# Discrete Integrals and Axiomatically Defined Functionals

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem 1.1**Let ϕ be continuous on an interval $\left[a,b\right]$, and let ${I}_{u}^{v}$ be defined for $a\le u\le v\le b$. Suppose that

- (A)
- $${I}_{a}^{x+\Delta x}={I}_{a}^{x}+{I}_{x}^{x+\Delta x}$$

- (B)
- $$\left(\underset{\left[x,x+\Delta x\right]}{min}\phi \right)\Delta x\le {I}_{x}^{x+\Delta x}\le \left(\underset{\left[x,x+\Delta x\right]}{max}\phi \right)\Delta x$$

“That is to say, the hypotheses from which one starts become ever more abstract and more remote from experience. But in return one comes closer to the preeminent goal of science, that of encompassing a maximum of empirical contents through logical deduction with a minimum of hypotheses or axioms.”

## 2. Capacities and $[0,1]$-Valued Discrete Universal Integrals

**Definition 2.1**A capacity (or fuzzy measure) on X is a non-decreasing set function $m:{2}^{X}\to [0,1]$ which satisfies the boundary conditions $m(\varnothing )=0$ and $m\left(X\right)=1$.

**Definition 2.2**A function $\mathbf{I}:{\bigcup}_{n\in \mathbb{N}}\left({\mathcal{M}}_{n}\times {\mathcal{F}}_{n}\right)\to [0,1]$ is called a $[0,1]$-valued discrete universal integral if it satisfies the following axioms:

- (A1)
**I**is non-decreasing in each component;- (A2)
- $\mathbf{I}(m,{\mathbf{1}}_{E})=m\left(E\right)$ for all $X=\{1,2,\cdots ,n\}$, $m\in {\mathcal{M}}_{n}$, and $E\subseteq X$;
- (A3)
- $\mathbf{I}(m,c\xb7{\mathbf{1}}_{X})=c$ for all $X=\{1,2,\cdots ,n\}$, $m\in {\mathcal{M}}_{n}$, and $c\in [0,1]$;
- (A4)
- $\mathbf{I}({m}_{1},{f}_{1})=\mathbf{I}({m}_{2},{f}_{2})$ for all pairs $({m}_{1},{f}_{1})\in {\mathcal{M}}_{{n}_{1}}\times {\mathcal{F}}_{{n}_{1}}$ and $({m}_{2},{f}_{2})\in {\mathcal{M}}_{{n}_{2}}\times {\mathcal{F}}_{{n}_{2}}$ satisfying ${m}_{1}\left(\{{f}_{1}\ge t\}\right)={m}_{2}\left(\{{f}_{2}\ge t\}\right)$ for each $t\in [0,1]$.

**Proposition 2.3.**Let

**I**be a $[0,1]$-valued discrete universal integral. Then there exists a semicopula ⊗ such that we have $\mathbf{I}(m,c\xb7{\mathbf{1}}_{E})=c\otimes m\left(E\right)$ for all $n\in \mathbb{N}$, $m\in {\mathcal{M}}_{n}$, $c\in [0,1]$, and $E\subseteq \{1,2,\cdots ,n\}$.

## 3. Special $[0,1]$-Valued Discrete Universal Integrals

**Theorem 3.1**Consider a function $\mathbf{J}:{\bigcup}_{n\in \mathbb{N}}{\mathcal{F}}_{n}\to [0,1]$. Then, for each $n\in \mathbb{N}$ there is a capacity $m\in {\mathcal{M}}_{n}$ such that for all $f\in {\mathcal{F}}_{n}$ we have $\mathbf{J}\left(f\right)=\mathbf{Ch}(m,f)$ if and only if

- (M)
- $\mathbf{J}$ is non-decreasing;
- (C1)
- $\mathbf{J}\left({\mathbf{1}}_{X}\right)=1$ for all $n\in \mathbb{N}$ and $X=\{1,2,\cdots ,n\}$;
- (C2)
- $\mathbf{J}$ is comonotone additive, i.e., for each $n\in \mathbb{N}$ and all $f,g\in {\mathcal{F}}_{n}$ satisfying $f+g\in {\mathcal{F}}_{n}$ and $f\sim g$ we have $\mathbf{J}(f+g)=\mathbf{J}\left(f\right)+\mathbf{J}\left(g\right)$.

