research article nonstandard methods in measure...

Research ArticleNonstandard Methods in Measure Theory

Grigore Ciurea

Mathematics Department Academy of Economic Studies Piata Romana 6 010374 Bucharest Romania

Correspondence should be addressed to Grigore Ciurea grigoreciureagmailcom

Received 14 January 2014 Accepted 28 April 2014 Published 26 June 2014

Academic Editor Geraldo Botelho

Copyright copy 2014 Grigore Ciurea This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Ideas and techniques from standard and nonstandard theories of measure spaces and Banach spaces are brought together to give anew approach to the study of the extension of vector measures Applications of our results lead to simple new proofs for theoremsof classical measure theory The novelty lies in the use of the principle of extension by continuity (for which we give a nonstandardproof) to obtain in an unified way some notable theorems which have been obtained by Fox Brooks Ohba Diestel and othersThemethods of proof are quite different from those used by previous authors and most of them are realized by means of nonstandardanalysis

Dedicated to Professor Solomon Marcus in honour of the 90th anniversary of his birthday

1 Introduction

Let Ω be a nonempty fixed set and ] a real-valued positivemeasure on a ringR of subsets ofΩ the measure is assumedto be countably additive in the sense that if (119864

119899) is a sequence

of disjoint members of R and if ⋃infin119899=1119864119899is also in R then

](⋃infin119899=1119864119899) = suminfin

119899=1](119864119899) A fundamental problem inmeasure

theory is that of finding conditions under which a countablyadditive measure on a ringR can be extended to a countablyadditive measure on a wider class of sets containing R Thisproblem is essentially solved by the Caratheodory processof generating an outer measure ]lowast and taking the family of]lowast-measurable sets (see [1]) then the original measure canbe extended to a 120590-ringΣ which contains the 120590-ringΣ(R)generated byR

Suppose instead that ] is no longer real-valued but itis a set function on R taking values in a Banach space119883 If ] is countably additive in the above sense in whatcircumstances is it still possible to extend ] to Σ(R) Anobvious necessary condition is that ] should be bounded overR that is sup](119864) 119864 isin R should be finite for ifthe extension ]

1onto Σ(R) exists then ]

1as a finite-valued

measure on a sigma ring is well known to be bounded over

its domain [2 III 45] so that ] is a fortiori bounded overR So if ] is a bounded set function on a ring 119877 with valuesin a Banach space what are the possibilities to obtain anextension

One of the simplest methods is to consider the family(120582119909lowast) of signed measure on R obtaining from each element

119909lowast of the topological dual 119883lowast by the following real-valued

mapping on R 120582119909lowast(119864) = ⟨](119864) 119909lowast⟩ 119864 isin R Such set

functions 120582119909lowast are bounded and countably additive over R

and as such can be subjected to the Jordan decompositionThus the problem is reduced to the Caratheodory procedureand the extension ]

1of ] is defined in terms of the elements

of 119883lowast with the following properties ]1takes values in the

algebraic dual of 119883lowast and is countably additive overR in theweak topology So the extension problem reduces thereforeto finding circumstances in which the range of ]

1is119883 (identi-

fying119883with its natural embedding into119883lowastlowast) Whenever thisis the case for instance in the case of reflexive spaces the setfunction is countably additive in view of Pettisrsquo theorem [2]This result was obtained for the first time by Fox [3]

Since vector-valued measures are important tools inintegral representation as well as disintegration of measuresmany authors have considered the extension problem when

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 851080 10 pageshttpdxdoiorg1011552014851080

2 Abstract and Applied Analysis

the range of ] is contained in a vector space 119883 In the lit-erature there are two main approaches to proving theoremsconcerning the extension of vector measures

The first approach is due to the properties which have tobe satisfied by themeasure which follows to be extended If119883is a Banach space we can mention the solutions due to Gaina[4] (in the case when ] has finite variation) Dinculeanu [5 6](if ] is regular and of bounded variation) Arsene and Stratila[7] (when ] is bounded above in norm by a positivemeasure)Dinculeanu andKluvanek [8] (in the casewhen ] is absolutelycontinuous with respect to a positive measure) and Fox [9](] satisfies a monotone-convergence condition)

The second approach relies on the conditions which haveto be satisfied by the range of ] In this category we have theresults of Fox [3] Kluvanek [10] Ohba [11] or Gould [12]Generalizing the notion of outer measure to vector-valuedmeasures and imitating the ]lowast-measurability procedure inorder to obtain a Lebesgue extension of ] Gould [12] showedthat a necessary and sufficient condition for ] to have aLebesgue extension is that the following property should holdfor the image space119883 of ](119860) If (119909

119899) is a sequence in119883whose norms have a positive

lower bound then there exists for arbitrary positive119870 a finitesubsequence (119909

119899119903

) such that sum119903119909119899119903

gt 119870This suggests a connection between weak completeness

and property (119860) and it is shown in [12] that all weaklycomplete spaces satisfy property (119860) In Section 3 we presentanother proof of this result It is easy to verify that 119897infin doesnot hold property (119860)Thus 119897infin is not weakly complete Moregenerally Banach spaces which are infinite-dimensionalfunction spaces with a supremum norm fail to satisfy prop-erty (119860) and therefore are not weakly complete For Hilbertspaces property (119860) is satisfied A direct proof of the fact thatproperty (119860) is satisfied by 119897119901 (1 le 119901 lt infin) is much harder

For a masterful study of measures with values in a topo-logical group we refer the reader to Sion [13] or Drewnowski[14ndash16] When 119883 is a commutative complete topologicalgroup and ] is of bounded variation then there is a very niceextension theorem by Takahashi [17]

The starting point in nonstandard theory of measurespaces is a paper [18] by Loeb He gave a way to construct newrich standard measure spaces from internal measure spacesThis construction has been used in recent years to establishnew standard results in a variety of different areas Some ofthese results can be found in [19] or [20]

Also nonstandard analysis has proved to be a naturalframework for studying vector measures and Banach spacesThe central construction in this approach is the notion ofnonstandard hull introduced by Luxemburg [21]This notionis not only a useful tool in studying vector measures andBanach spaces but also a construction arising naturallythroughout nonstandard analysis For a deeper discussion ofnonstandard hulls and their applications we refer the readerto the survey paper [22] of Henson and Moore

Zivaljevic [23] has pursued the extension problem usingthe nonstandard hull of 119883 Osswald and Sun [24] treated thesame problem from a different point of view the extensionof additive vector measures has been made using the internal

control measures Furthermore the authors present a differ-ent approach of Loebrsquos [18] in order to construct a countablyadditive vector measure from internal locally convex space-valued measure

In this work we study the extension of vector valued setfunctions in the framework of nonstandard analysis

The plan of this paper is as followsSection 2 is devoted to some preliminary results on stan-

dard vector measures Using concurrent relations we alsoobtain a result concerning the concentration of 119904-boundedvectormeasures on a specific set from the nonstandard exten-sion ofR Moreover the principle of extension by continuity[2] will be proved using nonstandard techniques

In Section 3 we give a nonstandard characterization ofthe absolute continuity and an extension theorem for vectormeasures The proof still uses nonstandard arguments Sincereflexive spaces are weakly complete and the weakly completespaces satisfy property (119860) we can rederive the Foxrsquos result[3] Some results of Gould [12] will be reproved in a differentmanner For this we use a result of Diestel et al (see [25] or[26]) on 119904-bounded measures To obtain these results Gouldhas used Pettisrsquo theorem Our approach does not use thisresult

In Section 4 we address the issues of the existing controlmeasures We tackle this subject by using the extension of setfunctions and linking these extensions to control measures

Section 5 deals with the extension of set functions withfinite semivariation The results in this section (for whichwe present a nonstandard proof) were originally obtained byLewis [27]

The last section shows that the extension of set functionswith finite variation is a particular case of the extension of setfunctions with finite semivariation

We adopt the nonstandard framework of [28] Thenonstandard model used in this paper is assumed to besufficiently saturated for our needs In what follows N

infin

denotes the set lowastN N where lowastN is the extension of N in ourmodel

2 Preliminaries

In this section we collect some basic definitions notationsand elementary results about standard vector measures Alsothe principle of extension by continuity will be proved usingnonstandard techniques

The terminology concerning families of sets set func-tions and so forth will be in general that of [2] or [1] LetR+denote the nonnegative reals and let N denote the set of

positive integers Sets are denoted as 119860 119861 119862 0means theempty set Notation for set operations is that commonly usedin particular119860Δ119861means (119860 119861)cup (119861 119860) Everywhere in thesequel R denotes a ring of subsets of a nonempty fixed setΩ the casesR should be a 120575-ring or 120590-ring will be explicitlyspecified The complement (in Ω) of a set 119860 is denoted by119860119888 Symbols and for sequences of sets or of reals have

their usual meaning A set function 120583 R rarr [0infin] will becalled a submeasure (subadditivemeasure in the terminologyof Orlicz [29]) if 10 120583(0) = 0 20 120583(119860 cup 119861) le 120583(119860) + 120583(119861)whenever 119860 119861 isin R and 119860 cap 119861 = 0 30 120583(119860) le 120583(119861) if

Abstract and Applied Analysis 3

119860 119861 isin R and 119860 sube 119861 The ring R is an abelian group withrespect to the symmetric difference operation and eachsubmeasure 120583 generates a semimetric on the group (R )by the Frechet-Nikodym ecart 120588(119860 119861) = 120583(119860 119861) Thissemimetric is invariant in the sense that 120588(119860 119861) = 120588(119860 119862 119861 119862) for sets 119860 119861 119862 isin R Therefore any submeasure120583 generates a topology on the group (R ) and a base ofneighborhoods is given by the family of sets119873(119860

0 120598) = 119860 isin

R 120583(119860 1198600) lt 120598 In the semimetric space (R 120588) the set

operations are continuous [5] (we also present a nonstandardproof of this result) Requiring from a topology in a ring topossess this property one obtains the topological ring of setsa natural generalization of the so-called spaces of measurablesets introduced by M Frechet and O Nikodym in which adistance between two sets is defined as the measure of theirsymmetric difference

Let (119883 sdot ) be a Banach space and let ] be a set functionfrom R to 119883 We say that ] is a finitely additive vectormeasure or simply a vector measure if whenever 119864

1and 119864

2

are disjoint members ofR then ](1198641cup 1198642) = ](119864

1) + ](119864

2)

If in addition ](⋃infin119899=1119864119899) = suminfin

119899=1](119864119899) in the norm topology

of 119883 for all disjoint sequences (119864119899) of members of R such

that ⋃infin119899=1119864119899isin R then ] is termed a countably additive

vector measure or simply ] is countably additive We usethe terminology from [30] and call a set function ] fromR to 119883 strongly bounded (often abbreviated 119904-bounded) if](119864119899) rarr 0 whenever (119864

119899) is a disjoint sequenceWe say that

] is order continuous (119900119888) if for each sequence (119864119899) sub R

such that 119864119899 0 we have ](119864

119899) rarr 0 We will denote by

120591(R) the class of all subsets 119864 of Ω which have the propertythat 119860 cap 119864 isin R for every 119860 isin R Then R forms an idealin 120591(R) and 120591(R) is an algebra We will denote byP(Ω) thepower set ofΩ and for anyC sube P(Ω)we put C = ⋃infin

