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## Material Information- Title:
- Generalizations of the Vitali-Hahn-Saks and Nikodym theorems
- Creator:
- Jewett, Robert Stanley, 1945-
- Place of Publication:
- Gainesville, Fla
- Publisher:
- University of Florida
- Publication Date:
- 1971
- Copyright Date:
- 1971
- Language:
- English
- Physical Description:
- v, 62 leaves : ; 28 cm.
## Subjects- Subjects / Keywords:
- Absolute convergence ( jstor )
Banach space ( jstor ) Integers ( jstor ) Mathematical sequences ( jstor ) Mathematical theorems ( jstor ) Mathematical vectors ( jstor ) Mathematics ( jstor ) Perceptron convergence procedure ( jstor ) Series convergence ( jstor ) Topological theorems ( jstor ) Dissertations, Academic -- Mathematics -- UF Mathematics thesis Ph. D Measure theory ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis - University of Florida.
- Bibliography:
- Bibliography: leaves 60-61.
- Additional Physical Form:
- Also available on World Wide Web
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 022050111 ( AlephBibNum )
13440789 ( OCLC ) ACY4608 ( NOTIS )
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Generalizations of the Vitali-Hahn-Saks and Nikodym Theorems By Robert Stanley Jewett A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1971 ACKNOWLEDGMENTS The author would like to thank his advisor Dr. James K. Brooks and the members of his supervisory committee, Dr. David Drake, Dr. Gene Hemp, Dr. Steve Saxon, and Dr. Zoran Pop-Stojanovic for their help in writing this dissertation. TABLE OF CONTENTS Abstract ----------------------------------------------------------iv Introduction ----------------------------------------------------- Chapter 1: Unconditional Convergence ------------------------------6 Chapter 2: Strongly Bounded Set Functions -------------------------1 Chapter 3: Extending the Schur Theorem ----------------------------20 Chapter b: Extensions of the Vitali-Hahn-Saks and Nikodym Theorems ---------------------------------------35 Chapter 5: A Counterexample ---------------------------------------56 References --------------------------------------------------------60 Biographical Sketch ------------------------------------------ 62 iii Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy GENERALIZATIONS OF THE VITALI-HAHN-SAKS AND NIKODYM THEOREMS By Robert Stanley Jewett August, 1971 Chairman: Dr. James K. Brooks Major Department: Mathematics The Vitali-Hahn-Saks and Nikodym theorems are two very closely related results concerning sequences of countably additive, real valued measures. The purpose of this dissertation is to improve on these theorems both in the statements and the proofs. That is, stronger theorems will be shown to be true, and the proofs will be easier that those usually given. The author and J. K. Brooks extend the Vitali-Hahn-Saks and Nikodym theorems to the finitely additive vector case. In 1969 J. K. Brooks used a result of Schur concerning uniform convergence of a double sequence of real numbers to derive simple, direct proofs of the Vitali-Hahn-Saks and Nikodym theorems. In Chapter 3 of this dissertation the above mentioned theorem of Schur is extended to the case where the double sequence is contained in a Banach space, and then this new result is used to obtain a short proof of the Nikodym theorem with the measures taking their values iv in a Banach space. However, the Vitali-Hahn-Saks and Nikodym theorems do not generalize directly to the case where the measures are finitely additive, bounded, and Banach valued, and the additional hypothesis of "strongly bounded" must be assumed. With this assumption, the theorems may be extended, and the proofs follow the same general line as those of Brooks'for the countably additive, scalar case. Instead of the Schur theorem, a generalized form of a result of Phillips is used, and the Nikodym theorem follows, but the proof of the Vitali-Hahn-Saks theorem requires a difficult construction with measurable sets. At the end of Chapter 4 a discussion is given to indicate how the theorems may be extended even further to the case where the measures take values in a locally convex linear topo- logical space instead of a Banach space. INTRODUCTION This dissertation concerns the Vitali-Hahn-Saks and Nikodym theorems (7). In 1907 Vitali (21) proved the following theorem: if f: [o, I R are Lebesgue integrable functions converging almost everywhere to then ., j f KcS and Jc exist and are o 0o equal if and only if the indefinite integrals of the fn are uni- formly absolutely continuous with respect to Lebesgue measure. Hahn (10) then proved in 1922 that if -: [0,1]-) f are Lebesgue integrable functions and if ~l, fd t exists for every measur- able set E then the indefinite integral of the fh are uniformly absolutely continuous with respect to Lebesgue measure, and they converge to a set