**Example 3.2**The requirement (M) in Theorem 3.1 cannot be omitted, as this counterexample shows: put $n=3$ and identify each function $f\in {\mathcal{F}}_{3}$ with the triplet $({x}_{1},{x}_{2},{x}_{3})\in {[0,1]}^{3}$, i.e., $f\left(i\right)={x}_{i}$. Then the function $\mathbf{J}:{\mathcal{F}}_{3}\to [0,1]$ defined by

**Theorem 3.3**Consider a function $\mathbf{J}:{\bigcup}_{n\in \mathbb{N}}{\mathcal{F}}_{n}\to [0,1]$. Then, for each $n\in \mathbb{N}$ there is a capacity $m\in {\mathcal{M}}_{n}$ such that for all $f\in {\mathcal{F}}_{n}$ we have $\mathbf{J}\left(f\right)=\mathbf{Su}(m,f)$ if and only if

- (S1)
- $\mathbf{J}\left({\mathbf{1}}_{X}\right)=1$ for all $n\in \mathbb{N}$ and $X=\{1,2,\cdots ,n\}$;
- (S2)
**J**is comonotone maxitive, i.e., for each $n\in \mathbb{N}$ and all $f,g\in {\mathcal{F}}_{n}$ with $f\u223eg$ we have $\mathbf{J}(max(f,g\xb7))=max\left(\mathbf{J}\right(f),\mathbf{J}(g\left)\right)$;- (S3)
**J**is min-homogeneous, i.e., for each $n\in \mathbb{N}$ and $X=\{1,2,\cdots ,n\}$, for all $f\in {\mathcal{F}}_{n}$ and for all $c\in [0,1]$ we have $\mathbf{J}(min(f,c\xb7{\mathbf{1}}_{X}))=min\left(\mathbf{J}\right(f),c)$.

- (SY)
- $\mathbf{J}\left(f\right)=\mathbf{J}(f\circ \sigma )$ for all $n\in \mathbb{N}$, $f\in {\mathcal{F}}_{n}$ and $\sigma \in {S}_{n}$,

**Corollary 3.4**A function $\mathbf{J}:{\bigcup}_{n\in \mathbb{N}}{\mathcal{F}}_{n}\to [0,1]$ is

- (i)
- an OWA operator if and only if it satisfies the axioms (C1), (C2), (M) and (SY);
- (ii)
- an OWMax operator if and only if it satisfies the axioms (S1), (S2), (S3) and (SY).

**Corollary 3.5**Consider a function $\mathbf{J}:{\bigcup}_{n\in \mathbb{N}}{\mathcal{F}}_{n}\to [0,1]$. Then, for each $n\in \mathbb{N}$ there is a probability measure $m\in {\mathcal{M}}_{n}$ such that for all $f\in {\mathcal{F}}_{n}$ we have

**J**satisfies the axioms (C1) and (C2

^{*})

**J**is additive.

**J**follows from the additivity (C2

^{*}) and the fact that $\text{Ran}\left(\mathbf{J}\right)\subseteq [0,1]$.