119899=1119864119899

119864119899isin C 119899 = 1 2 If ] is a vector measure from R to 119883 we call (ΩR ])

a vector measure space The inner quasi-variation ](119860) of anarbitrary subset 119860 ofΩ is defined by ](119860) = sup](119864) 119864 isinR 119864 sube 119860 a set119860 is ]-bounded if ](119860) is finite [12] If ](Ω) isfinite then ]will be called a bounded vectormeasure Clearlythe inner quasi-variation is a submeasure onR

Abstraction of the condition of strong boundedness ona ring is the concept of the Rickart submeasure Thus asequence (119860

119899) sub R is called dominated if there exists a set

119861 isin R such that 119860119899sube 119861 for 119899 = 1 2 The submeasure

] is said to be Rickart on the ring R if for each dominateddisjoint sequence (119860

119899) sub R we have lim

119899](119860119899) = 0

Note that every finite additive submeasure on the ring R isRickart

An extensive research of topological rings of sets gener-ated by Rickart families of submeasures with applications tovector measures was initiated by Oberle [31] and developedby Bogdanowicz and Oberle [32] The topological point ofview was realized by Drewnowski [14ndash16]

On the other hand essential properties of finite or count-ably additive vector measures are reflected on the prop-erties of corresponding submeasures This enables us touse submeasures as a convenient tool in various questionsconcerning vector measures For instance Walker [33] has

used the corresponding submeasures to study uniform sigmaadditivity or equicontinuity

Throughout this section we assume that ] is countablyadditive and 119904-bounded Then we know that ] is a boundedvector measure and ] is 119904-bounded Furthermore for anysequences 119864

119899 0 of members of P(Ω) we have ](119864

119899) 0

so the countable additivity of ] implies order continuity of thesubmeasure ] [14ndash16 26 34] For any set 119860 sube Ω Γ(119860) = 119861 isin120591(R) 119860 sube 119861 is a directed set where 119861

1le 1198612if and only

if 1198611supe 1198612for 1198611 1198612isin Γ(119860) Then the generalized sequence

](119861) 119861 isin Γ(119860) is a Cauchy net By the completeness of Rwe put for any subset 119860 of Ω the outer quasi-variation of 119860given by ](119860) = lim

119861isinΓ(119860)](119861) Consequently there exists a

unique set function ] P(Ω) rarr R+such that for every set

119860 isin 120591(R) we have ](119860) = ](119860) We may refer to ] and ] asthe inner and the outer measures generated by ]

Since ] is bounded we can define on P(Ω) the ecartfunction 120588(119860 119861) = ](119860 119861) where 119860 119861 is the symmetricdifference of 119860 and 119861 Then (P(Ω) 120588) is a semimetric spaceWe denote by R the closure of R in the space (P(Ω) 120588) if(119879 120591) is a topological space and 119860 sube 119879 the closure of 119860 in(119879 120591) is denoted by 119860 For 119860 a subset of R and 119899 a positiveinteger let 119899119860 = sum119899

119894=1119909119894 119909119894isin 119860 for 119894 = 1 119899 If 119880

is a neighbourhood of 0 in R we put = (119860 119861) 119860 119861 isinP(Ω) ](119860 119861) isin 119880

The below proposition is straightforward so we omit itsproof

Definition 1 A ringR is called ]-dense if for any subset 119860 ofΩ and for any positive real number 120576 there is 119864 isin R 119864 sube 119860such that ](119860 119864) lt 120576

Definition 2 The sets 119860 119861 of lowastR will be called equivalent(119860 asymp 119861) if 120588(119860 119861) asymp 0 where 120588 is the ecart function 120588(119860 119861) =](119860 119861)

Proposition 3 Let ] R rarr 119883 be a vector measure If ] is119904-bounded thenR is ]-dense

Lemma 4 (119894) Let 119860 sube 119861 subsets of Ω Then ](119860) le ](119861) and](119860) le ](119861)(119894119894) For 119860 subset of Ω we have ](119860) le ](119860) and ](119860) =

](119860) on 120591(R)(119894119894119894) Let 119860 isin 120591(R) and 119861 subset of Ω Then ](119860 cup 119861) le

](119860) + ](119861)(119894119907) If (119864

119899) sub P(Ω) and (119864

119899) 0 then ](119864

119899) 0

(119907) If (119864119899) sub 120591(R) and (119864

119899) 119864 then ](119864

119899) ](119864)

(119907119894) If (119864119899) sub 120591(R) then ](⋃infin

119899=1119864119899) le suminfin

119899=1](119864119899)

Lemma 5 (119894) If (119864119899) sub P(Ω) then ](⋃infin

119899=1119864119899) le sum

infin

119899=1](119864119899)

(119894119894) If (119864119899) sub P(Ω) (119864

119899) 119864 then ](119864

119899) ](119864)

The above properties of the inner and the outer measuresgenerated by ] are well known Now we are going to prove aresult which is needed in the sequel

Theorem 6 (119894)R is a sigma algebra and 120591(R) subR


(119894119894) If (119860119899)119899subeR and119860 = ⋃infin

119899=1119860119899then120588(119860 119860

119899) = ](119860

119860119899) 0(119894119894119894) If (119860

119899)119899subeR and (119860

119899)119899 0 then ](119860

119899) 0

Proof (i) Let 119860 isin R There is a set 119861 isin lowastR such that 119860 asymp 119861Then 119860119888 asymp 119861119888 SinceR is ]-dense there is 119864 isin lowastR such that119864 sube 119861

119888 and ](119861119888 119864) asymp 0 But ](119861119888 119864) = ](119861119888 119864) becauseof 119861119888 119864 isin lowast120591(R)

It is easy to see that119860119888119864 sube (119860119888119861119888) cup (119861119888 119864) Hence119860119888

asymp 119864 so 119860119888 isinRLet now 119860

1 1198602isin R There are 119861

1 1198612isinlowast

R such that1198601asymp 11986111198602asymp 1198612 We have119860

1cup1198602asymp 1198611cup1198612 1198611cup1198612isinlowast

Rso 1198601cup 1198602isinR ThereforeR is algebra

Suppose (119860119899)119899is a sequence of elements ofR and set119860 =

⋃infin

119899=1119860119899 It remains to prove that 119860 isin R SinceR is algebra

we can take (119860119899) Choose119880 and119881 neighbourhoods of zero

in R such that 119881 + 119881 sube 119880Let (119881

119899)119899be a sequence of neighbourhoods of zero inR so

thatsum119899119896=1119881119896sube 119881 for all 119899 For every 119899 we can choose 119864

119899isinR

such that (119860119899 119864119899) isin 119881119899 If 119864 = ⋃infin

119899=1119864119899 a trivial verification

shows that 119860 119864 sube ⋃infin119899=1119860119899 119864119899 By Lemma 5(i) we have

] (119860 119864) leinfin

sum

119896=1

] (119860119899 119864119899) (1)

whence ](119860119864) isin 119881 Setting 119865119899= ⋃119899

119896=1119864119896 we have 119865

119899isinR

(119865119899) and 119860

119899 119865119899sube ⋃119899

119896=1119860119896 119864119896 Lemma 5(i) implies

] (119860119899 119865119899) le

119899

sum

119896=1

] (119860119896 119864119896) (2)

so ](119860119899 119865119899) isin 119881 Moreover 119864 119865

119899isin 120591(R) and by

Lemma 4(ii)

] (119864 119865119899) = ] (119864 119865

119899) 0 (3)

according to (119865119899) It follows that ](119864 119865

119899) isin 119881 for 119899 large

enoughWe observe that119860119865119899sube (119860119864)cup (119864119865

119899) for each

119899 and on account of Lemma 5(i) we conclude that

] (119860 119865119899) le ] (119860 119864) + ] (119864 119865

119899) (4)

Thus for 119899 large enough ](119860 119865119899) isin 119881 + 119881 sube 119880 so 119860 isinR

Let 119860 isin 120591(R) and fix 120576 gt 0 Proposition 3 now gives 119864 isinR so that 119864 sube 119860 and ](119860 119864) lt 120576 According to Lemma 4(ii)since 119860 119864 isin 120591(R) we get

] (119860 119864) = ] (119860 119864) lt 120576 (5)

Thus 119860 isinR(ii) If (119860

119899)119899subeR and119860 = ⋃infin

119899=1119860119899 we verify immediately

that

119860 119860119899sube (119860 119865

119899)⋃ (119860

119899 119865119899) (6)

Lemma 5(i) implies

] (119860 119860119899) le ] (119860 119865

119899) + ] (119860

119899 119865119899) (7)

From this we deduce for 119899 large enough that ](119860 119860119899) isin

119880 + 119881 sube 119880 + 119880 = 2119880 so 120588(119860 119860119899) 0 as 119899 rarr infin

(iii) From (119860119899)119899 0 we see that (119860119888

119899)119899 Ω According

to (i) and (ii) (119860119888119899)119899is a sequence of elements of R and

120588(Ω119860119888

119899) 0 as 119899 rarr infin We remark at once that 120588(Ω119860119888

119899) =

](119860119899) and the proof is complete

Because the main tool in our approach is the principle ofextension by continuity we give a nonstandard proof of it Fora standard proof we refer the reader to [2]

Theorem 7 (principle of extension by continuity) Let119883 and119884 be metric spaces and let 119884 be complete If119860 is a dense subsetof 119883 and 119891 119860 rarr 119884 is uniformly continuous then 119891 has aunique uniformly continuous extension 119892 119883 rarr 119884

Proof Let 119909 isin 119883 then there is a 119910 isin lowast119860 with 119910 asymp 119909Moreover there is a sequence of points (119886

119899)119899of 119860 with 119886

119899rarr

119909 For each 120579 isin Ninfin

we have 119886120579asymp 119909 But 119891 is uniformly

continuous on 119860 so 119886120579asymp 119910 implies 119891(119886

120579) asymp 119891(119910)

For 120576 a positive real number we put 119881120576= 119899 isin

lowast

N

119889119884(119891(119886119899) 119891(119910)) lt 120576 Then 119881

120576is internal by the definition

principle and each 120579 isin Ninfin

belongs to 119881120576 In particular for

some 119899 isin N we have 119899 isin 119881120576 Hence 119891(119910) is a prenearstandard

point and so 119891(119910) is nearstandard Let 119892(119909) = ∘119891(119910) where∘ is the standard part map Then 119892 is obviously well definedand extends 119891 To verify that 119892 is uniformly continuous let 120576be an arbitrary positive real number The assumption impliesthat there is a positive real number 120575 such that

[119889119883(1199101 1199102) lt 120575 997904rArr 119889

119884(119891 (1199101) 119891 (119910

2)) lt

120576

2]

(forall1199101 1199102isin 119860)

(8)

By the transfer principle

[119889119883(1199101 1199102) lt 120575 997904rArr 119889

119884(119891 (1199101) 119891 (119910

2)) lt

120576

2]

