A new necessary and sufficient condition for the Egoroff theorem in non-additive measure theory (Mathematics for Uncertainty and Fuzziness)

A

new necessary

and sufficient condition for

the

Egoroff theorem

in

non-additive

measure

theory

MasayukiTakahashi,ToshiakiMurofushi, ShnAsahina

Departmentof Computational Intelligence and SystemsSciences,

TokyoInstituteofTechnology

E-mail:masayuh@&.dis.titech.ac.jp

Abstract: This paper is a briefsummary of“M.Takahashi, T.MuroMhi, S.Asahina, $A$

new

necessary and sufficient condition for the Egorofftheorem in non-additive

measure

theory,

Fuzzy Sets and Systems, to appear.” This

paper

statesthat

a

newlydefinedcondition, called

con-dition(M),is

a

necessalyand sufficient condition for the Egorofftheoreminnon-additive

measure

theory. The existingnecessaryandsufficient conditions for the Egorofftheorem

are

described by

a

doubly-indexedsequenceofmeasurablesets,whilecondition(M)is described by

a

singly-indexed

sequenceofmeasurablesets.

1 INTRODUCTION

SinceSugeno[5]introduced the concept ofnon-additive measure, which he called

a

$b\eta$measure,

non-additive

measure

theoryhasbeen constructedalongthe lines oftheclassical

measure

theory[1,

4, 7]. Generally,theorems intheclassical

measure

theory

no

longer holdinnon-additive

measure

theory,

so

that to find necessary and/or _{sufficient conditions for such theorems to hold is} $ve\iota y$

importantfor theconstructionof non-additive

measure

theory.

TheEgorofftheorem, which assertsthat almost everywhereconvergence implies almost

uni-form convergence, is

one

ofthe most important convergence theorems in the classical

measure

theory. Innon-additive

measure

theory,thistheorem does not holdwithoutadditionalconditions.

Necessaryand sufficient conditions for the Egorofftheoremtoholdin non-additive

measure

theory

have beenproposed, forinstance,theEgoroffcondition [3]or

a

certaincondition calledcondition (E) in [2]. Howeverboth conditions

are

complex since they

are

described by a doubly-indexed

sequence ofmeasurable sets. This paper gives

a

new

simpler condition (M) which is described

bya singly-indexedsequenceofmeasurablesetsandshowsthatcondition(M)is equivalenttothe

Egoroffcondition.

Thispaperisabriefsummaryofaforthcomingpaper[6].

2 Preliminaries

Througout thepaper, $(X, \mathscr{S})$is assumedtobeameasurablespaceand$\mathbb{N}$denotesthesetofpositive

integers. Inaddition,every measurable function$f$is assumedto befinite realvalued, i.e., $-\infty<$

$f(x)<\infty$for all$x\in X.$

Definition 1 Anon-additivemeasure on$\mathscr{S}$isa

setfunction$\mu:\mathscr{S}arrow[0,\infty]$ satisfyingthe

follow-ingtwoconditions:

(i) $\mu(\emptyset)=0,$

(ii) $A,B\in \mathscr{S},$ $A\subset B\Rightarrow\mu(A)\leq\mu(B)$

.

数理解析研究所講究録

Unless stated otherwise, all subsets

are

supposed to belong to $\mathscr{S}$

and$\mu$ is assumedto be a

non-additive

measure

on$\mathscr{S}.$

Definition2 Let$\{f_{n}\}$be asequenceof measurablefunctions,and$f$beameasurable function.

(i) $\{f_{n}\}$issaid to convergeto_$f$almost everywhere, written$f_{n}arrow fae$, ifthereexists$N$suchthat

$\mu(N)=0$and$\{f_{n}(x)\}$ convergesto$f(x)$ forall$x\in X\backslash N.$

(ii) $\{f_{n}\}$issaid toconvergeto$f$almostuniformly,written$f_{n}arrow fau$,ifforevery$\epsilon>0$thereexists

$N_{\epsilon}$ such that$\mu(N_{\epsilon})<\epsilon$and$\{f_{n}\}$ convergesto$f$uniformlyon$X\backslash N_{\epsilon}.$

Definition3 $\mu$ is said to satisfy the Egoroffconditionif,foreverydoubly-indexedsequence$E_{m,n}$

such that$E_{m,n}\supset E_{m’,n’}$ for$m\geq m’$ and$n\leq n’$ and$\mu(\bigcup_{m=1}^{\infty}\bigcap_{n=1}^{\infty}E_{m,n})=0$, and foreverypositive

number$\epsilon$,thereexistsasequence_{$\{n_{m}\}$} ofpositiveintegerssuchthat_{$\mu(\bigcup_{m=1}^{\infty}E_{m,n_{m}})<\epsilon.$} $[3]$ Murofushi et al. [3] show that the Egoroffcondition is a necessary and sufficient condition for the Egorofftheorem, i.e.,the Egoroff conditionissatisfiediffalmosteverywhereconvergence

implies almost uniformconvergence.

3 TheEgorofftheorem

Inthis section, wedefine condition(M),whichisequivalent tothe Egoroff condition.

Definition4 $\mu$ issaidtosatisfy condition$(w$if$\mu(\bigcup_{n=1}^{\infty}\bigcap_{i=n}^{\infty}E_{i})=0$implies that forevery

posi-tivenumber$\epsilon$ thereexists

a

sequence _{$\{m_{n}\}$} ofpositiveintegerssuch that $\mu(\bigcup_{n=1}^{\infty}\bigcap_{i=n}^{m_{n}}E_{i})<\epsilon.$ The following lemma is necessary to show that condition (M) is equivalent to the Egoroff

condition.