function which is also absolutely contirious with respect to Lebesgue measure. In 1933 Nikodym (12) generalized these two results when he proved that if (S, ,/ ) is a measure space and if n is a sequence of /u -continuous measures such that ALM, X(E) exists for all E in 1- then the An are uni- formly absolutely continuous with respect to / A little later Saks (19) gave another proof of Nikodym's theorem, and it became known as the Vitali-Hahn-Saks theorem. Using the countable addi- tivity of the control measure, Saks defined a complete metric on the measurable sets, and then applied the Baire Category theorem. If the control measure had been only finitely additive, the metric would not necessarily have been complete, and this technique could not have been used. Nikodym (12) proved tht if Un is a sequence of measures defined on a 0--algebra such that /uE) : ,, (E) exists for all E then /u is countably additive and the countable additivity of the /u~ is uniform. This is known as the Nikodym theorem, and in most books it is proved as a consequence of the Vitali-Hahn-Saks theorem. Rickart (18) and Phillips (16) extended these theorems to the case where the measures are Banach space valued. The proofs of these results rely heavily on the fact that the set functions in question are countably additive, and there seems to be no way to generalize their techniques in order to prove Vitali-Hahn-Saks and Nikodym theorems for finitely additive, vector valued set functions. However, in 1969 Brooks (3) gave a short proof of the Nikodym theorem and then used that result to prove the Vitali-Hahn-Saks theorem. His proof used a difficult construction with measurable sets along with a theorem of Schur (20) concerning the equivalence of weak and strong convergence in Il In the same paper he gave a short proof that the truth of the two theorems in the scalar case would imply their truth in the vector case. An extension of the Vitali-Hahn-Saks theorem to the finitely additive, scalar case has been proved by Ando (1), but his theorem was not extended to the case where the measures are vector valued. In 1970 Brooks and the author proved Vitali-Hahn-Saks ana Aikodym theorems for finitely additive, strongly bounded, vector valued set functions, where strongly bounded is defined as in (17). The strongly bounded hypothesis is especially needed in the Nikodym theorem, because the statement would not even make sense if the measures were only bounded; the strongly bounded hypothesis may be dropped in the Vitali-Hahn-Saks theorem if the control measure is assumed to be finite. The proofs of the theorems are built around a generalization of a theorem of Phillips (15) and an extension of a technique of Darst (6). The first two chapters provide the background material for the results proved later in the dissertation. However, Theorems 1.3 and 1.h are stated only for completeness, and are not used in subsequent proofs. Although most of the theorems in these capters may be found in Hilderbrandt (11) or Rickart (17), new proof tre given by the author. The purpose of Chapter 3 is to generalize a theorem of Schur (20) and to use the new result to construct a direct proof of the vector case of the Nikodym theorem. Since all of the measures are assumed to be countably additive, the theorems rely only on the results of Chapter 1, and not of Chapter 2. Chapters 3 and L, re independent, and no theorem of Chapter 3 is used in a proof in Chapter 4. The most important theorems are proved in Chapter 4. Both the Vitali-Hahn-Saks and Nikodym theorems are extended to the finitely additive case, and it is shown how the newer version of the Nikodym theorem implies the usual version when the measures are countably additive. The concept of "strongly bounded" is of utmost importance here, but Corollary 4.5 and the remark following the statement of Theorem h.7 indicate when the condition may be dropped. CHAPTER I UNCONDITIONAL CONVERGENCE Throughout this dissertation the following notation will be used. 7 is a O'-algebra of subsets of a set 5 (l) is the power set of the natural numbers and E and A are generic notations for sets belonging to t and (M) respectively. R denotes.the real numbers. X is a Banach space over the real or complex numbers, and I)( is the norm of E 6 3 is the conjugate space of with the norm of an element in defined in the usual way. Let n,n E 7 with n Define (71,>]=- ,7+I ...j7 , f7,oo)r ny,7+I,... j If 4 is a real valued measure defined on a T -algebra 2 , then ful is the total variation of / which is defined in the standard way. In this chapter we define the term "unconditional convergence and establish other conditions equivalent to it. Definition: Let X K ,n= f,,, If t: 7r - is a one-to-one, onto mapping, then XT(n) inI is called a rearrangement of Xr 0 The series tn is said to be unconditionally convergent if the sum of every rearrangement of its terms converges. Remark: A series that converges absolutely will converge unconditionally, but the converse is not necessarily true. In fact, Dvoretzky and Rogers (8) proved in 1950 that the two con- ditions are equivalent if and only if the Banach space in question is finite dimensional. The difference between absolute and uncon- ditional convergence was further demonstrated by Brouxs (L) in 1969 when he showed that weakly and strongly integrable functions with range in a Banach space correspond to unconditionally and absolutely convergent series in the space. 