(forall1199101 1199102isinlowast

119860)

(9)

Let 1199091 1199092isin 119883 with 119889

119883(1199091 1199092) lt 1205752 There are 119910

1 1199102isin

lowast

119860 such that 1199101asymp 1199091 1199102asymp 1199092 Then 119889

119883(1199101 1199102) le

119889119883(1199101 1199091) + 119889

119883(1199091 1199092) + 119889

119883(1199092 1199102) so 119889

119883(1199101 1199102) lt 120575

Hence 119889119884(119891(1199101) 119891(119910

2)) lt 1205762 By the definition of 119892 we have

119892(1199091) asymp 119891(119910

1) 119892(119909

2) asymp 119891(119910

2) Since we also have

119889119884(119892 (1199091) 119892 (119909

2))

le 119889119884(119892 (1199091) 119891 (119910

1)) + 119889

119884(119891 (1199101) 119891 (119910

2))

+ 119889119884(119891 (1199102) 119892 (119909

2))

(10)

we get that 119889119884(119892(1199091) 119892(1199092)) lt 120576 whence 119892 is uniformly con-

tinuousLet 119892

1be another function with the same properties

and 119909 isin 119883 Then there is 119910 isin lowast119860 with 119910 asymp 119909 Since119892 1198921are uniformly continuous we have 119892(119910) asymp 119892(119909) and

1198921(119910) asymp 119892

1(119909) By the transfer principle 119892(119910) = 119892

1(119910) so

that 119892(119909) asymp 1198921(119909) As 119892(119909) and 119892

1(119909) are standard it follows

that 119892(119909) = 1198921(119909) and the proof is complete


The section closes with an outcome about the concentra-tion of 119904-bounded measures on a set of lowastR By 119904119886(R 119883) wedenote the set of all 119904-bounded additive measures fromR to119883

Proposition 8 There exists a set 119860 of lowastR so that for all 119861 oflowast

R 119861 disjoint of 119860 and for every ] belonging to 119904119886(R 119883) wehave ](119861) asymp 0

Proof Let Ξ be a relation on (119904119886(R 119883)timesR+)timesR defined by

((] 120598) 119864) isin Ξ if and only if for all 119865 isin R 119865 disjoint of 119864 wehave ](119865) lt 120598 We see at once that dom(Ξ) is 119904119886(R 119883) timesR+ which is clear from Proposition 3 We verify that Ξ is a

concurrent relation Indeed if (]1 1205981) (]

119899 120598119899) isin dom(Ξ)

there exists 1198641 119864

119899isinR such that for all 119865 isinR 119865 disjoint

of 119864119894we have ]

119894(119865) lt 120598 119894 = 1 119899 Setting 119864 = ⋃119899

119894=1119864119894 we

have 119864 isin R and ((]119894 120598119894) 119864) isin Ξ for all 119894 = 1 119899 which is

our assertion By the concurrence theorem [35] there is a set119860 isinlowastR such that ((] 120598) 119860) isinlowast Ξ for all (] 120598) isin dom(Ξ) Fix

] isin 119904119886(R 119883) Then for all 119861 isin lowastR 119861 disjoint of 119860 we have](119861) lt 120598 Let us regard 120598 as ran and the proof is complete

3 Extension of a Vector Valued Measure

We adopt here the main framework of nonstandard analysisfrom [28] We also give the definition nonstandard formu-lation and some of the basic properties of the absolutelycontinuous concept In this section 120583 denotes a submeasurefrom R to R

+and 120588 stands for the Frechet-Nikodym ecart

120588(119860 119861) = 120583(119860 119861) associated with 120583 We have seen inSection 2 that (R 120588) is a semimetric space We recall that thesets 119860 119861 of lowastR are equivalent (119860 asymp 119861) if 120588(119860 119861) asymp 0

Definition 9 A vector measure ] is called absolutely contin-uous with respect to 120583 or simply 120583-continuous if for anypositive real number 120576 there is a positive real number 120575 suchthat for any 119860 isin R ](119860) lt 120576 if 120583(119860) lt 120575 In this case wesay that 120583 is a control submeasure of ] and we denote that by] ≪ 120583

A nonstandard formulation of absolutely continuousmaybe stated as follows

Lemma 10 A vector measure ] on (ΩR) is absolutely con-tinuous with respect to 120583 if and only if for all 119860 isin lowastR with120583(119860) asymp 0 we have ](119860) in the monad of zero in 119883

Proof Assume 120583 is a control submeasure of ] and let 120576 be anarbitrary positive real number If 119860 is some element in theinternal algebra lowastR with 120583(119860) asymp 0 we have by the transferprinciple that ](119860) lt 120576 Since 120576 is arbitrary we obtain](119860) asymp 0

To prove the converse fix119860 isin lowastR with 120583(119860) asymp 0 and fora given positive real number 120576 let 120575 be an infinitesimal suchthat 120583(119860) lt 120575 Then for any 119861 isin lowastR with 120583(119861) lt 120575 we get](119861) lt 120576 Thus we have shown

(exist120575 isinlowast

R+) (forall119861 isin

lowast

R) [120583 (119861) lt 120575 997904rArr ] (119861) lt 120576] (11)

Now apply the transfer principle to obtain the desired condi-tion for absolute continuity

Remark 11 As we havementioned the set operations are con-tinuous [5] in the semimetric space (R 120588) Here a nonstan-dard proof is given

Lemma 12 The maps 119891 119892 ℎ R timesR rarr R defined by theequalities

119891 (119860 119861) = 119860⋃119861 119892 (119860 119861) = 119860⋂119861

ℎ (119860 119861) = 119860 119861

(12)

are uniformly continuous

Proof We denote by ∘ one of the operations cap cup For everyset 1198601 1198602 1198611 1198612of lowastR we have (119860

1∘ 1198611) (119860

2∘ 1198612) sube

(11986011198602) cup (119861

1 1198612) Hence

120588 (1198601∘ 1198611 1198602∘ 1198612) le 120588 (119860

1 1198602) + 120588 (119861

1 1198612) (13)

If1198601asymp 1198602 1198611asymp 1198612 then 120588(119860

1 1198602) asymp 0 120588(119861

1 1198612) asymp 0Thus

we conclude 120588(1198601∘ 1198611 1198602∘ 1198612) asymp 0 and the set operations


Lemma 13 A vector measure ] on (ΩR) is absolutelycontinuous with respect to 120583 if and only if ] is uniformlycontinuous on (R 120588)

Proof Suppose 119860 119861 isin lowastR and 119860 asymp 119861 By Lemma 10 ](119860 119861) asymp 0 and ](119861 119860) asymp 0 But ] is a vector measure so ](119860) =](119860 119861) + ](119860 cap 119861) asymp ](119860 cap 119861) and ](119861) = ](119861 119860) + ](119860 cap119861) asymp ](119860 cap 119861) Therefore ](119860) asymp ](119861) which means that ] isuniformly continuous

Conversely let119860 isin lowastR with 120583(119860) asymp 0 In this case119860 asymp 0and by uniform continuity of ] we have ](119860) in the monad ofzero in119883 Then the result follows by Lemma 10

Theorem 14 LetA be a ring of subsets ofΩ such thatA subeR

and let ] be a vector measure on (ΩA) Suppose A is densein (R 120588) and ] is absolutely continuous with respect to 120583Then ] has a unique vector measure extension ]

1 R rarr

119883 Furthermore this extension is uniformly continuous on(ΩR)

Proof On account ofTheorem 7 there is a unique uniformlycontinuous map ]

1 R rarr 119883 which extends ] It remains

to prove that whenever 1198641and 119864

2are disjoint members ofR

then ]1(1198641cup 1198642) = ]1(1198641) + ]1(1198642) By assumption we can

choose 1198651 1198652isinlowastA such that 119864

1asymp 1198651 1198642asymp 1198652 According

to Lemma 12 we have 1198641cup 1198642asymp 1198651cup 1198652 1198641 1198642asymp 1198651 1198652

Uniform continuity of ]1implies ]

1(1198641cup 1198642) asymp ]1(1198651cup 1198652)

]1(1198641 1198642) asymp ]1(1198651 1198652) and ]

1(1198642) asymp ]1(1198652) The transfer

principle leads to ] that is finitely additive on lowastA and

]1(1198641⋃1198642) asymp ] (119865

1⋃1198652) = ] (119865

1 1198652) + ] (119865

2)

= ]1(1198651 1198652) + ]1(1198652)

]1(1198651 1198652) + ]1(1198652) asymp ]1(1198641 1198642) + ]1(1198642)

(14)


Now

]1(1198641 1198642) + ]1(1198642) = ]1(1198641) + ]1(1198642) (15)

which is due to the fact that 1198641and 119864

2are disjoint

We conclude from (14) and (15) that

]1(1198641⋃1198642) asymp ]1(1198641) + ]1(1198642) (16)

which clearly forces

]1(1198641⋃1198642) = ]1(1198641) + ]1(1198642) (17)

This completes the proof

Theorem 15 Let ] R rarr 119883 be a 119904-bounded and countablyadditive vector measure Then there exists a unique countablyadditive vector measure ]

1R rarr 119883 extending ]

Proof We claim that ] is absolutely continuous with respectto ] Indeed if 119860 isin lowastR and ](119860) asymp 0 the transfer principleleads to ](119860) asymp 0 Since ](119860) le ](119860) we have ](119860) in themonad of zero in119883 which is our assertionTheorem 14 showsthat there exists a vector measure ]

1 R rarr 119883 absolutely

continuous with respect to ] which extends ] What is leftto prove is that ]

1is countably additive Consider (119860

119899)119899a

sequence of elements of R such that (119860119899)119899 0 Therefore

](119860119899) 0 by Theorem 6(iii) Then for 120579 isin N

infin ](119860120579) asymp

0 Therefore applying Theorem 14 and Lemma 10 we have]1(119860120579) in the monad of zero in 119883 Then ]

1(119860119899) rarr 0 as

119899 rarr infin which completes the proof

The following corollary can be found in [11Theorem 1] or[36]

Corollary 16 LetR be a ring and 120590(R) the 120590-ring generatedby R A countably additive vector measure ] R rarr 119883 canbe extended uniquely to a countably additive vector measure]1 120590(R) rarr 119883 if and only if ] is 119904-bounded

The next two results were proved by Gould [12] and theyare important consequences of Corollary 16 He has shownthat in every weakly complete Banach space property (119860)holds [12 Theorem 31] Even though our construction isadapted from [12] Proposition 17 yields an elegant proof ofthis result

Proposition 17 Let119883 be a weakly complete Banach space and] a bounded set function from R to 119883 If ] is a countablyadditive vectormeasure then ] has a unique countably additiveextension ]

1 120590(R) rarr 119883

Proof We begin by proving that ] is 119904-bounded If ] werenot 119904-bounded we would have 119883 that contains a subspaceisomorphic to 119888

0(see for instance [26 page 20]) Since

every closed linear subspace of weakly complete Banachspace is weakly complete and 119888

