Inner and Outer Choquet integral representation (Advanced Study of Applied Functional Analysis and Information Sciences)

184

Inner

and

Outer

Choquet

integral

representation

桐朋学園 成川 康男 (YasuoNarukawa)

TohoGakuen

1

Introduction

The Choquetintegral is commonly usedasthe integralforanon-additive setfunction,because the Choquetintegral coincides with expectation inprobability theory whenthe setfunctionis probability

measure.

The mostimportantproperties of the Choquet integral are

comonotonic

additivityand monotonieity(forshortwecall$\mathrm{c}.\mathrm{m}.$).

Considering the topology, variousregularities

are

proposedand studied [4, 5, 12, 17, 16,8, 11, 14], especially, [16] propose their regularity in relationto Choquet integral, thatis,

a

$\mathrm{c}\mathrm{m}$.

functional

on

theclass ofcontinuousfunctionswithcompact support is represented byChoquet integralwithrespecttoan$0$-regular non-additive

measure.

In this paper,

we

assume

the universal set $X$ tobe

a

locally compact Hausdorff space.We

saythat therepresentation using $0$-regular non-additive

measure

defined by the

supremum

of

the functional is

an

outer representation. On the otherhand, therepresentationthat

uses

the

infinimum of thefunctional, that is called an inner representation,

can

be considered. Inthis

paper

we

study $\mathrm{i}$-regular non-additive

measure

andinner representation. We make clarify the

difference between$\mathrm{o}$regularity and

$\mathrm{i}$ regularity

or

betw

een an

outerrepresentationand

an

inner

representation. Thestructure of this

paper

isasfolows: Insection2wepresentbasicdefinition$\mathrm{s}$

and properties without topological assumption of non-additive

measure

and Choquet integral

withrespect to

a

non-additive

measure as

a preliminaries. In Section 3 we

assume

that the

universal

space

$X$ is locally compact

space.

We define $0$-regular and $\mathrm{i}$-regular non-additive

measure

and show

some

properties.

In section 4, we study the representation of a $\mathrm{c}\mathrm{m}$. functional. We show that the inner

representationis possible, although the method is somewhat differentfrom the outer

one

andthe

proofis rather complex. The induced$\mathrm{i}$-regular non-additive

measure

coincides with o-regular

one

on

theclassofcompact sets.

Insection5,

we

finish with

some

concludingremark.

2

Non-additive

measure

and

Choquet

integral

Inthissection,

we

present

some

basic definition andproperties of non-additive

measure

theory.

$X$denotes theuniversalsetand $B$denotes its 0- algebra. Notopologicalassumption is needed

in thissection.

In this

paper, we

distinguishtheterm“fuzzy

measure

from“non-additivemeasur\"e. Sugeno’$\mathrm{s}$

original axioms [15]for

a

fuzzy

measure

has

some

continuity. Onthe otherhand,

some

authors

define

a

fuzzy measure, thatis monotone setfunction vanishing at 0, and is not assumed any continuity. Inorderto avoidconfusion,in thispaPer,according toDenneberg’s monograph [2],

we say it a non-additive

measure.

Definition2,1. _{A non-additive}

measure

$\mu$is

an

extendedreal valuedsetfunction,

$\mu$:$tB$

$arrow\overline{R^{+}}$

with the followingproperties;

$\mu(\emptyset)=0$,and$\mathrm{j}\mathrm{j}(\mathrm{A})\leq\mu(B)$ whenever$A\subset B$,$A,B\in B$where

$\overline{R^{+}}=[0,\infty]$is thesetof extended

nonnegativereal numbers. In this

paper

we

assume

that$\mu$isfinite,thatis,$\mu(X)<\infty$.

186

Definition2,2. [1, 2] Let$\mu$beanon-additive

measure

on

(X, B).

1. The Choquet integral of

_f

$\in M^{+}$ with respectto$\mu$is definedby

(C)$ffd \mu=\int_{0}^{\infty}\mu_{f}(r)dr$,

when$\mu f(r)=\mu(\{x|f(x) \geq r\})$.

Definition2.3. [3] Let $f$,$g\in M^{+}$. We say that $f$and$g$ are comonotonic if$f(x)$ $<f(x’)\Rightarrow$

$g(x)$ $\leq g(x’)$ for$x$,$x’\in X$

.

We denote$f\sim g$,when$f$and$g$

are

comonotonic.

TheChoquet integral of$f\in M$ with respectto anon-additive

measure

has the next basic properties.

Theorem2.3. [2] Letf, g$\in M^{+}$.

1. (Monotonicity)

If

f

$\leq g$, then

(C)$\int fd\mu\leq(C)\int gd\mu$

2. (Comonotonic additivity)

Iff

$\sim g$, then

(C)$) \int(f+g)d\mu=(C)$$I^{fd\mu+}(\backslash c)\prime gd\mu$.

