}\ne 0$, $d_{2}\ne 0$. Since

$$\begin{aligned} \frac{\lambda }{\left| d\right| _{k}}=\frac{\lambda _{1}e_{1}+\lambda _{2}e_{2}}{\left| d_{1}\right| e_{1}+\left| d_{2}\right| e_{2}}=\frac{\lambda _{1}}{\left| d_{1}\right| } e_{1}+\frac{\lambda _{2}}{\left| d_{2}\right| }e_{2} \end{aligned}$$

(3.2)

for any $\lambda \in {\mathbb {D}}$ and $d\cdot u\left( x\right) =\left( d_{1}u_{1}\left( x\right) \right) e_{1}+\left( d_{2}u_{2}\left( x\right) \right) e_{2}$ for any $x\in \Omega$, we have

$$\begin{aligned} D_{du}^{\mathbb{B}\mathbb{C}}\left( \lambda \right)= & {} \vartheta _{1}\left\{ x\in \Omega :\left| d_{1}u_{1}\left( x\right) \right|>\lambda _{1}\right\} e_{1}+\vartheta _{2}\left\{ x\in \Omega :\left| d_{2}u_{2}\left( x\right) \right|>\lambda _{2}\right\} e_{2} \\= & {} \vartheta _{1}\left\{ x\in \Omega :\left| u_{1}\left( x\right) \right|>\frac{\lambda _{1}}{\left| d_{1}\right| }\right\} e_{1}+\vartheta _{2}\left\{ x\in \Omega :\left| u_{2}\left( x\right) \right| >\frac{\lambda _{2}}{\left| d_{2}\right| }\right\} e_{2} \\= & {} D_{u_{1}}\left( \frac{\lambda _{1}}{\left| d_{1}\right| }\right) e_{1}+D_{u_{2}}\left( \frac{\lambda _{2}}{\left| d_{2}\right| } \right) e_{2} \\= & {} D_{u}^{\mathbb{B}\mathbb{C}}\left( \frac{\lambda }{\left| d\right| _{k}} \right) \end{aligned}$$

by (3.2).

(d) Let $u=u_{1}e_{1}+u_{2}e_{2},~v=v_{1}e_{1}+v_{2}e_{2}$ be any two measurable, ${\mathbb {B}} {\mathbb {C}}$-valued functions and $\alpha =\alpha _{1}e_{1}+\alpha _{2}e_{2},~\gamma =\gamma _{1}e_{1}+\gamma _{2}e_{2}$ be non-negative hyperbolic numbers. By using the definition of $D_{u}^{\mathbb{B}\mathbb{C}}\left( \cdot \right)$,

$$\begin{aligned} D_{u+v}^{\mathbb{B}\mathbb{C}}\left( \alpha +\gamma \right)= & {} D_{u_{1}+v_{1}}\left( \alpha _{1}+\gamma _{1}\right) e_{1}+D_{u_{2}+v_{2}}\left( \alpha _{2}+\gamma _{2}\right) e_{2} \nonumber \\= & {} \vartheta _{1}\left\{ x\in \Omega :\left| \left( u_{1}+v_{1}\right) \left( x\right) \right|>\alpha _{1}+\gamma _{1}\right\} e_{1} \nonumber \\{} & {} \quad +\vartheta _{2}\left\{ x\in \Omega :\left| \left( u_{2}+v_{2}\right) \left( x\right) \right| >\alpha _{2}+\gamma _{2}\right\} e_{2} \end{aligned}$$

(3.3)

is written. Since we have the following inclusion

$$\begin{aligned} {}\left\{ x\in \Omega :\left| \left( u_{j}+v_{j}\right) \left( x\right) \right|>\alpha _{j}+\gamma _{j}\right\} \subset \left\{ x\in \Omega :\left| u_{j}\left( x\right) \right|>\alpha _{j}\right\} \cup \left\{ x\in \Omega :\left| v_{j}\left( x\right) \right| >\gamma _{j}\right\} , \end{aligned}$$

we can write that

$$\begin{aligned} \vartheta _{j}\left\{ x\in \Omega :\left| \left( u_{j}+v_{j}\right) \left( x\right) \right|>\alpha _{j}+\gamma _{j}\right\}< & {} \vartheta _{j}\left\{ x\in \Omega :\left| u_{j}\left( x\right) \right|>\alpha _{j}\right\} \\{} & {} \quad +\vartheta _{j}\left\{ x\in \Omega :\left| v_{j}\left( x\right) \right| >\gamma _{j}\right\} \end{aligned}$$

and so

$$\begin{aligned} D_{u_{j}+v_{j}}\left( \alpha _{j}+\gamma _{j}\right) <D_{u_{j}}\left( \alpha _{j}\right) +D_{v_{j}}\left( \gamma _{j}\right) \end{aligned}$$

(3.4)

for all $j\in 1,2$. By combining (3.3) and (3.4),

$$\begin{aligned} D_{u+v}^{\mathbb{B}\mathbb{C}}\left( \alpha +\gamma \right)= & {} D_{u_{1}+v_{1}}\left( \alpha _{1}+\gamma _{1}\right) e_{1}+D_{u_{2}+v_{2}}\left( \alpha _{2}+\gamma _{2}\right) e_{2} \\\preceq & {} \left( D_{u_{1}}\left( \alpha _{1}\right) +D_{v_{1}}\left( \gamma _{1}\right) \right) e_{1}+\left( D_{u_{2}}\left( \alpha _{2}\right) +D_{v_{2}}\left( \gamma _{2}\right) \right) e_{2} \\= & {} \left( D_{u_{1}}\left( \alpha _{1}\right) e_{1}+D_{u_{2}}\left( \alpha _{2}\right) e_{2}\right) +\left( D_{v_{1}}\left( \gamma _{1}\right) e_{1}+D_{v_{2}}\left( \gamma _{2}\right) e_{2}\right) \\= & {} D_{u}^{\mathbb{B}\mathbb{C}}\left( \alpha \right) +D_{v}^{\mathbb{B}\mathbb{C}}\left( \gamma \right) \end{aligned}$$

can be obtained.