0is not weakly complete [2

page 339] we obtain a contradiction Now it suffices to applyCorollary 16 and the proof is complete

Remark 18 The reader should observe the similarity betweenthis proposition and some of the criteria established byGould [12 Theorem 25 page 688] for the extension of setfunctions with values in weakly complete Banach spaces Itis worth noting that in the work of Gould [12] it suffices torequire that ] be locally bounded over R and the extensionis made onto a family Σ

1which is a 120590(R)-hereditary ring

containingR (if 119878 is a given family of sets a ringR is said tobe 119878-hereditary if everymember of 119878which is a subset of somemember ofR is also a member ofR) Our requirements arestronger but the final conclusion is somewhat more generalMoreover we do not use Pettisrsquo theorem for the proof

We are now ready to prove [12 Theorem 31 page 689]

Proposition 19 If 119883 is a weakly complete Banach space thenproperty (119860) holds

Proof Suppose that the proposition was false Then we couldfind positive 120575 119870 and a sequence (119909

119899) in 119883 so that 119909

119899 ge 120575

for all 119899 and sum119903119909119899119903

le 119870 for every finite subsequence (119909119899119903

)

of (119909119899) LetR be the ring of finite sets of positive integers and

let ] denote the set function taking each finite set (119899119903) into the

vectorsum119903119909119899119903

Clearly ] is a bounded vector measure fromRto119883 Furthermore ] is countably additive since there are noinfinite disjoint nonempty sequences (119864

119899) inR whose union

is inR Proposition 17 yields that there is a countably additiveextension ]

1onto 120590(R) which extends ] In particular the

set N of all the positive integers belongs to 120590(R) Hence]1(N) = sum

119903119909119903 and the convergence of this series contradicts

the hypothesis that 119909119899 ge 120575 for all 119899

For Banach spaces reflexivity and semireflexivity areequivalent and either implies weak completeness Thusthe rederivation of Foxrsquos theorem [3] is a consequence ofProposition 17 (see also [11 Corollary 2 page 65])

Corollary 20 A bounded countably additive vector measureon a field taking its values in a reflexive Banach spaceextends uniquely to a countably additive vector measure on thegenerated sigma-field

Theorem 21 Let 119883 be a Banach space for which property (119860)holds Assume ] R rarr 119883 is a countably additive vectormeasure Then ] has a countably additive extension ]

1

120590(R) rarr 119883 if and only if ] is bounded

ProofNecessity Assume that ] has a countably additive extension]1 120590(R) rarr 119883 Then ]

1as a finite-valued measure on a 120590-

ring is bounded over its domain [2 III 45] so ] is boundedoverR

To prove the sufficiency for such ] we verify that ] is 119904-bounded Indeed if ] were not 119904-bounded we would havea sequence 119864

119899 of disjoint members of R and 120575 gt 0 such

that ](119864119899) ge 120575 for all 119899 By property (119860) for all positive 119870

there is a finite subsequence ](119864119899119903

) such that sum119903](119864119899119903

) =


](⋃119903119864119899119903

) ge 119870 This contradicts that ] is bounded and thetheorem follows

4 Existence of Control Measure

In [8] the authors show that the Bartle-Dunford-Schwartztheorem [37] does not work if we replace the 120590-ring by aring and ask whether the result remains valid for 120575-ringsThe theorem states that for every countably additive measure] defined on sigma algebra there exists a positive controlmeasure 120583 such that 120583(119864) rarr 0 if and only if ](119864) rarr 0where ] is the semivariation of ] By a counterexampleit is shown in [38] that this result could not work even ifthe measure is defined on 120575-rings So we have to imposeadditional conditions for obtaining this goal In [38] one alsoshows that the theorem works if the space 119883 is separableNow if we pass to the conditions imposed on the measure] and no the space119883 in which it has values we will prove thefollowing two Brooksrsquo results [36]

Theorem 22 Let ] R rarr 119883 be a countably additive vectormeasure Then ] is 119904-bounded if and only if there exists a pos-itive countably additive bounded set function 120583 defined on Rsuch that

lim120583(119864)rarr0

] (119864) = 0 (18)

To prove this theorem Brooks uses Orlicz-Pettis theoremand two results of Porcelli [39] about some embeddingtheorems and their implications inweak convergence respec-tively compactness in the space of finitely additive measureHe also uses a result of Leader [40] from the theory of 119871119901spaces for finitely additive measures Brooks and Dinculeanuin [41] by extending a result of Dieudonne [42] prove theassertion for finitely additive and locally strongly additivemeasures Traynor [43] gave an elementary proof of this resultfor strongly additive measures Using this result he showsthat for strongly additive and countably additive measures onalgebra the existence of a finite control measure is equivalentto the relatively weak compactness of range of measureswhich is equivalent to the existence of a countably additiveextension on the sigma algebra generated byR [11 26]

There is some interest in the extension measure theoreticapproach given here The proof that we will give uses theextension of ] to 120590-ring generated by R and then weapply the Bartle-Dunford-Schwartz theorem Thus we avoidsome deep results in vector measures and unconditionallyconvergent series

Proof of Theorem 22 First assume that ] is countably additiveand 119904-boundedTheorem 15 gives a countably additive vectormeasure ]

1 R rarr 119883 which extends ] We know that R

is 120590-algebra and R sube R (see Theorem 6(i)) By the Bar-tle-Dunford-Schwartz theorem [37] there exists a positivecountably additive bounded set function 120595 onR such that

lim120595(119864)rarr0

]1(119864) = 0 (19)

Define 120583(119860) = 120595(119860) if 119860 isinR and

lim120583(119864)rarr0

] (119864) = 0 (20)

as claimedFor the converse let (119864

119899) be a sequence of pairwise dis-

joint members of R Since 120583 is bounded and countablyadditive there exists119872 ge 0 such that for all 119899

119899

sum

119896=1

120583 (119860119896) le 119872 (21)

so suminfin119896=1120583(119860119896) is convergent Then 120583(119860

119896) rarr 0 as 119896 rarr infin

and

lim120583(119864)rarr0

] (119864) = 0 (22)

implies ](119860119896) rarr 0 as 119896 rarr infin which completes the proof

Corollary 23 Let ] R rarr 119883 be countably additive Thenthere is a countably additive bounded set function 120583 defined onRwhich is a control measure for ] if and only if ] is 119904-bounded

The next result can be found in [11 Theorem 2] or in [26Theorem 2 page 27]

Corollary 24 LetR be a ring and 120590(R) the 120590-ring generatedby R Every countably additive vector measure ] R rarr 119883

can be extended uniquely to a vector measure ]1 120590(R) rarr 119883

if and only if one of the following conditions is satisfied

(i) there exists a positive bounded measure 120583 on R suchthat

lim120583(119864)rarr0

] (119864) = 0 119864 isinR (23)

(ii) ] is 119904-bounded(iii) the range of the vector measure ] is relatively weakly

compact

Proof The equivalence of (i) and (ii) is given byTheorem 22To prove (ii)rArr (iii) we note by Corollary 16 that there is

a countably additive vector measure ]1 120590(R) rarr 119883 which

extends ] Applying the Bartle-Dunford-Schwartz theorem[37 Theorem 29] the set ]

1(119860) 119860 isin 120590(R) is relatively

weakly compact and so is the set ](119860) 119860 isinRThe implication (iii)rArr (ii) is proved by Kluvanek in [44

Theorem 53]

5 Extension of Set Function withFinite Semivariation

Lewis in [27] used Caratheodoryrsquos method about the exten-sion of set functions to perform the extension of set func-tions with finite semivariation While the circumstances aresomewhat similar to the extension of set functions with finitevariation the techniques employed from there have carriedover to that situation studied by Dinculeanu [5] In this


section we give a nonstandard proof of the central result ofLewis This in turn is applied in Section 6 to achieve theextension of set functions with finite variation so we canunify the extension of set function with finite semivariationwith the extension of set function with finite variation

Let R be a ring of subsets of a universal space Ω For 119864119865 Banach spaces we denote by 119871(119864 119865) the Banach space ofall bounded linear operators 119891 119864 rarr 119865 and let ] R rarr

119871(119864 119865) be a set function with finite semivariation That is if119860 isinR we assume that

sup 10038171003817100381710038171003817sum ] (119860119894) 119909119894

10038171003817100381710038171003817(24)

is finite where we take the supremum over all finite sub-divisions (119860

119894) of 119860 which consist of elements of R and all

elements 119909119894of unit norm in 119864 Let 119867(R) be the hereditary

120590-ring generated by R We use the semivariation ] of ]to define an outer measure ]lowast on 119867(R) in the obvious wayThus ]lowast(119860) for 119860 isin 119867(R) is

infsum] (119860119894) (25)

where the infimum is taken over all countableR-coverings of119860 Clearly ]lowast is an outer measure on 119867(R) Let Ω(]) be theset of all elements 119860 in119867(R) so that if 119861 isin 119867(R) we have

]lowast (119861) = ]lowast (119861⋂119860) + ]lowast (119861⋂119860119888) (26)

By virtue of awell-known resultΩ(]) is a sigma ring [5]R(])will be the largest class of subsets of Ω so that Ω(]) forms anideal inR(]) that is 119860 isin R(]) if for each 119861 isin Ω(]) we have119861 cap 119860 isin Ω(]) For 119860 isinR(]) define

120583lowast

(119860) = sup ]lowast (119861) (27)

where the supremum is taken over all 119861 isin Ω(]) such 119861 sube 119860Σ(]) will be the 120575-ring of all elements in R(]) with finite 120583lowastmeasure The main result which will be proved by nonstan-dard means is the extension theorem of ] to a uniquely setfunction ]

1defined onΣ(])with values in 119871(119864 119865) A standard

proof can be found in [27]

Definition 25 We say that a finitely additive set function ] R rarr 119871(119864 119865) is variationally semiregular provided that if(119860119899) is a decreasing sequence of sets inRwhose intersection

is empty and ](1198601) lt infin then lim

119899rarrinfin](119860

119899) = 0

Halmos [1] says that ] is continuous from above at 0

For 119860 119861 in Σ(]) define 120588(119860 119861) to be 120583lowast(119860 119861) Then 120588defines a semimetric onΣ(]) In [27] the following two resultsare proved which we mention without proof

Lemma 26 Suppose that ] R rarr 119871(119864 119865) is variationallysemiregular with finite semivariation onR Let ]lowast be the outermeasure on119867(R) Then ] = ]lowast onR

Lemma 27 IfR sube Ω(]) thenR is 120588-dense in Σ(])

We now move on to the nonstandard proof of ourproblem

Theorem 28 Let ] R rarr 119871(119864 119865) be variationally semireg-ular If ] is with finite semivariation onR andR sube Ω(]) thenthere exists a unique extension ]

1 Σ(]) rarr 119871(119864 119865) of ] such

that ]1satisfies the following conditions

(a) ]1is countably additive on Σ(])

(b) ]1 = 120583lowast on Σ(])

(c) ]1has finite semivariation ]

1

(d) ]1 extends ]