3

Properties of

$\mathrm{i}$

-regular

non-additive

measure

In thissection weshow

some

propertiesof0-regularnon-additive

measure

and$\mathrm{i}$-regular

non-additive

measure.

The properties of$0$-regular non-additive

measure

are

shownin [6,7, 9, 10, 8,

11].

Inthe followingwe

assume

that$X$is

a

locally compact Hausdorff

space,

$lB$theclass of Borel

subsets of$X$, $O$theclass ofopensubsetsof$X$and$C$the classofcompact subsets of$X$

.

$C^{+}(X)$denotesthe classof non-negativecontinuousfunctionswithcompact support$\mathrm{a}\mathrm{n}\mathrm{d}C_{1}^{+}(X)$

Definition3.1. Let$\mu$be

a

non-additive

measure

onthe measurablespace $(X, B)$

.

1. $\mu$issaidtobe $0$-continuous frombelowif

$O_{n}\uparrow O\Rightarrow\mu(O_{n})\uparrow\mu(O)$ where$n=1,2,3$,$\cdots$

and both$O_{n}$ and$O$

are

opensets

.

2. $\mu$is saidtobe$c$-continuousfromabove if

$C_{n}\downarrow C\Rightarrow\mu(C_{n})\downarrow\mu(C)$ wheren$=1,$2,\cdots and

both$C_{n}$and Carecompactsets.

First,

we

define theregular non-additive

measures.

Definition3.2. Let$\mu$bea non-additive

measure

on measurable space (X,B). $\mu$ issaidto be

inner regular

_if

$\mu(B)=\sup\{\mu(C)|C\in C,C\subset B\}$

for

allB $\in B$

.

Innerregular non-additive

measure

is called$\mathrm{i}-regular$

if

$\mu(C)=\inf\{\mu(O)|O\in O, C\subset O\}$

for

all$C\in C$

.

$\mu$is saidtobeouterregular

if

$\mu(B)=\inf\{\mu(O)|O\in O, O\supset B\}$

for

all $B\in B$

.

Outer regular non-additive

measure

iscalled$\mathit{0}$-regular

if

$\mu(O)=\sup\{\mu(C)|C\in O,C\subset O\}$

for

all$O\in O$.

The next lemmasfollow fromthedefinition immediately.

Lemma3.3. Let$\mu i$bean

$\mathrm{i}$-regular non-additive

measure

and$\mu_{\mathit{0}}$ beano-regularnon-additive

measure.

$\mu_{i}(O)=\mu_{\mathit{0}}(O)$

for

$0\in O$

if

andonly

if

$\mu_{i}(C)=\mu_{\mathit{0}}(C)$

for

C$\in C$

.

Lemma3.1. Let$\mu_{i}$be an

$\mathrm{i}$-regular non-additive

measure

and$\mu_{\mathit{0}}$bean$0$-regularnon-additive

measure

such that$\mu_{i}(C)=\mu_{\mathit{0}}(C)$

for

C$\in C$. Then

we

have$\mu_{i}(A)\leq\mu_{\mathit{0}}(A)$

for

all A $\in B$

.

The next tworesults follow fromthedefinition.

Proposition3.5, [6]Let$\mu$bea o- regularnon-additive

measure.

188

2. ptis$c$-continuous

from

above.

RemarkProposition 3.5(2)istrue whennon-additive

measure

are

outer regularnon-additive

measures.

Wehave thenextlemmafrom the definitionandUlysohn’slemma, although it isstill

open

whetherthe similarcontinuity to the lemma above holds in thecase of$\mathrm{i}$-regular non-additive

measure.

Proposition3.6. Let$\mu$bean

$\mathrm{i}$-regularnon-additivemeasure. For everycompactsetC$\in C$there

existsa sequence $\{O_{n}\}$

of

opensetssuch that$\mu(O_{n})\downarrow\mu(C)$.

Inrelationto the Choquetintegral,

we

have the nextpropositions.

Proposition3.7. [6]Let$\mu_{1}$ and$\mu_{2}$be$c$-continuous

from

above.

Iffor

all$f \in K(C)\int fd\mu_{1}=(C)\int fd\mu_{2}$,then$\mu 1(C)=\mu 2(C)$

for

all$C\in C$.

Inthe

case

of$\mathrm{i}$-regular,wehave the nextsimilar result.

Proposition3.8. Let$\mu$bean

$\mathrm{i}$-regularnon-additive

measure.

Iffor

all$f \in K(C)\int fd\mu_{1}=(C)\int fd\mu_{2}$, then$\mu_{1}(C)=\mu 2(C)$

for

all$C\in C$

.