(e) Let $u,~v,~\alpha$ and $\gamma$ be as in (d). Then $uv=u_{1}v_{1}e_{1}+u_{2}v_{2}e_{2}$ and $\alpha \gamma =\alpha _{1}\gamma _{1}e_{1}+\alpha _{2}\gamma _{2}e_{2}$. By the definition of $D_{\left( \cdot \right) }^{\mathbb{B}\mathbb{C}}\left( \cdot \right)$, we can write that

$$\begin{aligned} D_{uv}^{\mathbb{B}\mathbb{C}}\left( \alpha \gamma \right)= & {} D_{u_{1}v_{1}}\left( \alpha _{1}\gamma _{1}\right) e_{1}+D_{u_{2}v_{2}}\left( \alpha _{2}\gamma _{2}\right) e_{2} \\= & {} \vartheta _{1}\left\{ x\in \Omega :\left| \left( u_{1}v_{1}\right) \left( x\right) \right|>\alpha _{1}\gamma _{1}\right\} e_{1} \\{} & {} ~~~~~~~~~\ \ \ \ \ \ \ \ \ \ \ \ \ ~~+\vartheta _{2}\left\{ x\in \Omega :\left| \left( u_{2}v_{2}\right) \left( x\right) \right| >\alpha _{2}\gamma _{2}\right\} e_{2}. \end{aligned}$$

Since the inclusion

$$\begin{aligned} \left\{ x{\in } \Omega :\left| \left( u_{j}v_{j}\right) \left( x\right) \right| {>}\alpha _{j}\gamma _{j}\right\} \subset \left\{ x\in \Omega :\left| u_{j}\left( x\right) \right| {>}\alpha _{j}\right\} \cup \left\{ x\in \Omega :\left| v_{j}\left( x\right) \right| >\gamma _{j}\right\} \end{aligned}$$

exits for all $j\in 1,2$, it is easy to see that

$$\begin{aligned} D_{u_{j}v_{j}}\left( \alpha _{j}\gamma _{j}\right) <D_{u_{j}}\left( \alpha _{j}\right) +D_{v_{j}}\left( \gamma _{j}\right) \end{aligned}$$

by the monotonocity of measure for all $j\in 1,2$. Therefore

$$\begin{aligned} D_{uv}^{\mathbb{B}\mathbb{C}}\left( \alpha \gamma \right)= & {} D_{u_{1}v_{1}}\left( \alpha _{1}\gamma _{1}\right) e_{1}+D_{u_{2}v_{2}}\left( \alpha _{2}\gamma _{2}\right) e_{2} \\\preceq & {} \left( D_{u_{1}}\left( \alpha _{1}\right) +D_{v_{1}}\left( \gamma _{1}\right) \right) e_{1}+\left( D_{u_{2}}\left( \alpha _{2}\right) +D_{v_{2}}\left( \gamma _{2}\right) \right) e_{2} \\= & {} \left( D_{u_{1}}\left( \alpha _{1}\right) e_{1}+D_{u_{2}}\left( \alpha _{2}\right) e_{2}\right) +\left( D_{v_{1}}\left( \gamma _{1}\right) e_{1}+D_{v_{2}}\left( \gamma _{2}\right) e_{2}\right) \\= & {} D_{u}^{\mathbb{B}\mathbb{C}}\left( \alpha \right) +D_{v}^{\mathbb{B}\mathbb{C}}\left( \gamma \right) . \end{aligned}$$

(f) Let $\lambda \succ 0$ be fixed, $D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) =\vartheta _{1}\left\{ x\in \Omega :\left| u_{1}\left( x\right) \right|>\lambda _{1}\right\} e_{1} +\vartheta _{2}\left\{ x\in \Omega :\left| u_{2}\left( x\right) \right| >\lambda _{2}\right\} e_{2}$, $E_{j}=\left\{ x\in \Omega :\left| u_{j}\left( x\right) \right| >\lambda _{j}\right\}$ and $E_{j}^{n} =\left\{ x\in \Omega :\left| u_{j}^{n}\left( x\right) \right| >\lambda _{j}\right\}$ for $j=1,2$ and all $n\in {\mathbb {N}}$. Since $\left| u\right| _{k}\preceq$ ${liminf}_{{\mathbb {D}}}\left| u_{n}\right| _{k}$ $\vartheta -$a.e. by the hypothesis, we have $E_{1}\subset \bigcup \limits _{i=1}^{\infty }\left( \bigcap \limits _{n>i}E_{1}^{n}\right)$ and $E_{2}\subset \bigcup \limits _{i=1}^{\infty }\left( \bigcap \limits _{n>i}E_{2}^{n}\right)$ by Remark 1. Therefore

$$\begin{aligned} \vartheta _{j}\left( \bigcap \limits _{n>i}E_{j}^{n}\right) \le \inf _{n>i}\vartheta _{j}\left( E_{j}^{n}\right) \le \sup _{i}\left( \inf _{n>i}\vartheta \left( E_{j}^{n}\right) \right) =\underset{n\rightarrow \infty }{liminf}\vartheta _{j}\left( E_{j}^{n}\right) \end{aligned}$$

(3.5)

for $j=1,2$ and all $i\in {\mathbb {N}}$. As a consequence of monotone convergence theorem and the fact that $\bigcap \limits _{n>i}E_{j}^{n}\subset \bigcap \limits _{n>i+1}E_{j}^{n}$ for $j=1,2$, we can get

$$\begin{aligned} \vartheta _{j}\left( E_{j}\right) \le \vartheta _{j}\left( \bigcup \limits _{i=1}^{\infty }\left( \bigcap \limits _{n>i}E_{j}^{n}\right) \right) =\underset{i\rightarrow \infty }{lim }\vartheta _{j}\left( \bigcap \limits _{n>i}E_{j}^{n}\right) \le \underset{n\rightarrow \infty }{ liminf }\vartheta _{j}\left( E_{j}^{n}\right) \end{aligned}$$

by (3.5). So

$$\begin{aligned} D_{u}^{\mathbb {B}\mathbb {C}}\left( \lambda \right) =&\,\, D_{u_{1}}\left( \lambda _{1}\right) e_{1}+D_{u_{2}}\left( \lambda _{2}\right) e_{2} \\=&\,\, \vartheta _{1}\left\{ x\in \Omega :\left| u_{1}\left( x\right) \right|>\lambda _{1}\right\} e_{1}+\vartheta _{2}\left\{ x\in \Omega :\left| u_{2}\left( x\right) \right| >\lambda _{2}\right\} e_{2} \\=&\,\, \vartheta _{1}\left( E_{1}\right) e_{1}+\vartheta _{2}\left( E_{2}\right) e_{2} \\\preceq&\,\, {liminf}\vartheta _{1}\left( E_{1}^{n}\right) e_{1}+{liminf }\vartheta _{2}\left( E_{2}^{n}\right) e_{2} \\=&\,\, {liminf}_{{\mathbb {D}}}\left( \vartheta _{1}\left( E_{1}^{n}\right) e_{1}+\vartheta _{2}\left( E_{2}^{n}\right) e_{2}\right) \\=&\,\,{liminf}_{{\mathbb {D}}}D_{u_{n}}^{\mathbb {B}\mathbb {C}}\left( \lambda \right) . \end{aligned}$$