Proof (a) Let 120582 = 120583lowast|Σ(]) Obviously 120582 is finite and countably

additive on Σ(]) From this we obtain that if (119860119899)119899is a

sequence of members of Σ(]) such that 119860119899 0 then

120582(119860120579) asymp 0 for all 120579 isin N

infin By Lemma 27 the assumptions

of Theorem 14 are satisfied so there exists a unique finiteadditive extension ]

1 Σ(]) rarr 119871(119864 119865) of ] which is

absolutely continuous with respect to 120582 Then ]1is uniformly

continuous on Σ(]) (Lemma 13) Lemma 10 now leads to]1(119860120579) asymp 0 for all 120579 isin N

infin which is our claim

(b) Let 119860 isin Σ(]) According to Lemma 27 there exists119861 isin

lowast

R so that 119861 asymp 119860 It is well known that if ]1

is absolutely continuous with respect to 120582 then ]1 is

absolutely continuous with respect to 120582 We conclude fromLemma 13 that ]

1 is uniformly continuous hence that

]1(119860) asymp ]

1(119861) Since ] and ]

1agree on R ]

1 and

] agree on R By using the transfer principle they agreeon lowastR so that ]

1(119861) = ](119861) We also have that ] and

]lowast agree on R (Lemma 26) so a repeated application of thetransfer principle enables us to write ] = ]lowast on lowastR We getthat ]lowast and 120583lowast agree on lowastR because they agree on Ω(]) andR sube Ω(]) (apply again the transfer principle) It follows that]lowast(119861) = 120583lowast(119861) and finally that ]

1(119860) asymp 120583

lowast

(119861) Now 119861 asymp 119860implies 120583lowast(119861) asymp 120583lowast(119860) so ]

1(119860) asymp 120583

lowast

(119860) with both partsbeing standard this clearly forces ]

1 = 120583lowast on Σ(])

(c) On account of (b) we have that ]1has finite semivari-

ation on Σ(])(d) Since 120583lowast is countably additive on Σ(]) (b) we obtain

that ]1 is countably additive on Σ(]) As ]lowast is countably

additive on Ω(]) and R sube Ω(]) we have by Lemma 26 that] = ]lowast = 120583lowast on R Thus ] is countably additive on RTherefore ]

1 extends ] both being countably additive on

Σ(]) respectively and onR

6 Extension of Set Function withFinite Variation

Finally we deal with the extension of set functions withfinite variation As mentioned in Section 1 we are concernedin this section with the study of how the extension of setfunctions with finite semivariation implies the extension ofset functions with finite variation We use the same notationsas in the previous section and we will reduce the problem tothe previous case It is known that for any finite additive setfunction ] R rarr 119883with ](0) = 0 we can choose the spaces119864 and 119865 so that the semivariation of ] relative to these spacesis equal to the variation of ] If119883 is a normed space we have awell-known connection between semivariation and variation[5]


Proposition 29 The semivariation of the set function ] R rarr 119883 sub 119871(119864C) relatively to the spaces 119864 and C is equalto the variation of ]

This result will be used in the next theorem But first wemake some additional observations related to the precedencetheorems and lemmas Note that if we choose 119864 so that119883 is embedded in 119871(119864C) not only the variation and thesemivariation are equal but alsoR sube Ω(])

Lemma 27 of Section 5 shows a density property of thering 119877 in the 120575-ring Σ(]) for the topology induced by thesemimetric 120588(119860 119861) We now expand sigma additive vectormeasures with finite variation from the ringR to 120575-ring Σ(])which is wider than the 120575-ring Σ(R) generated byR More-over if the function is with finite variation and countablyadditive it is automatically variationally semiregular ThenTheorem 28 leads to the following result [5]

Theorem 30 Let 119883 be a Banach space R a ring of setsand ] R rarr 119883 a countably additive measure with finitevariationThen ] can be extended uniquely to a vector measure]1 Σ(]) rarr 119883 such that(a) ]1is countably additive on Σ(])

(b) |]1| = 120583lowast on Σ(])

(c) ]1has finite variation

(d) |]1| extends |]|

Remark 31 In the construction presented by Lewis [27] theauthor uses a technique similar to that of the extension ofset functions with finite variation Basically this techniquecarries over to that of Caratheodory process Here notingthat variation and semivariation are equal in some particularcases we could get the extension of set functions with finitevariation as a particular case of the extension of set func-tions with finite semivariation In addition to these resultsnonstandard proofs of these classical measure theory resultsare found to be more intuitive and easier than the standardproofs

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author wishes to express his gratitude to the anonymousreferee for a number of helpful comments

References

[1] P R HalmosMeasureTheory D van Nostrand Company NewYork NY USA 1950

[2] N Dunford and J T Schwartz Linear Operators I GeneralTheory Interscience Publishers New York NY USA 1958

[3] G Fox ldquoExtension of a bounded vector measure with values ina reflexive Banach spacerdquo Canadian Mathematical Bulletin vol10 pp 525ndash529 1967

[4] S Gaina ldquoExtension of vector measuresrdquo Revue Roumaine deMathematique Pures et Appliquees vol 8 pp 151ndash154 1963

[5] N Dinculeanu Vector Measures Hochschulbucher fur Mathe-matik VEB Deutscher Berlin Germany 1966

[6] N Dinculeanu ldquoOn regular vector measuresrdquoActa ScientiarumMathematicarum vol 24 pp 236ndash243 1963

[7] Gr Arsene and S Stratila ldquoProlongement des mesures vecto-riellesrdquo Revue Roumaine de Mathematique Pures et Appliqueesvol 10 pp 333ndash338 1965

[8] N Dinculeanu and I Kluvanek ldquoOn vector measuresrdquo Proceed-ings of the LondonMathematical SocietyThird Series vol 17 pp505ndash512 1967

[9] G Fox ldquoInductive extension of a vector measure under a con-vergence condition rdquoCanadian Journal of Mathematics vol 20pp 1246ndash1255 1968

[10] I Kluvanek ldquoOn the theory of vector measures IIrdquo Matemat-icko-Fyzikalny Casopis Slovenskej Akademie Vied vol 16 pp76ndash81 1966 (Russian)

[11] SOhba ldquoExtensions of vectormeasuresrdquoYokohamaMathemat-ical Journal vol 21 pp 61ndash66 1973

[12] G G Gould ldquoExtensions of vector-valued measuresrdquo Proceed-ings of the London Mathematical Society vol 3 no 16 pp 685ndash704 1966

[13] M Sion ldquoOuter measures with values in a topological grouprdquoProceedings of the London Mathematical Society vol 19 pp 89ndash106 1969

[14] L Drewnowski ldquoTopological rings of sets continuous set func-tions integration Irdquo Bulletin de lrsquoAcademie Polonaise des Sci-ences Serie des Sciences Mathematiques Astronomiques etPhysiques vol 20 pp 269ndash276 1972

[15] L Drewnowski ldquoTopological rings of sets continuous setfunctions integration IIrdquo Bulletin de lrsquoAcademie Polonaise desSciences Serie des Sciences Mathematiques Astronomiques etPhysiques vol 20 pp 277ndash286 1972

[16] L Drewnowski ldquoTopological rings of sets continuous set func-tions integration IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sci-ences Serie des Sciences Mathematiques Astronomiques etPhysiques vol 20 p 445 1972

[17] M Takahashi ldquoOn topological-additive-group-valued mea-suresrdquo Proceedings of the Japan Academy vol 42 pp 330ndash3341966

[18] P A Loeb ldquoConversion from nonstandard to standardmeasurespaces and applications in probability theoryrdquo Transactions ofthe American Mathematical Society vol 211 pp 113ndash122 1975

[19] S Albeverio R Hoslashegh-Krohn J E Fenstad and T LindstroslashmNonstandard Methods in Stochastic Analysis and MathematicalPhysics vol 122 Academic Press Orlando Fla USA 1986

[20] K D Stroyan and J M Bayod Foundations of InfinitesimalStochastic Analysis vol 119 North-Holland Publishing CoAmsterdam The Netherlands 1986

[21] W A J Luxemburg ldquoA general theory of monadsrdquo in Applica-tions of Model Theory to Algebra Analysis and Probability pp18ndash86 Holt Rinehart and Winston New York NY USA 1969

[22] C W Henson and L C Moore Jr ldquoNonstandard analysis andthe theory of Banach spacesrdquo in Nonstandard AnalysismdashRecentDevelopments vol 983 of Lecture Notes in Mathematics pp 27ndash112 Springer Berlin Germany 1983

[23] R T Zivaljevic ldquoLoeb completion of internal vector-valuedmeasuresrdquo Mathematica Scandinavica vol 56 no 2 pp 276ndash286 1985


[24] H Osswald and Y Sun ldquoOn the extensions of vector-valuedLoeb measuresrdquo Proceedings of the American MathematicalSociety vol 111 no 3 pp 663ndash675 1991

[25] J Diestel and B Faires ldquoOn vector measuresrdquo Transactions ofthe American Mathematical Society vol 198 pp 253ndash271 1974

[26] J Diestel and J J Uhl Jr Vector Measures vol 15 ofMathemat-ical Surveys American Mathematical Society Providence RIUSA 1977

[27] P W Lewis ldquoExtension of operator valued set functions withfinite semivariationrdquo Proceedings of the AmericanMathematicalSociety vol 22 pp 563ndash569 1969

[28] A E Hurd and P A Loeb An Introduction to Nonstandard RealAnalysis vol 118 of Pure and Applied Mathematics AcademicPress Orlando Fla USA 1985

[29] WOrlicz ldquoAbsolute continuity of vector-valued finitely additiveset functions Irdquo Studia Mathematica vol 30 pp 121ndash133 1968

[30] C E Rickart ldquoDecomposition of additive set functionsrdquo DukeMathematical Journal vol 10 pp 653ndash665 1943

[31] R A Oberle ldquoCharacterization of a class of equicontinuous setsof finitely additivemeasureswith an application to vector valuedBorel measuresrdquo Canadian Journal of Mathematics vol 26 pp281ndash290 1974

[32] W M Bogdanowicz and R A Oberle ldquoDecompositions offinitely additive vector measures generated by bands of finitelyadditive scalar measuresrdquo Illinois Journal of Mathematics vol19 pp 370ndash377 1975

[33] H D Walker ldquoUniformly additive families of measuresrdquo Bul-letin Mathematique de la Societe des Sciences Mathematiques dela Republique Socialiste de Roumanie vol 18 no 1-2 pp 217ndash222 1974

[34] G G Gould ldquoIntegration over vector-valued measuresrdquo Pro-ceedings of the London Mathematical Society vol 3 no 15 pp193ndash225 1965

[35] M Davis Applied Nonstandard Analysis Pure and AppliedMathematics John Wiley amp Sons New York NY USA 1977

[36] J K Brooks ldquoOn the existence of a control measure for stronglybounded vector measuresrdquo Bulletin of the American Mathemat-ical Society vol 77 pp 999ndash1001 1971

[37] R G Bartle N Dunford and J Schwartz ldquoWeak compactnessand vector measuresrdquo Canadian Journal of Mathematics vol 7pp 289ndash305 1955