ItfollowsfromProposition 3.6that

we

have

$\mu_{i}(\{x|f^{n}(x)\geq\alpha\})\downarrow\mu_{i}(\{x|1_{C}(x)\geq\alpha\})$

as

$narrow\infty$for$0<\alpha\leq 1$ and$\mathrm{i}=1,2$

.

The nexttheorem isthemain theorem of this chapter. Theorem3,9. _Let$\mu_{i}$be

$\mathrm{i}$-regular non-additive measureand

$\mu_{\mathit{0}}$ be$0$-regularnon-additive

mea-sures.

_Iffor

all$f \in C^{+}(C)\int fd\mu_{i}=(C)\int fd\mu_{\mathit{0}}$, then

2. $\mathrm{t}\mathrm{n}(\mathrm{O})=\mu_{\mathit{0}}(O)$

for

all$C\in O$,

3. $\mu_{i}(A)\leq\mu_{\mathit{0}}(A)$

for

all$A$$\in B$

.

4

Representation

theorems

The Choquet integral is

a

comonotonicallyadditive and monotonefunctional. Conversely, we considera$\mathrm{c}$

.

$\mathrm{m}$

.

functionalonthe class ofcontinuousfunctionswithcompact support.

Definition4.1. LetIbea real valued

_functional

on$C^{+}(X)$.

We say that I is comonotonicallyadditive

iff

$f\sim g\Rightarrow I(f+g)=I(f)+I(g)$

for

$f,g\in$ $C^{+}(X)$, andthatIismonotone

iff

$f\leq g\Rightarrow I(f)\leq I(g)$

for

$f,g\in C^{+}(X)$.

$Ifa$

functional

Iis comonotonicallyadditive andmonotone,wesaythat I isa$\mathrm{c}\mathrm{m}$

.

functional.

First, wereviewtheouterrepresentation.

Lemma4.2. [

16J

Let Ibe ac.m.

_functional

on $C^{+}$

.

We put$f_{I}(O)= \sup$

{

$I(f)|f\in C_{1}^{+}$,supp(f)\subset O}

for

$O\in O$

and$\mu_{I}^{o}(B)=\inf\{f_{I}(O)|O\in O, O\supset B\}$

for

$B\in B$. Then$\mu_{I}^{o}$ isanouterregular non-additive

measure.

Weshallsaythatthis outerregularnon-additive

measure

$\mu_{I}^{o}$is

an

outernon-additive

measure

inducedby

a cm.

functional I.

Proposition4.3. [16]Let$\mu_{I}^{o}$beanouternon-additive

measure

inducedbya

cm.

functional

I.

1.

_If

$f\in C_{7}^{+}A\subset\{x|f(x)\geq 1\}$,$andA\in B$, then $\mathrm{I}(\mathrm{g})$ $\leq I(f)$

2.

_If

C isacompactset, then$\mu l(C)<\infty$.

1ao

Therefore

the non-additive

measure

inducedbya c.m-

_functional

is o-regular.

Fromtheabovelemma,

we

obtainthe following theorem [16].

Theorem4.4. (Outerrepresentationtheorem) [16]

Let Ibea$\mathrm{c}\mathrm{m}$.

functional

on

$C^{+}$

.

If

$lP_{I}$ isanouterregular non-additivemeasure inducedby

$I$, thenwehave$I(f)=(C) \int fd\mu_{I}^{O}$

for

all$f\in C^{+}(X)$.

Inthe following,

we

present

some

preliminary resultsforinnerrepresentation. Theproofs

are

in $[16, 6]$

.

Let$\mathit{0}_{n,k}=\{x|f(x)>\frac{k-1}{n}\}$

:$CU|k= \{x|f(x)\geq\frac{k}{n}\}$ where$f\in C^{+}$,$k$and$n$is

a

positive integer

and $1\leq k\leq n$. Then for all$n$ and $k$, $C_{n,k}$ is acompact set, $O_{n,k}$ is an open set, and $O_{n_{7}k+1}\subset$

$C_{n,k}\subset O_{n,k}\subset supp(f)$. Since$X$isalocallycompact Hausdorffspace,forall$n,k$thereexists

$f_{n,k}\in C_{1}^{+}$ suchthat$f_{n,k}(x)=1$ when$x$ $\in C_{n,k}$and$supp(f_{n,k})\subset \mathit{0}_{n,k}$.

Thefunctions$f_{n,k}$havethe followingproperties.

Lemma4.5.

fl6f

1. Forall positive integer n,k and j such that $1\leq k\leq n$and $1\leq j\leq n$, $f_{n,k}$ and fnj are

comonotonic.