(g) Let $\left| u^{\left( n\right) }\right| _{k}$ be a ${\mathbb {D}}$-increasing and $\mathbb {D-}$convergent sequence to $\left| u\right|$. Then $\left| u^{\left( 1\right) }\right| _{k}\preceq \left| u^{\left( 2\right) }\right| _{k}\preceq \left| u^{\left( 3\right) }\right| _{k}\preceq \cdots$ implies that the following four sequences: $\left| u_{1}^{\left( n\right) }\right| ,~\left| u_{2}^{\left( n\right) }\right|$ and

$$\begin{aligned} E_{1}^{\left( n\right) }=\left\{ x\in \Omega :\left| u_{1}^{\left( n\right) }\left( x\right) \right|>\lambda _{1}\right\} ,\text { } E_{2}^{\left( n\right) }=\left\{ x\in \Omega :\left| u_{2}^{\left( n\right) }\left( x\right) \right| >\lambda _{2}\right\} \end{aligned}$$

are increasing. Therefore

$$\begin{aligned} E_{1}=\left\{ x\in \Omega :\left| u_{1}\left( x\right) \right| >\lambda _{1}\right\} =\bigcup \limits _{n=1}^{\infty }E_{1}^{\left( n\right) } \end{aligned}$$

and

$$\begin{aligned} E_{2}=\left\{ x\in \Omega :\left| u_{2}\left( x\right) \right| >\lambda _{2}\right\} =\bigcup \limits _{n=1}^{\infty }E_{2}^{\left( n\right) }. \end{aligned}$$

As a result,

$$\begin{aligned} D_{u}^{\mathbb {B}\mathbb {C}}\left( \lambda \right) =&\,\, {} D_{u_{1}}\left( \lambda _{1}\right) e_{1}+D_{u_{2}}\left( \lambda _{2}\right) e_{2} \\=&\,\, {} \vartheta _{1}\left( E_{1}\right) e_{1}+\vartheta _{2}\left( E_{2}\right) e_{2} \\=&\,\, {} \vartheta _{1}\left( \bigcup \limits _{n=1}^{\infty }E_{1}^{\left( n\right) }\right) e_{1}+\vartheta _{2}\left( \bigcup \limits _{n=1}^{\infty }E_{2}^{\left( n\right) }\right) e_{2} \\=&\,\, {} {lim} \vartheta _{1}\left( E_{1}^{\left( n\right) }\right) e_{1}+{lim} \vartheta _{2}\left( E_{2}^{\left( n\right) }\right) e_{2} \\=&\,\, {}{lim}_{{\mathbb {D}}}\left( D_{u_{1}^{\left( n\right) }}\left( \lambda _{1}\right) e_{1}+D_{u_{2}^{\left( n\right) }}\left( \lambda _{2}\right) e_{2}\right) \\=&\,\, {}{lim}_{{\mathbb {D}}}D_{u_{n}}^{\mathbb {B}\mathbb {C}}\left( \lambda \right) . \end{aligned}$$

$\square$

4 $\mathbb {D-}$Decreasing Rearrangement

In this section, by using the notion of $\mathbb {D-}$distribution function we will introduce the $\mathbb {D-}$decreasing rearrangement and show some fundamental properties of it.

Definition 5

Let $\lambda \in {\mathbb {D}}^{+}\cup \left\{ 0\right\}$ and u be a $\mathbb{B}\mathbb{C}$ function in ${\mathfrak {F}}\left( \Omega ,{\mathfrak {M}} \right)$. The $\mathbb {D-}$decreasing rearrangement of u, $u_{\mathbb{B}\mathbb{C} }^{*}$ is the function $u_{\mathbb{B}\mathbb{C}}^{*}:{\mathbb {D}}^{+}\cup \left\{ 0\right\} \rightarrow {\mathbb {D}}^{+}\cup \left\{ 0\right\}$ defined by

$$\begin{aligned} u_{\mathbb {B}\mathbb {C}}^{*}\left( t\right) =&\,\, {} {inf}_{{\mathbb {D}}}\left\{ \alpha \succeq 0:D_{u}^{\mathbb {B}\mathbb {C}}\left( \alpha \right) \preceq t\right\} \\=&\,\, {}{inf}\left\{ \alpha _{1}\ge 0:D_{u_{1}}\left( \alpha _{1}\right) \le t_{1}\right\} e_{1}+{inf}\left\{ \alpha _{2}\ge 0:D_{u_{2}}\left( \alpha _{2}\right) \le t_{2}\right\} e_{2} \\=&\,\, {} u_{1}^{*}\left( t_{1}\right) e_{1}+u_{2}^{*}\left( t_{2}\right) e_{2} \end{aligned}$$

where inf$_{{\mathbb {D}}}\varnothing ={\infty }_{\mathbb {D}}$.

According to [5, Example 2.2], since

$$\begin{aligned} \left\| u\right\| _{\infty }^{{\mathbb {D}}}={inf}_{\mathbb { D}}\left\{ \alpha \succeq 0:\vartheta \left\{ x\in \Omega :\left| u\left( x\right) \right| _{k}\succ \alpha \right\} =0\right\} , \end{aligned}$$

and $\left\| u_{1}\right\| _{\infty },~\left\| u_{2}\right\| _{\infty }\preceq \left\| u\right\| _{\infty }^{{\mathbb {D}}}$, one can write that $\left\| u\right\| _{\infty }^{{\mathbb {D}}}=\left\| u_{1}\right\| _{\infty }e_{1}+\left\| u_{2}\right\| _{\infty }e_{2}$ and so

$$\begin{aligned} u_{\mathbb {B}\mathbb {C}}^{*}\left( 0\right) =&\,\, {}{inf}_{{\mathbb {D}} }\left\{ \alpha \succeq 0:D_{u}^{\mathbb {B}\mathbb {C}}\left( \alpha \right) =0\right\} \\=&\,\, {}{inf}_{{\mathbb {D}}}\left\{ \alpha \succeq 0:\vartheta _{j}\left\{ x\in \Omega :\left| u_{j}\left( x\right) \right| >\alpha _{j}\right\} =0,j=1,2\right\} \\=&\,\, {} \left\| u\right\| _{\infty }^{{\mathbb {D}}}\text{. } \end{aligned}$$