[38] S Ohba ldquoOn vector measures Irdquo Proceedings of the JapanAcademy vol 46 pp 51ndash53 1970

[39] P Porcelli ldquoTwo embedding theorems with applications toweak convergence and compactness in spaces of additive typefunctionsrdquo Journal of Mathematics and Mechanics vol 9 pp273ndash292 1960

[40] S Leader ldquoThe theory of 119871119901-spaces for finitely additive setfunctionsrdquo Annals of Mathematics vol 58 pp 528ndash543 1953

[41] J K Brooks and N Dinculeanu ldquoWeak compactness and con-trolmeasures in the space of unboundedmeasuresrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 69 pp 1083ndash1085 1972

[42] J Dieudonne ldquoSur les espaces de Kotherdquo Journal drsquoAnalyseMathematique vol 1 pp 81ndash115 1951

[43] T Traynor ldquoS-bounded additive set functionsrdquo in Vector andOperator Valued Measures and Applications pp 355ndash365 Aca-demic Press New York NY USA 1973

[44] I Kluvanek ldquoIntegrale vectorielle de Daniellrdquo Matematicko-Fyzikalny Casopis Slovenskej Akademie Vied vol 15 pp 146ndash161 1965 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the range of ] is contained in a vector space 119883 In the lit-erature there are two main approaches to proving theoremsconcerning the extension of vector measures

The first approach is due to the properties which have tobe satisfied by themeasure which follows to be extended If119883is a Banach space we can mention the solutions due to Gaina[4] (in the case when ] has finite variation) Dinculeanu [5 6](if ] is regular and of bounded variation) Arsene and Stratila[7] (when ] is bounded above in norm by a positivemeasure)Dinculeanu andKluvanek [8] (in the casewhen ] is absolutelycontinuous with respect to a positive measure) and Fox [9](] satisfies a monotone-convergence condition)

The second approach relies on the conditions which haveto be satisfied by the range of ] In this category we have theresults of Fox [3] Kluvanek [10] Ohba [11] or Gould [12]Generalizing the notion of outer measure to vector-valuedmeasures and imitating the ]lowast-measurability procedure inorder to obtain a Lebesgue extension of ] Gould [12] showedthat a necessary and sufficient condition for ] to have aLebesgue extension is that the following property should holdfor the image space119883 of ](119860) If (119909

119899) is a sequence in119883whose norms have a positive

lower bound then there exists for arbitrary positive119870 a finitesubsequence (119909

119899119903

) such that sum119903119909119899119903

gt 119870This suggests a connection between weak completeness

and property (119860) and it is shown in [12] that all weaklycomplete spaces satisfy property (119860) In Section 3 we presentanother proof of this result It is easy to verify that 119897infin doesnot hold property (119860)Thus 119897infin is not weakly complete Moregenerally Banach spaces which are infinite-dimensionalfunction spaces with a supremum norm fail to satisfy prop-erty (119860) and therefore are not weakly complete For Hilbertspaces property (119860) is satisfied A direct proof of the fact thatproperty (119860) is satisfied by 119897119901 (1 le 119901 lt infin) is much harder

For a masterful study of measures with values in a topo-logical group we refer the reader to Sion [13] or Drewnowski[14ndash16] When 119883 is a commutative complete topologicalgroup and ] is of bounded variation then there is a very niceextension theorem by Takahashi [17]

The starting point in nonstandard theory of measurespaces is a paper [18] by Loeb He gave a way to construct newrich standard measure spaces from internal measure spacesThis construction has been used in recent years to establishnew standard results in a variety of different areas Some ofthese results can be found in [19] or [20]

Also nonstandard analysis has proved to be a naturalframework for studying vector measures and Banach spacesThe central construction in this approach is the notion ofnonstandard hull introduced by Luxemburg [21]This notionis not only a useful tool in studying vector measures andBanach spaces but also a construction arising naturallythroughout nonstandard analysis For a deeper discussion ofnonstandard hulls and their applications we refer the readerto the survey paper [22] of Henson and Moore

Zivaljevic [23] has pursued the extension problem usingthe nonstandard hull of 119883 Osswald and Sun [24] treated thesame problem from a different point of view the extensionof additive vector measures has been made using the internal

control measures Furthermore the authors present a differ-ent approach of Loebrsquos [18] in order to construct a countablyadditive vector measure from internal locally convex space-valued measure

In this work we study the extension of vector valued setfunctions in the framework of nonstandard analysis

The plan of this paper is as followsSection 2 is devoted to some preliminary results on stan-

dard vector measures Using concurrent relations we alsoobtain a result concerning the concentration of 119904-boundedvectormeasures on a specific set from the nonstandard exten-sion ofR Moreover the principle of extension by continuity[2] will be proved using nonstandard techniques

In Section 3 we give a nonstandard characterization ofthe absolute continuity and an extension theorem for vectormeasures The proof still uses nonstandard arguments Sincereflexive spaces are weakly complete and the weakly completespaces satisfy property (119860) we can rederive the Foxrsquos result[3] Some results of Gould [12] will be reproved in a differentmanner For this we use a result of Diestel et al (see [25] or[26]) on 119904-bounded measures To obtain these results Gouldhas used Pettisrsquo theorem Our approach does not use thisresult

In Section 4 we address the issues of the existing controlmeasures We tackle this subject by using the extension of setfunctions and linking these extensions to control measures

Section 5 deals with the extension of set functions withfinite semivariation The results in this section (for whichwe present a nonstandard proof) were originally obtained byLewis [27]

The last section shows that the extension of set functionswith finite variation is a particular case of the extension of setfunctions with finite semivariation

We adopt the nonstandard framework of [28] Thenonstandard model used in this paper is assumed to besufficiently saturated for our needs In what follows N

infin

denotes the set lowastN N where lowastN is the extension of N in ourmodel

2 Preliminaries

In this section we collect some basic definitions notationsand elementary results about standard vector measures Alsothe principle of extension by continuity will be proved usingnonstandard techniques

The terminology concerning families of sets set func-tions and so forth will be in general that of [2] or [1] LetR+denote the nonnegative reals and let N denote the set of

positive integers Sets are denoted as 119860 119861 119862 0means theempty set Notation for set operations is that commonly usedin particular119860Δ119861means (119860 119861)cup (119861 119860) Everywhere in thesequel R denotes a ring of subsets of a nonempty fixed setΩ the casesR should be a 120575-ring or 120590-ring will be explicitlyspecified The complement (in Ω) of a set 119860 is denoted by119860119888 Symbols and for sequences of sets or of reals have

their usual meaning A set function 120583 R rarr [0infin] will becalled a submeasure (subadditivemeasure in the terminologyof Orlicz [29]) if 10 120583(0) = 0 20 120583(119860 cup 119861) le 120583(119860) + 120583(119861)whenever 119860 119861 isin R and 119860 cap 119861 = 0 30 120583(119860) le 120583(119861) if



0 120598) = 119860 isin




1and 119864

2


1) + ](119864

2)








such that 119864119899 0 we have ](119864



119899=1119864119899








119899](119860119899) = 0







119899) 0









119894=1119909119894 119909119894isin 119860 for 119894 = 1 119899 If 119880








](119860) + ](119861)(119894119907) If (119864

119899) sub P(Ω) and (119864

119899) 0 then ](119864

119899) 0

(119907) If (119864119899) sub 120591(R) and (119864

119899) 119864 then ](119864

119899) ](119864)


119899=1119864119899) le suminfin

119899=1](119864119899)


119899=1119864119899) le sum

infin

119899=1](119864119899)

(119894119894) If (119864119899) sub P(Ω) (119864

119899) 119864 then ](119864

119899) ](119864)





119899=1119860119899then120588(119860 119860

119899) = ](119860

119860119899) 0(119894119894119894) If (119860

119899)119899subeR and (119860

119899)119899 0 then ](119860

119899) 0






1 1198612isinlowast





⋃infin






119899isinR




] (119860 119864) leinfin

sum

119896=1

] (119860119899 119864119899) (1)


119896=1119864119896 we have 119865

119899isinR

(119865119899) and 119860

119899 119865119899sube ⋃119899

119896=1119860119896 119864119896 Lemma 5(i) implies

] (119860119899 119865119899) le

119899

sum

119896=1

] (119860119896 119864119896) (2)

so ](119860119899 119865119899) isin 119881 Moreover 119864 119865

119899isin 120591(R) and by

Lemma 4(ii)

] (119864 119865119899) = ] (119864 119865

119899) 0 (3)


119899) isin 119881 for 119899 large


119899) for each


] (119860 119865119899) le ] (119860 119864) + ] (119864 119865

119899) (4)



] (119860 119864) = ] (119860 119864) lt 120576 (5)


119899)119899subeR and119860 = ⋃infin


that

119860 119860119899sube (119860 119865

119899)⋃ (119860

119899 119865119899) (6)

Lemma 5(i) implies

] (119860 119860119899) le ] (119860 119865

119899) + ] (119860

119899 119865119899) (7)






120588(Ω119860119888


119899) =





119899)119899of 119860 with 119886

119899rarr




120579) asymp 119891(119910)


lowast

N

119889119884(119891(119886119899) 119891(119910)) lt 120576 Then 119881






[119889119883(1199101 1199102) lt 120575 997904rArr 119889

119884(119891 (1199101) 119891 (119910

2)) lt

120576

2]

(forall1199101 1199102isin 119860)

(8)


[119889119883(1199101 1199102) lt 120575 997904rArr 119889

119884(119891 (1199101) 119891 (119910

2)) lt

120576

2]


119860)

(9)

Let 1199091 1199092isin 119883 with 119889

119883(1199091 1199092) lt 1205752 There are 119910

1 1199102isin

lowast


119883(1199101 1199102) le

119889119883(1199101 1199091) + 119889

119883(1199091 1199092) + 119889

119883(1199092 1199102) so 119889

119883(1199101 1199102) lt 120575

Hence 119889119884(119891(1199101) 119891(119910


119892(1199091) asymp 119891(119910

1) 119892(119909

2) asymp 119891(119910


119889119884(119892 (1199091) 119892 (119909

2))

le 119889119884(119892 (1199091) 119891 (119910

1)) + 119889

119884(119891 (1199101) 119891 (119910

2))

+ 119889119884(119891 (1199102) 119892 (119909

2))

(10)


tinuousLet 119892



1198921(119910) asymp 119892


1(119910) so

that 119892(119909) asymp 1198921(119909) As 119892(119909) and 119892










119899 120598119899) isin dom(Ξ)



of 119864119894we have ]

119894(119865) lt 120598 119894 = 1 119899 Setting 119864 = ⋃119899

119894=1119864119894 we















lowast

R) [120583 (119861) lt 120575 997904rArr ] (119861) lt 120576] (11)




119891 (119860 119861) = 119860⋃119861 119892 (119860 119861) = 119860⋂119861

ℎ (119860 119861) = 119860 119861

(12)