**Corollary 3.6**Suppose that a function $\mathbf{J}:{\bigcup}_{n\in \mathbb{N}}{\mathcal{F}}_{n}\to [0,1]$ satisfies the axioms (S1), (S3) and (S2

^{*})

**J**is maxitive.

**I**satisfying, for each $n\in \mathbb{N}$, each $m\in {\mathcal{M}}_{n}$, each $E\subseteq \{1,\cdots ,n\}$ and each $c\in [0,1]$, the equality $\mathbf{I}(m,c\xb7{\mathbf{1}}_{E})=c\otimes m\left(E\right)$ is denoted by ${\mathbf{I}}_{\otimes}$ (see [8]). The explicit formula of ${\mathbf{I}}_{\otimes}$ is

**Theorem 3.7**Let ⊗ be a semicopula and consider a function $\mathbf{J}:\bigcup _{n\in \mathbb{N}}{\mathcal{F}}_{n}\to [0,1]$. Then, for each $n\in \mathbb{N}$ there is a capacity $m\in {\mathcal{M}}_{n}$ such that for each $f\in {\mathcal{F}}_{n}$ we have $\mathbf{J}\left(f\right)={\mathbf{I}}_{\otimes}(m,f)$ if and only if $\mathbf{J}$ satisfies the axioms (S1), (S2) and

- (S3
_{⊗}) **J**is ⊗-homogeneous on characteristic functions, i.e., for each $n\in \mathbb{N}$, for all $E\subseteq \{1,\cdots ,n\}$ and for all $c\in [0,1]$ we have $\mathbf{J}(c\otimes {\mathbf{1}}_{E})=c\otimes \mathbf{J}\left({\mathbf{1}}_{E}\right)$.

**Remark 3.8**As a consequence of Theorem 3.7, the axiom (S3) of min-homegeneity in the axiomatic characterization of the Sugeno integral in Theorem 3.3 can be replaced by the weaker axiom (S${3}_{\wedge}$), requiring the min-homogeneity for characteristic functions only.

## 4. Copula-Based $[0,1]$-Valued Discrete Universal Integrals

**Definition 4.1**A function $C:{[0,1]}^{2}\to [0,1]$ is called a (2-dimensional) copula if it is a 2-increasing semicopula, i.e., if for all $x,y,{x}^{*},{y}^{*}\in [0,1]$ with $x\le {x}^{*}$ and $y\le {y}^{*}$ we have

**Theorem 4.2**Consider a function $\mathbf{J}:{\bigcup}_{n\in \mathbb{N}}{\mathcal{F}}_{n}\to [0,1]$. Then, for each $n\in \mathbb{N}$ there is a capacity $m\in {\mathcal{M}}_{n}$ and a copula C such that for all $f\in {\mathcal{F}}_{n}$ we have $\mathbf{J}\left(f\right)={\mathbf{I}}_{C}(m,f)$ if and only if $\mathbf{J}$ satisfies the axioms (M) and

- (K1)
- $\mathbf{J}(c\xb7{\mathbf{1}}_{X})=c$ for all $n\in \mathbb{N},X=\{1,2,\cdots ,n\}\text{}and\text{}c\in [0,1];$
- (K2)
**J**is comonotone modular, i.e., for each $n\in \mathbb{N}$ and for all $f,g\in {\mathcal{F}}_{n}$ with f ∼ g we have$$\mathbf{J}\left(\text{max}\right(f,g\left)\right)+\mathbf{J}\left(\text{min}\right(f,g\left)\right)=\mathbf{J}\left(f\right)+\mathbf{J}\left(g\right)$$- (K3)
- for all $n\in \mathbb{N},E,F\subseteq \{1,2,\cdots ,n\}\text{}and\text{}(u,v)\in {[0,1]}^{2}\text{}we\text{}have$
- (i)
- if $\mathbf{J}\left({\mathbf{1}}_{E}\right)=\mathbf{J}\left({\mathbf{1}}_{F}\right)$ then $\mathbf{J}(u\xb7{\mathbf{1}}_{E})=\mathbf{J}(u\xb7{\mathbf{1}}_{F})$;
- (ii)
- if $u\le v$ and $\mathbf{J}\left({\mathbf{1}}_{E}\right)\le \mathbf{J}\left({\mathbf{1}}_{F}\right)$ then $\mathbf{J}(u\xb7{\mathbf{1}}_{F})-\mathbf{J}(u\xb7{\mathbf{1}}_{E})\le \mathbf{J}(v\xb7{\mathbf{1}}_{F})-\mathbf{J}(v\xb7{\mathbf{1}}_{E})$.