On the other hand, $\mathbb {D-}$decreasing property of $D_{u}^{\mathbb{B}\mathbb{C} }\left( \cdot \right)$ implies that

$$\begin{aligned} u_{\mathbb {B}\mathbb {C}}^{*}\left( D_{u}^{\mathbb {B}\mathbb {C}}\left( t\right) \right) =&\,\, {}{inf}_{{\mathbb {D}}}\left\{ \alpha \succeq 0:D_{u}^{\mathbb {B}\mathbb {C} }\left( \alpha \right) \le D_{u}^{\mathbb {B}\mathbb {C}}\left( t\right) \right\} \\=&\,\, {}{inf}_{{\mathbb {D}}}\left\{ \alpha \succeq 0:\alpha \succ t\right\} =t \end{aligned}$$

or

$$\begin{aligned} u_{\mathbb {B}\mathbb {C}}^{*}\left( D_{u}^{\mathbb {B}\mathbb {C}}\left( t\right) \right) =&\,\, {}{inf}\left\{ \alpha _{1}\ge 0:D_{u_{1}}\left( \alpha _{1}\right) \le D_{u_{1}}\left( t_{1}\right) \right\} e_{1} \\{}&{} \quad +{inf}\left\{ \alpha _{2}\ge 0:D_{u_{2}}\left( \alpha _{2}\right) \le D_{u_{2}}\left( t_{2}\right) \right\} e_{2} \\=&\,\, {}{inf}\left\{ \alpha _{1}\ge 0:\alpha _{1}>t_{1}\right\} e_{1}+{inf}\left\{ \alpha _{2}\ge 0:\alpha _{2}>t_{2}\right\} e_{2} \\=&\,\, {} t_{1}e_{1}+t_{2}e_{2}=t \end{aligned}$$

and so $u_{\mathbb{B}\mathbb{C}}^{*}\left( \cdot \right)$ is the left ${\mathbb {D}}$-inverse of the distribution function $D_{u}^{\mathbb{B}\mathbb{C}}\left( \cdot \right)$. Now, let $u_{\mathbb{B}\mathbb{C}}^{*}\left( t\right) =\lambda =\lambda _{1}e_{1}+\lambda _{2}e_{2}\preceq \infty _{{\mathbb {D}}}$. Then by Definition 5, there exists a sequence $\lambda _{n}=\lambda _{n}^{\left( 1\right) }e_{1}+\lambda _{n}^{\left( 2\right) }e_{2}$ in ${\mathbb {D}}^{+}$ such that $\lambda _{n}^{\left( 1\right) }\downarrow \lambda _{1}$, $\lambda _{n}^{\left( 2\right) }\downarrow \lambda _{2}$, $D_{u_{1}}\left( \lambda _{n}^{\left( 1\right) }\right) \le t_{1}$ and $D_{u_{2}}\left( \lambda _{n}^{\left( 2\right) }\right) \le t_{2}$. By using the techniques used in the continuation of [2, Definition 4.4] and the right continuity of the usual distribution function, we get

$$\begin{aligned} D_{u}^{\mathbb {B}\mathbb {C}}\left( u_{\mathbb {B}\mathbb {C}}^{*}\left( t\right) \right) =&\,\, {} D_{u}^{\mathbb {B}\mathbb {C}}\left( \lambda \right) =D_{u_{1}}\left( \lambda _{1}\right) e_{1}+D_{u_{2}}\left( \lambda _{2}\right) e_{2} \\=&{} \left( {lim} D_{u_{1}}\left( \lambda _{n}^{\left( 1\right) }\right) \right) e_{1}+\left( {lim} D_{u_{2}}\left( \lambda _{n}^{\left( 2\right) }\right) \right) e_{2} \\\preceq&\,\, {} t_{1}e_{1}+t_{2}e_{2}=t. \end{aligned}$$

Therefore

$$\begin{aligned} u_{\mathbb{B}\mathbb{C}}^{*}\left( D_{u}^{\mathbb{B}\mathbb{C}}\left( \alpha \right) \right) \preceq \alpha \text { and }D_{u}^{\mathbb{B}\mathbb{C}}\left( u_{\mathbb{B}\mathbb{C} }^{*}\left( t\right) \right) \preceq t. \end{aligned}$$

(4.1)

In the following theorem we will give some basic properties of the $\mathbb {D }$-decreasing rearrangement function.

Theorem 3

The decreasing rearrangement $u_{\mathbb{B}\mathbb{C}}^{*}$ of a measurable function u has the following properties:

(a):

$u_{\mathbb{B}\mathbb{C}}^{*}$ is ${\mathbb {D}}$-decreasing;

(b):

$u_{\mathbb{B}\mathbb{C}}^{*}(t)\succ \lambda _{0}$ if and only if $D_{u}^{\mathbb{B}\mathbb{C}}(\lambda _{0})\succ t$;

(c):

u and $u_{\mathbb{B}\mathbb{C}}^{*}$ are ${\mathbb {D}}$ -equimeasurables, that is $D_{u}^{\mathbb{B}\mathbb{C}}(\lambda )=D_{u_{\mathbb{B}\mathbb{C} }^{*}}^{\mathbb{B}\mathbb{C}}(\lambda )$ for all $\lambda \succeq 0$;

(d):

If $u\in {\mathfrak {F}}(\Omega ,{\mathfrak {M}})$, then $u_{ \mathbb{B}\mathbb{C}}^{*}(t)=D_{D_{u}^{\mathbb{B}\mathbb{C}}}^{\mathbb{B}\mathbb{C}}(t)$ for all $t\succeq 0$;

(e):

$(\gamma u)_{\mathbb{B}\mathbb{C}}^{*}(t)=|\gamma |_{k}u_{ \mathbb{B}\mathbb{C}}^{*}(t)$ for any $\gamma \in {\mathbb {D}}-$ $\left\{ 0\right\}$;

(f):

If $\left| u_{n}\right| _{k}\uparrow |u|_{k}$ in the sense of ${\mathbb {D}}$, then $\left( u_{n}\right) _{\mathbb{B}\mathbb{C}}^{*}\uparrow u_{\mathbb{B}\mathbb{C}}^{*}$ in ${\mathbb {D}}$;