1∘ 1198611) (119860

2∘ 1198612) sube

(11986011198602) cup (119861

1 1198612) Hence

120588 (1198601∘ 1198611 1198602∘ 1198612) le 120588 (119860

1 1198602) + 120588 (119861

1 1198612) (13)


1 1198602) asymp 0 120588(119861









1 R rarr











1(1198641cup 1198642) asymp ]1(1198651cup 1198652)

]1(1198641 1198642) asymp ]1(1198651 1198652) and ]



]1(1198641⋃1198642) asymp ] (119865

1⋃1198652) = ] (119865

1 1198652) + ] (119865

2)

= ]1(1198651 1198652) + ]1(1198652)

]1(1198651 1198652) + ]1(1198652) asymp ]1(1198641 1198642) + ]1(1198642)

(14)


Now

]1(1198641 1198642) + ]1(1198642) = ]1(1198641) + ]1(1198642) (15)


2are disjoint


]1(1198641⋃1198642) asymp ]1(1198641) + ]1(1198642) (16)


]1(1198641⋃1198642) = ]1(1198641) + ]1(1198642) (17)








119899)119899a



infin ](119860120579) asymp


1(119860119899) rarr 0 as






1 120590(R) rarr 119883












119899) in 119883 so that 119909

119899 ge 120575

for all 119899 and sum119903119909119899119903


)



vectorsum119903119909119899119903











1









) such that sum119903](119864119899119903

) =


](⋃119903119864119899119903





lim120583(119864)rarr0

] (119864) = 0 (18)






lim120595(119864)rarr0

]1(119864) = 0 (19)


lim120583(119864)rarr0

] (119864) = 0 (20)




119899

sum

119896=1

120583 (119860119896) le 119872 (21)



and

lim120583(119864)rarr0

] (119864) = 0 (22)








lim120583(119864)rarr0

] (119864) = 0 119864 isinR (23)


compact






Theorem 53]







sup 10038171003817100381710038171003817sum ] (119860119894) 119909119894

10038171003817100381710038171003817(24)





infsum] (119860119894) (25)




120583lowast

(119860) = sup ]lowast (119861) (27)






119899rarrinfin](119860

119899) = 0







1 Σ(]) rarr 119871(119864 119865) of ] such



(b) ]1 = 120583lowast on Σ(])


1

(d) ]1 extends ]







1 Σ(]) rarr 119871(119864 119865) of ] which is





lowast





]1(119860) asymp ]

1(119861) Since ] and ]

1agree on R ]

1 and




1(119860) asymp 120583

lowast


1(119860) asymp 120583

lowast


1 = 120583lowast on Σ(])














(b) |]1| = 120583lowast on Σ(])






Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 120598) = 119860 isin




1and 119864

2


1) + ](119864

2)








such that 119864119899 0 we have ](119864



119899=1119864119899








119899](119860119899) = 0







119899) 0









119894=1119909119894 119909119894isin 119860 for 119894 = 1 119899 If 119880








](119860) + ](119861)(119894119907) If (119864

119899) sub P(Ω) and (119864

119899) 0 then ](119864

119899) 0

(119907) If (119864119899) sub 120591(R) and (119864

119899) 119864 then ](119864

119899) ](119864)


119899=1119864119899) le suminfin

119899=1](119864119899)


119899=1119864119899) le sum

infin

119899=1](119864119899)

(119894119894) If (119864119899) sub P(Ω) (119864

119899) 119864 then ](119864

119899) ](119864)





119899=1119860119899then120588(119860 119860

119899) = ](119860

119860119899) 0(119894119894119894) If (119860

119899)119899subeR and (119860

119899)119899 0 then ](119860

119899) 0






1 1198612isinlowast





⋃infin






119899isinR




] (119860 119864) leinfin

sum

119896=1

] (119860119899 119864119899) (1)


119896=1119864119896 we have 119865

119899isinR

(119865119899) and 119860

119899 119865119899sube ⋃119899

119896=1119860119896 119864119896 Lemma 5(i) implies

] (119860119899 119865119899) le

119899

sum

119896=1

] (119860119896 119864119896) (2)

so ](119860119899 119865119899) isin 119881 Moreover 119864 119865

119899isin 120591(R) and by

Lemma 4(ii)

] (119864 119865119899) = ] (119864 119865

119899) 0 (3)


119899) isin 119881 for 119899 large


119899) for each


] (119860 119865119899) le ] (119860 119864) + ] (119864 119865

119899) (4)



] (119860 119864) = ] (119860 119864) lt 120576 (5)


119899)119899subeR and119860 = ⋃infin


that

119860 119860119899sube (119860 119865

119899)⋃ (119860

119899 119865119899) (6)

Lemma 5(i) implies

] (119860 119860119899) le ] (119860 119865

119899) + ] (119860

119899 119865119899) (7)






120588(Ω119860119888


119899) =





119899)119899of 119860 with 119886

119899rarr




120579) asymp 119891(119910)


lowast

N

119889119884(119891(119886119899) 119891(119910)) lt 120576 Then 119881






[119889119883(1199101 1199102) lt 120575 997904rArr 119889

119884(119891 (1199101) 119891 (119910

2)) lt

120576

2]

(forall1199101 1199102isin 119860)

(8)


[119889119883(1199101 1199102) lt 120575 997904rArr 119889

119884(119891 (1199101) 119891 (119910

2)) lt

120576

2]


119860)

(9)

Let 1199091 1199092isin 119883 with 119889

119883(1199091 1199092) lt 1205752 There are 119910

1 1199102isin

lowast


119883(1199101 1199102) le

119889119883(1199101 1199091) + 119889

119883(1199091 1199092) + 119889

119883(1199092 1199102) so 119889

119883(1199101 1199102) lt 120575

Hence 119889119884(119891(1199101) 119891(119910


119892(1199091) asymp 119891(119910

1) 119892(119909

2) asymp 119891(119910


119889119884(119892 (1199091) 119892 (119909

2))

le 119889119884(119892 (1199091) 119891 (119910

1)) + 119889

119884(119891 (1199101) 119891 (119910

2))

+ 119889119884(119891 (1199102) 119892 (119909

2))

(10)


tinuousLet 119892



1198921(119910) asymp 119892


1(119910) so

that 119892(119909) asymp 1198921(119909) As 119892(119909) and 119892










119899 120598119899) isin dom(Ξ)



of 119864119894we have ]

119894(119865) lt 120598 119894 = 1 119899 Setting 119864 = ⋃119899

119894=1119864119894 we















lowast

R) [120583 (119861) lt 120575 997904rArr ] (119861) lt 120576] (11)




119891 (119860 119861) = 119860⋃119861 119892 (119860 119861) = 119860⋂119861

ℎ (119860 119861) = 119860 119861

(12)



1∘ 1198611) (119860

2∘ 1198612) sube

(11986011198602) cup (119861

1 1198612) Hence

120588 (1198601∘ 1198611 1198602∘ 1198612) le 120588 (119860

1 1198602) + 120588 (119861

1 1198612) (13)


1 1198602) asymp 0 120588(119861









1 R rarr











1(1198641cup 1198642) asymp ]1(1198651cup 1198652)

]1(1198641 1198642) asymp ]1(1198651 1198652) and ]



]1(1198641⋃1198642) asymp ] (119865

1⋃1198652) = ] (119865

1 1198652) + ] (119865

2)

= ]1(1198651 1198652) + ]1(1198652)

]1(1198651 1198652) + ]1(1198652) asymp ]1(1198641 1198642) + ]1(1198642)

(14)


Now

]1(1198641 1198642) + ]1(1198642) = ]1(1198641) + ]1(1198642) (15)


2are disjoint


]1(1198641⋃1198642) asymp ]1(1198641) + ]1(1198642) (16)


]1(1198641⋃1198642) = ]1(1198641) + ]1(1198642) (17)








119899)119899a



infin ](119860120579) asymp


1(119860119899) rarr 0 as






1 120590(R) rarr 119883












119899) in 119883 so that 119909

119899 ge 120575

for all 119899 and sum119903119909119899119903


)



vectorsum119903119909119899119903











1









) such that sum119903](119864119899119903

) =


](⋃119903119864119899119903





lim120583(119864)rarr0

] (119864) = 0 (18)






lim120595(119864)rarr0

]1(119864) = 0 (19)


lim120583(119864)rarr0

] (119864) = 0 (20)




119899

sum

119896=1

120583 (119860119896) le 119872 (21)



and

lim120583(119864)rarr0

] (119864) = 0 (22)








lim120583(119864)rarr0

] (119864) = 0 119864 isinR (23)


compact






Theorem 53]







sup 10038171003817100381710038171003817sum ] (119860119894) 119909119894

10038171003817100381710038171003817(24)





infsum] (119860119894) (25)




120583lowast

(119860) = sup ]lowast (119861) (27)






119899rarrinfin](119860

119899) = 0







1 Σ(]) rarr 119871(119864 119865) of ] such



(b) ]1 = 120583lowast on Σ(])


1

(d) ]1 extends ]







1 Σ(]) rarr 119871(119864 119865) of ] which is





lowast





]1(119860) asymp ]

1(119861) Since ] and ]

1agree on R ]

1 and




1(119860) asymp 120583

lowast


1(119860) asymp 120583

lowast


1 = 120583lowast on Σ(])














(b) |]1| = 120583lowast on Σ(])






Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899=1119860119899then120588(119860 119860

119899) = ](119860

119860119899) 0(119894119894119894) If (119860

119899)119899subeR and (119860

119899)119899 0 then ](119860

119899) 0






1 1198612isinlowast





⋃infin






119899isinR




] (119860 119864) leinfin

sum

119896=1

] (119860119899 119864119899) (1)


119896=1119864119896 we have 119865

119899isinR

(119865119899) and 119860

119899 119865119899sube ⋃119899

119896=1119860119896 119864119896 Lemma 5(i) implies

] (119860119899 119865119899) le

119899

sum

119896=1

] (119860119896 119864119896) (2)

so ](119860119899 119865119899) isin 119881 Moreover 119864 119865

119899isin 120591(R) and by

Lemma 4(ii)

] (119864 119865119899) = ] (119864 119865

119899) 0 (3)


119899) isin 119881 for 119899 large


119899) for each


] (119860 119865119899) le ] (119860 119864) + ] (119864 119865

119899) (4)



] (119860 119864) = ] (119860 119864) lt 120576 (5)


119899)119899subeR and119860 = ⋃infin


that

119860 119860119899sube (119860 119865

119899)⋃ (119860

119899 119865119899) (6)

Lemma 5(i) implies

] (119860 119860119899) le ] (119860 119865

119899) + ] (119860

119899 119865119899) (7)






120588(Ω119860119888


119899) =





119899)119899of 119860 with 119886

119899rarr




120579) asymp 119891(119910)


lowast

N

119889119884(119891(119886119899) 119891(119910)) lt 120576 Then 119881






[119889119883(1199101 1199102) lt 120575 997904rArr 119889

119884(119891 (1199101) 119891 (119910

2)) lt

120576

2]

(forall1199101 1199102isin 119860)

(8)


[119889119883(1199101 1199102) lt 120575 997904rArr 119889

119884(119891 (1199101) 119891 (119910

2)) lt

120576

2]