Lemma 1 Let $\{E_{m,n}\}$ be a doubly-indexedsequence, $E_{m,n}\supset E_{m’,n’}$

for

$m\geq m’$ and $n\leq n’$, and

$\bigcup_{m=1}^{\infty}\bigcap_{n=1}^{\infty}E_{m,n}=\emptyset$

.

Thenthere exists a sequence $\{A_{n}\}$such that$\bigcup_{n=1}^{\infty}\bigcap_{i=n}^{\infty}A_{j}=\emptyset$,

andfor

every

strictlyincreasingsequence$\{k_{n}\}$ ofpositive integers, there existsanon-decreasing sequence$\{n_{i}\}$

ofpositive integerssuch that$\bigcup_{m=1}^{\infty}\bigcap_{i=m}^{\infty}E_{i,n_{i}}\subset\bigcup_{n=1}^{\infty}\bigcap_{i=n}^{k_{n}}A_{i}.$

We

can

show Lemma 2, whichisageneralization of Lemma 1.

Lemma2 Foreverydoubly-indexedsequence$E_{m,n}$suchthat$E_{m,n}\supset E_{m’,n’}$

for

$m\geq m’$and$n\leq n’,$

there existsasequence$\{A_{n}\}$

ofsets

such that$\bigcup_{n=1}^{\infty}\bigcap_{i=n}^{\infty}A_{j}=\bigcup_{m=1}^{\infty}\bigcap_{n=1}^{\infty}E_{m,n}$,

andfor

everystrictly

increasingsequence$\{k_{n}\}_{J}$ there existsanon-decreasingsequence$\{n_{i}\}$such that$\bigcup_{m=1}^{\infty}\bigcap_{\dot{|}=m}^{\infty}E_{i.n_{i}}\subset$ $\bigcup_{n=1}^{\infty}\bigcap_{i=n}^{k_{n}}A_{j}.$

Itisnotdifficulttoshow that the Egoroffcondition implies condition(M), while it is noteasy

toshow the inverse implication. Ifitholdsthat foreverydoubly-indexedsequence$\{E_{m,n}\}$ such that

$E_{m,n}\supset E_{m’,n’}$for$m\geq m’$and$n\leq n’$, there exists

a

nondecreasingsequence $\{n_{i}\}$ofpositive integers

suchthat$\bigcup_{m=1}^{\infty}\bigcap_{i=m}^{\infty}E_{i,n_{i}}=\bigcup_{m=1}^{\infty}\bigcap_{n=1}^{\infty}E_{m,n}$,thenthat condition(M)impliestheEgoroffcondition

is obvious. Assume that the above if-part is true, and condition (M) is satisfied. Let $\{E_{m,n}\}$ be a doubly-indexed sequence, $E_{m,n}\supset E_{\sqrt{},n’}$ for $m\geq m’$ and$n\leq n’$, and $\mu(\bigcup_{m=1}^{\infty}\bigcap_{n=1}^{\infty}E_{m,n})=0.$

Sincethere existsa nondecreasingsequence $\{n_{j}\}$ ofpositive integerssuch that $\bigcup_{m=1}^{\infty}\bigcap_{i=m}^{\infty}E_{i,n_{l}}=$

$\bigcup_{m=1}^{\infty}\bigcap_{n=1}^{\infty}E_{m,n}$, foreachpositive integer$i$, let_{$A_{i}=E_{i,n_{j}}$}

.

Then, the Egoroff condition is satisfied

obviously. However, it doesnot necessarily hold that forevery doubly-indexed

sequence

$\{E_{m,n}\}$

such that$E_{m,n}\supset E$, for$m\geq m’$and$n\leq n’$,thereexists anondecreasingsequence $\{n_{i}\}$ofpositive

integerssuch that$\bigcup_{m}^{\infty}$ $\infty i=mE_{i,n_{j}}=\bigcup_{m=1}^{\infty}\bigcap_{n=1}^{\infty}E_{m,n}.$

ByusingLemma 2,

we

can

showthenexttheorem.

Theorem 1 Condition $(w$_isequivalent_tothe Egorvffcondition.

4 Concluding remarks

In this paper, we have defined condition (M), which is a necessary and sufficient condition for

the Egorofftheorem with respect to non-additive

measure

and is describedby a singly-indexed

sequenceofmeasurablesets.

It

seems

thatit is easiertotreat condition(M) than theEgoroffcondition since condition(M)

is describedbyasingly-indexed sequenceofmeasurable sets. However, concrete usefulness and

potentialapplicability

are

notknown. These

are

topics for future research.

REFERENCES

[1] D. Denneberg,Non-additiveMeasureandlntegral,2nded., Kluwer, Dordrecht, 1997.

[2] J. Li, A$fi_{1}$rtherinvestigation forEgoroff’s theorem with respect to monotone set functions,

Kybernetika, 39(2003)753-760.

[3] T.Murofushi,K. Uchino, and S.Asahina, Conditions forEgoroff’s theorem in non-additive

measure

theory,

_Fua

Sets and Systems, 146(2004) 135-146.

[4] E. Pap,Null-additiveSetFunctions,Kluwer, Dordrecht, 1995.

[5] M. Sugeno, TheoryofFuzzy Integrals andItsApplications,DoctoralThesis, Tokyolnstitute of Technology, 1974.

[6] M.Takahashi, T.Murofushi, and S.Asahina, Anew necessary and sufficient conditionfor the

Egorofftheorem in non-additive

measure

theory, Fuzzy SetsandSystems,toappear.

[7] Z. WangandG.J.Klir,Fuzy MeasureTheory,Plenum,NewYork, 1992.