(g):

If $|u|_{k}\preceq$liminf$_{{\mathbb {D}}}\left| u_{n}\right| _{k}$, then $u_{\mathbb{B}\mathbb{C}}^{*}\preceq$liminf$_{ {\mathbb {D}}}\left( u_{n}\right) _{\mathbb{B}\mathbb{C}}^{*}$;

(h):

If $|u|_{k}\preceq |v|_{k}$, then $u_{\mathbb{B}\mathbb{C}}^{*}(t)\preceq v_{\mathbb{B}\mathbb{C}}^{*}(t)$;

(i):

For any $E\in {\mathfrak {M}}$, $\left( \chi _{E}\right) _{ \mathbb{B}\mathbb{C}}^{*}(t)=\chi _{\left( 0,\vartheta (E)\right) }(t)=\chi _{\left( 0,\vartheta _{1}(E)\right) }(t)e_{1} +\chi _{\left( 0,\vartheta _{2}(E)\right) }(t)e_{2}$;

(j):

If $E\in {\mathfrak {M}}$, then $\left( u\chi _{E}\right) _{ \mathbb{B}\mathbb{C}}^{*}(t)\preceq u_{\mathbb{B}\mathbb{C}}^{*}(t)\chi _{\left( 0,\vartheta (E)\right) }(t)$;

(k):

If u belongs to ${\mathfrak {F}}(\Omega ,{\mathfrak {M}} ),\lambda \succ 0$ and $U{=}\chi _{E_{U}(\lambda )}$, then $U_{\mathbb{B}\mathbb{C} }^{*}(t){=}\chi _{E_{u_{\mathbb{B}\mathbb{C}}^{*}}(\lambda )}(t)$ where $E_{U}(\lambda )=\left\{ x\in \Omega :\left| U\left( x\right) \right| _{k}\succ \lambda \right\}$.

Proof

(a) Let $t,~t^{\prime }\succeq 0$ be any two hyperbolic numbers with $t\preceq t^{\prime }$ and $u=u_{1}e_{1}+u_{2}e_{2}$. Then $t_{1}^{\prime }\ge t_{1}$ and $t_{2}^{\prime }\ge t_{2}$ for $t=t_{1}e_{1}+t_{2}e_{2}$ and $t^{\prime }=t_{1}^{\prime }e_{1}+t_{2}^{\prime }e_{2}$. Since

$$\begin{aligned} \left\{ \lambda _{j}\ge 0:D_{u_{j}}(\lambda _{j})\le t_{j}\right\} \subset \left\{ \lambda _{j}\ge 0:D_{u_{j}}(\lambda _{j})\le t_{j}^{\prime }\right\} \end{aligned}$$

and

$$\begin{aligned} inf\left\{ \lambda _{j}\ge 0:D_{u_{j}}(\lambda _{j})\le t_{j}\right\} \ge inf\left\{ \lambda _{j}\ge 0:D_{u_{j}}(\lambda _{j})\le t_{j}^{\prime }\right\} \end{aligned}$$

for all $j=1,2$, we can write that

$$\begin{aligned} u_{\mathbb {B}\mathbb {C}}^{*}(t^{\prime })={} & {} {} inf_{{\mathbb {D}}}\left\{ \lambda \succeq 0:D_{u}^{\mathbb {B}\mathbb {C}}\left( \lambda \right) \preceq t^{\prime }\right\} \\={} & {} {} inf\left\{ \lambda _{1}\ge 0:D_{u_{1}}(\lambda _{1})\le t_{1}^{\prime }\right\} e_{1}+inf\left\{ \lambda _{2}\ge 0:D_{u_{2}}(\lambda _{2})\le t_{2}^{\prime }\right\} e_{2} \\\preceq{} & {} {} inf\left\{ \lambda _{1}\ge 0:D_{u_{1}}(\lambda _{1})\le t_{1}\right\} e_{1}+inf\left\{ \lambda _{2}\ge 0:D_{u_{2}}(\lambda _{2})\le t_{2}\right\} e_{2} \\={} & {} {} inf_{{\mathbb {D}}}\left\{ \lambda \succeq 0:D_{u}^{\mathbb {B}\mathbb {C}}\left( \lambda \right) \preceq t\right\} \\={} & {} {} u_{\mathbb {B}\mathbb {C}}^{*}(t). \end{aligned}$$

Therefore $u_{\mathbb{B}\mathbb{C}}^{*}(t)\succeq u_{\mathbb{B}\mathbb{C}}^{*}(t^{\prime })$.

(b) Let $\lambda _{0}=\lambda _{0}^{1}e_{1}+\lambda _{0}^{2}e_{2}\in {\mathbb {D}}$. If $\lambda _{0}\prec u_{\mathbb{B}\mathbb{C}}^{*}(t)$ then, by the definition of ${\mathbb {D}}$-infimum in Remark 1, we have $\lambda _{0}^{1}\notin \left\{ \lambda _{1}\ge 0:D_{u_{1}}(\lambda _{1})\le t_{1}\right\}$ and $\lambda _{0}^{2}\notin \left\{ \lambda _{2}\ge 0:D_{u_{2}}(\lambda _{2})\le t_{2}\right\}$ where $t=t_{1}e_{1}+t_{2}e_{2}$. Therefore $D_{u_{1}}(\lambda _{0}^{1})>t_{1}$, $D_{u_{2}}(\lambda _{0}^{2})>t_{2}$ and so $D_{u}^{\mathbb{B}\mathbb{C}}(\lambda _{0})\succ t$. Conversely, if $t\prec D_{u}^{\mathbb{B}\mathbb{C}}(\lambda _{0})$ for any $t\succeq 0$, then we get

$$\begin{aligned} u_{\mathbb{B}\mathbb{C}}^{*}(t)= & {} inf\left\{ \lambda _{1}\ge 0:D_{u_{1}}(\lambda _{0}^{1})\le t_{1}\right\} e_{1}+inf\left\{ \lambda _{2}\ge 0:D_{u_{2}}(\lambda _{0}^{2})\le t_{2}\right\} e_{2} \\\preceq & {} \lambda _{0}^{1}e_{1}+\lambda _{0}^{2}e_{2}=\lambda _{0}. \end{aligned}$$

By using the ${\mathbb {D}}$-decreasing property of distribution function and (4.1), we have $D_{u}^{\mathbb{B}\mathbb{C}}(\lambda _{0})\preceq D_{u}^{\mathbb{B}\mathbb{C}}\left( u_{\mathbb{B}\mathbb{C}}^{*}(t)\right) \preceq t$. This is a contradiction.