119860)

(9)

Let 1199091 1199092isin 119883 with 119889

119883(1199091 1199092) lt 1205752 There are 119910

1 1199102isin

lowast


119883(1199101 1199102) le

119889119883(1199101 1199091) + 119889

119883(1199091 1199092) + 119889

119883(1199092 1199102) so 119889

119883(1199101 1199102) lt 120575

Hence 119889119884(119891(1199101) 119891(119910


119892(1199091) asymp 119891(119910

1) 119892(119909

2) asymp 119891(119910


119889119884(119892 (1199091) 119892 (119909

2))

le 119889119884(119892 (1199091) 119891 (119910

1)) + 119889

119884(119891 (1199101) 119891 (119910

2))

+ 119889119884(119891 (1199102) 119892 (119909

2))

(10)


tinuousLet 119892



1198921(119910) asymp 119892


1(119910) so

that 119892(119909) asymp 1198921(119909) As 119892(119909) and 119892










119899 120598119899) isin dom(Ξ)



of 119864119894we have ]

119894(119865) lt 120598 119894 = 1 119899 Setting 119864 = ⋃119899

119894=1119864119894 we















lowast

R) [120583 (119861) lt 120575 997904rArr ] (119861) lt 120576] (11)




119891 (119860 119861) = 119860⋃119861 119892 (119860 119861) = 119860⋂119861

ℎ (119860 119861) = 119860 119861

(12)



1∘ 1198611) (119860

2∘ 1198612) sube

(11986011198602) cup (119861

1 1198612) Hence

120588 (1198601∘ 1198611 1198602∘ 1198612) le 120588 (119860

1 1198602) + 120588 (119861

1 1198612) (13)


1 1198602) asymp 0 120588(119861









1 R rarr











1(1198641cup 1198642) asymp ]1(1198651cup 1198652)

]1(1198641 1198642) asymp ]1(1198651 1198652) and ]



]1(1198641⋃1198642) asymp ] (119865

1⋃1198652) = ] (119865

1 1198652) + ] (119865

2)

= ]1(1198651 1198652) + ]1(1198652)

]1(1198651 1198652) + ]1(1198652) asymp ]1(1198641 1198642) + ]1(1198642)

(14)


Now

]1(1198641 1198642) + ]1(1198642) = ]1(1198641) + ]1(1198642) (15)


2are disjoint


]1(1198641⋃1198642) asymp ]1(1198641) + ]1(1198642) (16)


]1(1198641⋃1198642) = ]1(1198641) + ]1(1198642) (17)








119899)119899a



infin ](119860120579) asymp


1(119860119899) rarr 0 as






1 120590(R) rarr 119883












119899) in 119883 so that 119909

119899 ge 120575

for all 119899 and sum119903119909119899119903


)



vectorsum119903119909119899119903











1









) such that sum119903](119864119899119903

) =


](⋃119903119864119899119903





lim120583(119864)rarr0

] (119864) = 0 (18)






lim120595(119864)rarr0

]1(119864) = 0 (19)


lim120583(119864)rarr0

] (119864) = 0 (20)




119899

sum

119896=1

120583 (119860119896) le 119872 (21)



and

lim120583(119864)rarr0

] (119864) = 0 (22)








lim120583(119864)rarr0

] (119864) = 0 119864 isinR (23)


compact






Theorem 53]







sup 10038171003817100381710038171003817sum ] (119860119894) 119909119894

10038171003817100381710038171003817(24)





infsum] (119860119894) (25)




120583lowast

(119860) = sup ]lowast (119861) (27)






119899rarrinfin](119860

119899) = 0







1 Σ(]) rarr 119871(119864 119865) of ] such



(b) ]1 = 120583lowast on Σ(])


1

(d) ]1 extends ]







1 Σ(]) rarr 119871(119864 119865) of ] which is





lowast





]1(119860) asymp ]

1(119861) Since ] and ]

1agree on R ]

1 and




1(119860) asymp 120583

lowast


1(119860) asymp 120583

lowast


1 = 120583lowast on Σ(])














(b) |]1| = 120583lowast on Σ(])






Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119899 120598119899) isin dom(Ξ)



of 119864119894we have ]

119894(119865) lt 120598 119894 = 1 119899 Setting 119864 = ⋃119899

119894=1119864119894 we















lowast

R) [120583 (119861) lt 120575 997904rArr ] (119861) lt 120576] (11)




119891 (119860 119861) = 119860⋃119861 119892 (119860 119861) = 119860⋂119861

ℎ (119860 119861) = 119860 119861

(12)



1∘ 1198611) (119860

2∘ 1198612) sube

(11986011198602) cup (119861

1 1198612) Hence

120588 (1198601∘ 1198611 1198602∘ 1198612) le 120588 (119860

1 1198602) + 120588 (119861

1 1198612) (13)


1 1198602) asymp 0 120588(119861









1 R rarr











1(1198641cup 1198642) asymp ]1(1198651cup 1198652)

]1(1198641 1198642) asymp ]1(1198651 1198652) and ]



]1(1198641⋃1198642) asymp ] (119865

1⋃1198652) = ] (119865

1 1198652) + ] (119865

2)

= ]1(1198651 1198652) + ]1(1198652)

]1(1198651 1198652) + ]1(1198652) asymp ]1(1198641 1198642) + ]1(1198642)

(14)


Now

]1(1198641 1198642) + ]1(1198642) = ]1(1198641) + ]1(1198642) (15)


2are disjoint


]1(1198641⋃1198642) asymp ]1(1198641) + ]1(1198642) (16)


]1(1198641⋃1198642) = ]1(1198641) + ]1(1198642) (17)








119899)119899a



infin ](119860120579) asymp


1(119860119899) rarr 0 as






1 120590(R) rarr 119883












119899) in 119883 so that 119909

119899 ge 120575

for all 119899 and sum119903119909119899119903


)



vectorsum119903119909119899119903











1









) such that sum119903](119864119899119903

) =


](⋃119903119864119899119903





lim120583(119864)rarr0

] (119864) = 0 (18)






lim120595(119864)rarr0

]1(119864) = 0 (19)


lim120583(119864)rarr0

] (119864) = 0 (20)




119899

sum

119896=1

120583 (119860119896) le 119872 (21)



and

lim120583(119864)rarr0

] (119864) = 0 (22)








lim120583(119864)rarr0

] (119864) = 0 119864 isinR (23)


compact






Theorem 53]







sup 10038171003817100381710038171003817sum ] (119860119894) 119909119894

10038171003817100381710038171003817(24)





infsum] (119860119894) (25)




120583lowast

(119860) = sup ]lowast (119861) (27)






119899rarrinfin](119860

119899) = 0







1 Σ(]) rarr 119871(119864 119865) of ] such



(b) ]1 = 120583lowast on Σ(])


1

(d) ]1 extends ]







1 Σ(]) rarr 119871(119864 119865) of ] which is





lowast





]1(119860) asymp ]

1(119861) Since ] and ]

1agree on R ]

1 and




1(119860) asymp 120583

lowast


1(119860) asymp 120583

lowast


1 = 120583lowast on Σ(])














(b) |]1| = 120583lowast on Σ(])






Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Now

]1(1198641 1198642) + ]1(1198642) = ]1(1198641) + ]1(1198642) (15)


2are disjoint


]1(1198641⋃1198642) asymp ]1(1198641) + ]1(1198642) (16)


]1(1198641⋃1198642) = ]1(1198641) + ]1(1198642) (17)








119899)119899a



infin ](119860120579) asymp


1(119860119899) rarr 0 as






1 120590(R) rarr 119883












119899) in 119883 so that 119909

119899 ge 120575

for all 119899 and sum119903119909119899119903


)



vectorsum119903119909119899119903











1









) such that sum119903](119864119899119903

) =


](⋃119903119864119899119903





lim120583(119864)rarr0

] (119864) = 0 (18)






lim120595(119864)rarr0

]1(119864) = 0 (19)


lim120583(119864)rarr0

] (119864) = 0 (20)




119899

sum

119896=1

120583 (119860119896) le 119872 (21)



and

lim120583(119864)rarr0

] (119864) = 0 (22)








lim120583(119864)rarr0

] (119864) = 0 119864 isinR (23)


compact






Theorem 53]







sup 10038171003817100381710038171003817sum ] (119860119894) 119909119894

10038171003817100381710038171003817(24)





infsum] (119860119894) (25)




120583lowast

(119860) = sup ]lowast (119861) (27)






119899rarrinfin](119860

119899) = 0







1 Σ(]) rarr 119871(119864 119865) of ] such



(b) ]1 = 120583lowast on Σ(])


1

(d) ]1 extends ]







1 Σ(]) rarr 119871(119864 119865) of ] which is





lowast





]1(119860) asymp ]

1(119861) Since ] and ]

1agree on R ]

1 and




1(119860) asymp 120583

lowast


1(119860) asymp 120583

lowast


1 = 120583lowast on Σ(])














(b) |]1| = 120583lowast on Σ(])






Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










](⋃119903119864119899119903





lim120583(119864)rarr0

] (119864) = 0 (18)






lim120595(119864)rarr0

]1(119864) = 0 (19)


lim120583(119864)rarr0

] (119864) = 0 (20)




119899

sum

119896=1

120583 (119860119896) le 119872 (21)



and

lim120583(119864)rarr0

] (119864) = 0 (22)








lim120583(119864)rarr0

] (119864) = 0 119864 isinR (23)


compact






Theorem 53]







sup 10038171003817100381710038171003817sum ] (119860119894) 119909119894

10038171003817100381710038171003817(24)





infsum] (119860119894) (25)




120583lowast

(119860) = sup ]lowast (119861) (27)






119899rarrinfin](119860

119899) = 0







1 Σ(]) rarr 119871(119864 119865) of ] such



(b) ]1 = 120583lowast on Σ(])


1

(d) ]1 extends ]







1 Σ(]) rarr 119871(119864 119865) of ] which is





lowast





]1(119860) asymp ]

1(119861) Since ] and ]

1agree on R ]

1 and




1(119860) asymp 120583

lowast


1(119860) asymp 120583

lowast


1 = 120583lowast on Σ(])














(b) |]1| = 120583lowast on Σ(])






Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













sup 10038171003817100381710038171003817sum ] (119860119894) 119909119894

10038171003817100381710038171003817(24)





infsum] (119860119894) (25)




120583lowast

(119860) = sup ]lowast (119861) (27)






119899rarrinfin](119860

119899) = 0







1 Σ(]) rarr 119871(119864 119865) of ] such



(b) ]1 = 120583lowast on Σ(])


1

(d) ]1 extends ]







1 Σ(]) rarr 119871(119864 119865) of ] which is





lowast





]1(119860) asymp ]

1(119861) Since ] and ]

1agree on R ]

1 and




1(119860) asymp 120583

lowast


1(119860) asymp 120583

lowast


1 = 120583lowast on Σ(])














(b) |]1| = 120583lowast on Σ(])






Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(b) |]1| = 120583lowast on Σ(])






Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article nonstandard methods in measure...

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