2. Forallpositive integern andk suchthat$1\leq k\leq n$_,

$f_{n,1}+f_{n,2}+\cdots+f_{n,k}$and$f_{n,k+1}+f_{n,k+2}+\cdots+f_{n,n}$ arecomonotonic.

Lemma4.6. [16]

_Define

$f_{n}\in C_{1}^{+}for$n$=1,2$,\cdots by_{$f_{n}= \sum_{k=1}^{n}\frac{1}{n}f_{n,k}$}.

1. thereexists$F\in c_{1}^{+}$which

satisfies

the followingconditions.

(a) $0 \leq|f-f_{n}|\leq\frac{1}{n}F$ (b) $x\in$ supp(f) $\Rightarrow F(x)=1$

(c) $f_{n}\sim_{c}F$

for

all$n$

2. We have $||f-f_{n}|| \leq\frac{1}{n}$,where

||.||

is the $\sup$norm.

3. Suppose that$\frac{k}{n}\leq f(x)<\frac{k+1}{n}$, thenwehave $\frac{k}{n}\leq f_{n}(x)$ $\leq\frac{k+1}{n}$

4. $\lim_{narrow\infty}I(f_{n})=I(f)$

Nowwe

can

proceedtheinnerrepresentation. Thedefinition ofanon-additive

measure

is

a

littlebit different fromouterrepresentation. The proof is obviousfromthe definition.

Lemma4,7. _LetIbea$\mathrm{c}\mathrm{m}$.

functional

on

$C^{+}$.

Weput$\mu_{I}^{i}(C)=\inf\{I(f)|f\in C_{1}^{+},1_{C}\leq f\}$

for

$C\in C$and$\mu_{I}^{i}(B)=\sup\{\mu_{I}^{i}(C)|C\in C,C\subset B\}$

for

$B\in B$. Then$\mu_{I}^{i}$ isaninner regularnon-additive

measure.

We shall say that the non-additive$\mu_{I}^{i}$ isan innerregular non-additive

measure

inducedby a

$\mathrm{c}\mathrm{m}$

.

functional$I$

.

Proposition 4.8.

Let $\mu_{I}^{i}$ be an innernon-additive

measure

induced by a

$\mathrm{c}\mathrm{m}$.

functional

I. Suppose that

$f\in C_{1}^{+},A\supset sup(f)$ , and$A\in I\mathit{3}$

.

Thenwe have$\mu_{I}^{i}(A)\geq I(f)$.

Lemma4.9. Let$\mu_{I}^{i}$ beaninnernon-additive

measure

inducedbyac.m.

functional

I. Suppose

that

_f

$\in C_{1)}^{+}O\subset supp(f)$ , andO$\in O$. Then

we

have$\mu_{I}^{i}(O)\leq I(f)$

.

Proposition 4.8. Let$\mu_{I}^{i}$beaninnernon-additive

measure

inducedbya$c.m$

.

functional

I. Then

$\mu_{J}^{i}$ is$\mathrm{i}$-regulan thatis,$\mu_{I}^{i}(C)=\inf\{\mu_{I}^{i}(O)|O\in O,O\supset C\}$.

Applyingthe abovementioned lemmas

we can prove

theinnerrepresentationtheorem.

182

LetIbea $\mathrm{c}\mathrm{m}$

.

functional

on

$C^{+}$.

If

$\mu_{I}^{i}$ isan $\mathrm{i}$-regular non-additive

measure

inducedby$I$,

thenwehave

$I(f)=(C) \oint fd\mu_{I}^{i}$

for

all$f\in C_{1}^{+}$.

Thenexttheorem followsfromTheorem 3.9, Theorem4,4_{and Theorem}4.11.

Corollary 4.12. Let I bea cm.

_functional

on $C^{+}and$$f_{l}$ (resp. $\mu_{I}^{i}$) isan o- regular(resp. an

$\mathrm{i}$-regularnon-additivemeasureinducedbyI,

1. $\mu_{I}^{i}(C)=\mu_{I}^{o}(C)$

for

all$C\in C$,

2. $\mu_{I}^{i}(O)=\mu_{I}^{o}(O)$

for

all$C\in O$,

3. $\mu_{i}(A)\leq\mu_{\mathit{0}}(A)$

for

all$A\in B$

.

5

Conclusions

We have studied the properties ofan $i$-regular non-additive

measure

and shown that

an

inner

representationis possible. We showedthat an$0$-regularnon-additive

measure

andani-regular

non-additive

measure

induced by

a

$\mathrm{c}\mathrm{m}$

.

functional takes

same

value

on

the class ofcompact

sets andtheclass ofopen sets. It willbe future works whetherthey coincides

on

the class of Borel sets andunder what conditionthey coincides ifthey

are

not

same on

theclass ofBorel

sets.

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