(c) According to (b) we have

$$\begin{aligned} \left\{ t\succeq 0:u_{\mathbb{B}\mathbb{C}}^{*}(t)\succ \lambda \right\} =\left\{ \lambda \succeq 0:D_{u}^{\mathbb{B}\mathbb{C}}(\lambda )\succ t\right\} =\left( 0,D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) \right) _{{\mathbb {D}}} \end{aligned}$$

where $\left( 0,D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) \right) _{\mathbb {D }}=\left( 0,D_{u_{1}}\left( \lambda _{1}\right) \right) e_{1}+\left( 0,D_{u_{2}}\left( \lambda _{2}\right) \right) e_{2}$ and so

$$\begin{aligned} D_{u^{*}}^{\mathbb {B}\mathbb {C}}(\lambda )={} & {} {} m_{{\mathbb {D}}}\left\{ t\succeq 0:u_{\mathbb {B}\mathbb {C}}^{*}(t)\succ \lambda \right\} \\={} & {} {} m_{{\mathbb {D}}}\left( \left( 0,D_{u}\left( \lambda \right) \right) _{ {\mathbb {D}}}\right) =D_{u}^{\mathbb {B}\mathbb {C}}\left( \lambda \right) \end{aligned}$$

for all $\lambda \succeq 0$ where $m_{{\mathbb {D}}}\left( \left( 0,\alpha \right) _{{\mathbb {D}}}\right) =m\left( \left( 0,\alpha _{1}\right) \right) e_{1}+m\left( \left( 0,\alpha _{2}\right) \right) e_{2}$.

(d) By using (b) again, we have

$$\begin{aligned} \left\{ \lambda \succeq 0:D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) \succ t\right\} =\left( 0,u_{\mathbb{B}\mathbb{C}}^{*}(t)\right) _{{\mathbb {D}}} \end{aligned}$$

and so

$$\begin{aligned} u_{\mathbb{B}\mathbb{C}}^{*}(t)=m_{{\mathbb {D}}}\left( \left\{ \lambda \succeq 0:D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) \succ t\right\} \right) =D_{D_{u}^{\mathbb{B}\mathbb{C}}}^{\mathbb{B}\mathbb{C}}(t). \end{aligned}$$

(e) Let $u\in {\mathfrak {F}}(\Omega ,{\mathfrak {M}})$ and $\gamma \in \mathbb{B}\mathbb{C}$ with non-zero components. Then

$$\begin{aligned} \left( \gamma u\right) _{\mathbb {B}\mathbb {C}}^{*}\left( t\right) =&{} inf_{{\mathbb {D}}}\left\{ \alpha \succeq 0:D_{\gamma u}^{\mathbb {B}\mathbb {C} }\left( \alpha \right) \preceq t\right\} \\=&{} \left( \gamma _{1}u_{1}\right) ^{*}\left( t_{1}\right) e_{1}+\left( \gamma _{2}u_{2}\right) ^{*}\left( t_{2}\right) e_{2} \\=&{} inf\left\{ \alpha _{1}\ge 0:D_{\gamma _{1}u_{1}}\left( \alpha _{1}\right) \le t_{1}\right\} e_{1}\\{}&{}\quad \quad +inf\left\{ \alpha _{2}\ge 0:D_{\gamma _{2}u_{2}}\left( \alpha _{2}\right) \le t_{2}\right\} e_{2} \\=&{} inf\left\{ \alpha _{1}\ge 0:D_{u_{1}}\left( \frac{\alpha _{1}}{\gamma _{1}}\right) \le t_{1}\right\} e_{1}\\{}&{} \quad \quad +inf \left\{ \alpha _{2}\ge 0:D_{u_{2}}\left( \frac{\alpha _{2}}{\gamma _{2}} \right) \le t_{2}\right\} e_{2} \\=&{} \left| \gamma _{1}\right| inf\left\{ \alpha _{1}^{\prime }\ge 0:D_{u_{1}}\left( \alpha _{1}^{\prime }\right) \le t_{1}\right\} e_{1}\\{}&{} \quad \quad +\left| \gamma _{2}\right| inf \left\{ \alpha _{2}^{\prime }\ge 0:D_{u_{2}}\left( \alpha _{2}^{\prime }\right) \le t_{2}\right\} e_{2} \\=&{} \left( \left| \gamma _{1}\right| e_{1}+\left| \gamma _{2}\right| e_{2}\right) \left\{ {{inf}}\left\{ \alpha _{1}^{\prime }\ge 0:D_{u_{1}}\left( \alpha _{1}^{\prime }\right) \le t_{1}\right\} e_{1}\right. \\{}&{} \quad \quad \quad \left. +{{inf}}\left\{ \alpha _{2}^{\prime }\ge 0:D_{u_{2}}\left( \alpha _{2}^{\prime }\right) \le t_{2}\right\} e_{2}\right\} \\=&{} \left( \left| \gamma _{1}\right| e_{1}+\left| \gamma _{2}\right| e_{2}\right) \left( u_{1}^{*}\left( t_{1}\right) e_{1}+u_{2}^{*}\left( t_{2}\right) e_{2}\right) \\=&{} \left| \gamma \right| _{k}u_{\mathbb {B}\mathbb {C}}^{*}\left( t\right) \end{aligned}$$

where $\alpha _{j}=\alpha _{j}^{\prime }\gamma _{j}$ for $j=1,2$.