**Corollary 4.3**A function $\mathbf{J}:{\bigcup}_{n\in \mathbb{N}}{\mathcal{F}}_{n}\to [0,1]$ is an OMA operator if and only if it satisfies the axioms (K1), (K2) and (SY).

## 5. Generalizations of Choquet and Sugeno Integrals

**Theorem 5.1**Consider a function $\mathbf{J}:{\bigcup}_{n\in \mathbb{N}}{\mathcal{F}}_{n}\to [0,1]$. Then, for each $n\in \mathbb{N}$ there is a level dependent capacity $M={\left({m}_{t}\right)}_{t\in [0,1]}$ with ${m}_{t}\in {\mathcal{M}}_{n}$ for all $t\in [0,1]$, such that for all $f\in {\mathcal{F}}_{n}$ we have $\mathbf{J}\left(f\right)=\mathbf{Su}(M,f)$ if and only if $\mathbf{J}$ satisfies the axioms (K1) and (S2).

**Theorem 5.2**Fix $n\in \mathbb{N}$ and consider a function $\mathbf{J}:{\bigcup}_{n\in \mathbb{N}}{\mathcal{F}}_{n}\to [0,1]$. Then there is a capacity $m\in {\mathcal{M}}_{n}$ such that for all $f\in {\mathcal{F}}_{n}$ and for all $\mathbf{u}=({u}_{1},{u}_{2},\cdots ,{u}_{n})\in {[0,1]}^{n}$ we have $\mathbf{J}(\mathbf{u},f)=\mathbf{ICh}(m,\mathbf{u},f)$ if and only if $\mathbf{J}$ satisfies the axioms

- (IC1)
- $\mathbf{J}(\mathbf{u},{\mathbf{1}}_{\{1,2,\cdots ,n\}})=1$ for each $\mathbf{u}\in {[0,1]}^{n}$;
- (IC2)
- $\mathbf{J}(\mathbf{u},\xb7)$ is additive for each $\mathbf{u}\in {[0,1]}^{n}$;
- (IC3)
- $\mathbf{J}(\mathbf{u},{\mathbf{1}}_{E})=\mathbf{J}(\mathbf{v},{\mathbf{1}}_{E})\phantom{\rule{1.em}{0ex}}for\text{}all\text{}E\subseteq \{1,2,\cdots ,n\}\text{}and\text{}for\text{}all\text{}\mathbf{u},\mathbf{v}\in {[0,1]}^{n}\text{}satisfying\text{}{u}_{i}{u}_{j}\text{}and\text{}{v}_{i}{v}_{j}\text{}whenever\text{}i\in E\text{}and\text{}j\notin E;$
- (IC4)
- $\mathbf{J}(\mathbf{u},\xb7)=\mathbf{AM}\left\{\mathbf{J}\left(\right({\textstyle \frac{\sigma \left(1\right)}{n}},{\textstyle \frac{\sigma \left(2\right)}{n}},\cdots ,{\textstyle \frac{\sigma \left(n\right)}{n}}),\xb7\left)\right|\sigma \in {S}_{n},(\sigma \left(1\right),\sigma \left(2\right),\cdots ,\sigma \left(n\right))\sim \mathbf{u}\right\}.$

## 6. Concluding Remarks

## Acknowledgment

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Klement, E.P.; Mesiar, R.
Discrete Integrals and Axiomatically Defined Functionals. *Axioms* **2012**, *1*, 9-20.
https://doi.org/10.3390/axioms1010009

**AMA Style**

Klement EP, Mesiar R.
Discrete Integrals and Axiomatically Defined Functionals. *Axioms*. 2012; 1(1):9-20.
https://doi.org/10.3390/axioms1010009

**Chicago/Turabian Style**

Klement, Erich Peter, and Radko Mesiar.
2012. "Discrete Integrals and Axiomatically Defined Functionals" *Axioms* 1, no. 1: 9-20.
https://doi.org/10.3390/axioms1010009