(f) We know from Theorem 2(g) that if $\left| u_{n}\right| _{k}\uparrow |u|_{k}$ in the sense of ${\mathbb {D}}$, then ${lim}_{{\mathbb {D}}}D_{u_{n}}^{\mathbb {B}\mathbb {C}}(\lambda )=$ $D_{u}^{ \mathbb{B}\mathbb{C}}(\lambda )$. Now, let $U_{n}(t)=D_{u_{n}}^{\mathbb{B}\mathbb{C}}(t)$ and $U(t)=D_{u}^{\mathbb{B}\mathbb{C}}(t)$. Then, we get

$$\begin{aligned} \left( u_{n}\right) _{\mathbb {B}\mathbb {C}}^{*}(t)=&{} inf_{ {\mathbb {D}}}\left\{ \lambda \succeq 0:D_{u_{n}}^{\mathbb {B}\mathbb {C}}\left( \lambda \right) \preceq t\right\} \\=&{} D_{D_{u_{n}}^{\mathbb {B}\mathbb {C}}}^{\mathbb {B}\mathbb {C}}\left( t\right) =D_{U_{n}}^{ \mathbb {B}\mathbb {C}}\left( t\right) \end{aligned}$$

by (d). Since $\left| u_{n}\right| _{k}$ is ${\mathbb {D}}$-increasing, we have $D_{u_{n}}^{\mathbb{B}\mathbb{C}}(t)\preceq D_{u_{n+1}}^{\mathbb{B}\mathbb{C}}(t)$, $U_{n}(t)\preceq U_{n+1}(t)$ and so

$$\begin{aligned} E_{U_{1}}(t)\subseteq E_{U_{2}}(t)\subseteq \ldots \quad \text { and }\quad E_{U}(t)=\bigcup _{n=1}^{\infty }E_{U_{n}}(t) \end{aligned}$$

where $E_{U_{j}}(t)=\left\{ \lambda \succeq 0:D_{u_{j}}\left( \lambda \right) \succ t\right\}$. Thus lim$_{{\mathbb {D}}}D_{U_{n}}^{\mathbb { BC}}(t)=D_{U}^{\mathbb{B}\mathbb{C}}(t)$, i.e. ${lim}_{{\mathbb {D}}}\left( u_{n}\right) _{\mathbb {B}\mathbb {C}}^{*}(t)=u_{\mathbb {B}\mathbb {C}}^{*}(t)$.

(g) Assume $U_{n}(t)=\underset{m>n}{inf_{{\mathbb {D}}} }\left| u_{m}(t)\right| _{k}$ and observe that

$$\begin{aligned} U_{n}(t)\preceq U_{n+1}(t) \end{aligned}$$

for all $n\in {\mathbb {N}}$. If we take $v(t)=$ ${liminf}_{{\mathbb {D}} }\left| u_{n}(t)\right| _{k}=\underset{n\ge 1}{{sup}_{ {\mathbb {D}}}}U_{n}(t)$, then we obtain $\left( U_{n}\right) _{\mathbb{B}\mathbb{C} }^{*}\uparrow v_{\mathbb{B}\mathbb{C}}^{*}$ in ${\mathbb {D}}$ as $n\rightarrow \infty$ by (f) and the fact that $U_{n}\overset{{\mathbb {D}}}{\longrightarrow }v$. Since $\left| u\left( t\right) \right| _{k}\preceq$ ${liminf}_{{\mathbb {D}}}\left| u_{n}(t)\right| _{k}=v(t)$, hence we have

$$\begin{aligned} u_{\mathbb{B}\mathbb{C}}^{*}(t)\preceq v_{\mathbb{B}\mathbb{C}}^{*}(t)=\underset{n\ge 1}{sup_{{\mathbb {D}}}}\left( U_{n}\right) _{\mathbb{B}\mathbb{C}}^{*}(t). \end{aligned}$$

Again using the fact that $U_{n}\preceq \left| u_{m}\right| _{k}$ for $m\ge n$, it follows that $\left( U_{n}\right) _{\mathbb {B}\mathbb {C}}^{*}\preceq \underset{m>n}{{inf}_{{\mathbb {D}}}}\left( u_{n}\right) _{\mathbb {B}\mathbb {C}}^{*}(t)$ and so

$$\begin{aligned} u_{\mathbb {B}\mathbb {C}}^{*}(t)\preceq&{} v_{\mathbb {B}\mathbb {C}}^{*}(t)=\underset{ n\ge 1}{sup_{{\mathbb {D}}}}\left( U_{n}\right) _{\mathbb {B}\mathbb {C}}^{*}(t) \\\preceq&{} \underset{n\ge 1}{sup_{{\mathbb {D}}}}\left( \underset{m>n}{inf_{ {\mathbb {D}}}}\left( u_{m}\right) _{\mathbb {B}\mathbb {C}}^{*}(t)\right) ={ liminf}_{{\mathbb {D}}}\left( u_{n}\right) _{\mathbb {B}\mathbb {C}}^{*}(t). \end{aligned}$$

(h) Let $u,v\in {\mathfrak {F}}(\Omega ,{\mathfrak {M}})$ with $|u|_{k}\preceq |v|_{k}$ for all $x\in \Omega .$ Since $D_{u}^{\mathbb{B}\mathbb{C} }(\lambda )\preceq D_{v}^{\mathbb{B}\mathbb{C}}(\lambda )$ for all $\lambda \succeq 0$ , this yields

$$\begin{aligned} \left\{ \lambda \succeq 0:D_{v}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) \preceq t\right\} \subset \left\{ \lambda \succeq 0:D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) \preceq t\right\} , \end{aligned}$$

therefore

$$\begin{aligned} inf_{{\mathbb {D}}}\left\{ \lambda \succeq 0:D_{v}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) \preceq t\right\} \succeq inf_{{\mathbb {D}}}\left\{ \lambda \succeq 0:D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right) \preceq t\right\} \end{aligned}$$

and $u_{\mathbb{B}\mathbb{C}}^{*}(t)\preceq v_{\mathbb{B}\mathbb{C}}^{*}(t)$.

(i) Let $E\in {\mathfrak {M}}$ and $u=\chi _{E}$. Then, by the definition of $\mathbb{B}\mathbb{C}$-distribution and $\mathbb{B}\mathbb{C}$-rearrangement, we have

$$\begin{aligned} D_{u}^{\mathbb{B}\mathbb{C}}\left( \lambda \right)= & {} D_{\chi _{E}}^{\mathbb{B}\mathbb{C} }\left( \lambda \right) =D_{\chi _{E}}\left( \lambda _{1}\right) e_{1}+D_{\chi _{E}}\left( \lambda _{2}\right) e_{2} \\= & {} \vartheta _{1}\left\{ x\in \Omega :\left| \chi _{E}\left( x\right) \right|>\lambda _{1}\right\} e_{1}+\vartheta _{2}\left\{ x\in \Omega :\left| \chi _{E}\left( x\right) \right| >\lambda _{2}\right\} e_{2} \end{aligned}$$

and

$$\begin{aligned} u_{\mathbb {B}\mathbb {C}}^{*}\left( t\right) =&{} \left( \chi _{E}\right) _{\mathbb { BC}}^{*}(t)={inf}_{{\mathbb {D}}}\left\{ \alpha \succeq 0:D_{\chi _{E}}^{\mathbb {B}\mathbb {C}}\left( \alpha \right) \preceq t\right\} \\=&\,\, {} {inf}\left\{ \alpha _{1}\ge 0:D_{\chi _{E}}\left( \alpha _{1}\right) \le t_{1}\right\} e_{1}+{inf}\left\{ \alpha _{2}\ge 0:D_{\chi _{E}}\left( \alpha _{2}\right) \le t_{2}\right\} e_{2} \\=&\,\, {} \chi _{\left( 0,\vartheta _{1}\left( E\right) \right) }\left( t_{1}\right) e_{1}+\chi _{\left( 0,\vartheta _{2}\left( E\right) \right) }\left( t_{2}\right) e_{2} \end{aligned}$$

where $u=\chi _{E}e_{1}+\chi _{E}e_{2}$.

(j) Since $\left| \left( u\chi _{E}\right) \left( x\right) \right| _{k}\preceq \left| u\left( x\right) \right| _{k}$ for all $x\in \Omega$, we can write

$$\begin{aligned} D_{u\chi _{E}}^{\mathbb{B}\mathbb{C}}(\lambda )\preceq D_{u}^{\mathbb{B}\mathbb{C}}(\lambda ) \end{aligned}$$

for all $\lambda \succeq 0$ and then

$$\begin{aligned} \left\{ \lambda \succeq 0:D_{u\chi _{E}}^{\mathbb{B}\mathbb{C}}(\lambda )\succ t\right\} \subseteq \left\{ \lambda \succeq 0:D_{u}^{\mathbb{B}\mathbb{C}}(\lambda )\succ t\right\} . \end{aligned}$$

Therefore $D_{D_{u\chi _{E}}^{\mathbb{B}\mathbb{C}}}^{\mathbb{B}\mathbb{C}}(t)\preceq D_{D_{u}^{\mathbb{B}\mathbb{C}}}^{\mathbb{B}\mathbb{C}}(t)$ and so$\left( u\chi _{E}\right) _{ \mathbb{B}\mathbb{C}}^{*}(t)\preceq u_{\mathbb{B}\mathbb{C}}^{*}(t)\chi _{\left( 0,\vartheta (E)\right) }\left( t\right)$ for all $t\succeq 0$ where $\chi _{\left( 0,\vartheta (E)\right) }\left( t\right) =\chi _{\left( 0,\vartheta _{1}(E)\right) }\left( t\right) e_{1}+\chi _{\left( 0,\vartheta _{2}(E)\right) }\left( t\right) e_{2}$.

(k) Let $u\in {\mathfrak {F}}(\Omega ,{\mathfrak {M}})$, $E_{u}(\lambda )=\left\{ x\in \Omega :\left| u\left( x\right) \right| _{k}\succ \lambda \right\}$ and $U=\chi _{E_{u}(\lambda )}$. Then $U_{\mathbb{B}\mathbb{C} }^{*}\left( t\right) =\chi _{\left( 0,\vartheta \left( E_{u}(\lambda )\right) \right) }\left( t\right)$ by item (i) and

$$\begin{aligned} E_{u_{\mathbb{B}\mathbb{C}}^{*}}(\lambda )= & {} \left\{ t\ge 0:\left| u_{ \mathbb{B}\mathbb{C}}^{*}\left( t\right) \right| _{k}\succ \lambda \right\} \\= & {} \left\{ t\ge 0:D_{u}^{\mathbb{B}\mathbb{C}}(\lambda )\succ t\right\} \end{aligned}$$

by (b). Therefore we get $U_{\mathbb{B}\mathbb{C}}^{*}(t)=\chi _{E_{u_{\mathbb{B}\mathbb{C} }^{*}}(\lambda )}(t)$. $\square$

5 Conclusion

By using the properties of bicomplex numbers, bicomplex measure, $\mathbb{B}\mathbb{C}$- distribution and $\mathbb{B}\mathbb{C}$-rearrangement functions, one can easily define $\mathbb{B}\mathbb{C}$-integrable functions spaces. At first, the generalization of Lebesgue spaces into $\mathbb{B}\mathbb{C}$ will involve considering function spaces that extend beyond the classical Lebesgue spaces. These generalizations will provide a broader framework for studying functions with more intricate properties or behaviors. For example Orlicz Spaces, Morrey Spaces, Sobolev Spaces and Besov Spaces for $\mathbb{B}\mathbb{C}$ -valued functions can be built. These generalizations of $\mathbb{B}\mathbb{C}$ -Lebesgue spaces provide more nuanced and refined function spaces, allowing for a deeper understanding of various mathematical phenomena and facilitating the analysis of functions with specific growth, smoothness, or localization properties.

Since the Lorentz spaces are a class of function spaces that generalize the classical Lebesgue spaces, $\mathbb{B}\mathbb{C}$-Lorentz spaces will be the generalization of $\mathbb{B}\mathbb{C}$-Lebesgue spaces.If we use the notation $L_{ \mathbb{B}\mathbb{C}}\left( p,q,\left\| \cdot \right\| _{\mathbb{B}\mathbb{C} }^{p,q}\right) \left( X,\Sigma ,\vartheta \right)$ for $\mathbb{B}\mathbb{C}$ -Lorentz spaces, then the Lorentz norm $\left\| \cdot \right\| _{ \mathbb{B}\mathbb{C}}^{p,q}$ will measure the “size” or “spread” of a function f in terms of both its magnitude and distribution. The parameter p controls the decay of the function at infinity, while the parameter q governs the rate of decrease of the function’s tail. Variable Exponent Lorentz Spaces which is also known as Musielak–Orlicz spaces or Nakano spaces, Marcinkiewicz Spaces, Lorentz–Sobolev Spaces and Morrey–Lorentz Spaces for $\mathbb{B}\mathbb{C}$ -valued functions can be studied. A new generalization of Lebesgue, Lorentz, Zygmund, Lorentz–Zygmund and generalized Lorentz–Zygmund spaces is defined and called as Lorentz–Karamata spaces. It will be interesting to find properties of $\mathbb{B}\mathbb{C}$-Lorentz–Karamata spaces. These generalizations of Lorentz spaces for $\mathbb{B}\mathbb{C}$-valued functions will provide more nuanced and refined function spaces, allowing for a deeper understanding of various mathematical phenomena and facilitating the analysis of functions with specific growth, smoothness, or localization properties. They are essential tools in harmonic analysis, function theory, and the theory of partial differential equations.