00 Theorem 1.1: Let X) ,f :n 1,X,' The series Z X ,tI II converges unconditionally if and only if every subsequential 00 sum converges, that is, if L1 7 is a subsequence, then .-X converges. Proof: Let 1 XKV be a subsequence and assume the sum does not converge. Then there exist E>o and sequences fPiI and jVi where for each i P V K -l = {(- X :'K:Pi', j'"7 Then form the follow- ing rearrangement of X.I : Xn ... X7 V ,iXP ). )()I I ) nz P^ 3 X 3 ) V3 y*tp h - jnv', The partial sums of this sequence are not Cauchy, so the sum 00 does not exist; hence ZXn does not converge unconditionally. n=i Conversely, assume 19K is a rearrangement of ltI and that the sum of {Kj does not exist. Write p 4 yv if 9p precedes y when considered in the ordering of the sequence Xn . There exist 6>0 and sequences -m;) and I(n where for all i , )< n% and /6K >) Let P, be such that for all KrL,, h,] and all t> P , K 4 Rearrange IYj: KI= -,) *,71 so that the elements appear in the same order as they appear in PXnj and call this rearrangement X5n, ,.X I Let MA >P, Again rearrange i ( i: K ^ '"=- 'l appropriately, and label this arrangement YlX i", X. Since 9t I5 if r ,,n,] , S 1 'l -n we have that fX i,", Xn~ ,Xn ). xI"* XIwz has the same order as Xn . In this manner, define 71 K: IK=l,. and {[k) so that I wi*- for every i / K Then Xh K is a subsequence whose sum does not exist; thus the converse is proved. Q.E.D. 00 Theorem 1.2: Let Xh ,f7=I,X,* If rX converges unconditionally, then for all > O there exists an N such that for all L Ci',O0) IJ; I1< Proof: Assume the conclusion is false. Then there exists > 0 such that for all N there exists A C [J,0o) such that Let NI= I Then there exists A C.C oo) such that hI~I[ J> Also, there exists a finite subset of I , such that I > Let N = .:a There exists SC Nr+1,00) such that I X I so there exists a finite set rI c A such that l > Note that 1 n = 1 D where O denotes the null set. Also, for all SE t, c 5 f In a similar way define an infinite number of finite sets such that for all i 1Zf ll> and if LjJ and 5E ,f j F we have 5< . VE Lv P.L is a subsequence of .hj If m is any positive integer, there exists io such that for all nv E lo ' V, SV Writing f'L [lv, i > tv we have that Xvj I .> so is not a Cauchy sequence. Therefore the sum of the XAv does not exist and this contradicts the uncon- 00 ditional convergence of Z h . nr Theorem 1.3: Let >E (X 7= ,1,2,-- Then the series -Xn converges unconditionally if, and only if, 71=1 &(m, (X,)| = o uniformly in X where 1X The conclusion means that if 4 >0 there exists an N so that for all X) such that IX*1 / we have F IX()l< . coo Proof: Assume the series 2:- converges unconditionally, -Az 11 and let 6 )0 By Theorem 1.2 there exists N so that n 2C for every AC LNWo) Let X be chosen so that IX~ 1 I and define A,:fnE[f,oo): : x) Then A, UAI f ,oo) ,and (1/=0 Hence -X0) ntAr, m Ez. | Cnf | Note that for all > N /IvACi] I" therefore ifA i [ N/ X m) -I rN1 .[ It follows tat nCN,t~]t t ?n,6,n i .I Likewise, it may be shown that I i)l ; consequently IX(X) I I ) "- and the first implication of the theorem m= N is proved. Conversely, assume that the sum of the X) does not converge unconditionally. Then by Theorem 1.1 there is a subsequence PXKl where the sum of the XHK does not exist; hence there is an >O and sequences fPiL and fVyl where P < Vi P+I and for all , SCX > 6 By the Hahn-Banach theorem there exists X for each i so that IX l I and < iXi -- K= P K P(; ^ ( ^)1.K, I For every N there is a value of L so that Vj ip "'l CN, Co) therefore Jx(X )I x(Xk)\ 6 . Consequently the limit in the conclusion of the theorem does not converge uniformly in X where IXYI I and the second implication is proved. Q.E.D. Definition: Let 1E, 3X : I,,,''" and let G be the collection of all finite sets c contained in the natural numbers. Define a partial order on E by ", 2 if G, 2 We say oo the series Z(h converges to a Moore-Smith limit Xo if for every > O there exists Oo such that for all (02 Go , Shf- o< ( .< Since X is complete, an equivalent condition would be for every > 0 there exists Go so that if GC, 2 we have In K x 6 oo Theorem 1.h: Let ZgE X =1:I,Z,-" The series ZIX converges unconditionally if, and only if, it converges a Moore-Smith limit. oo Proof: Assume ZX converges unconditionally. Let >O . By Theorem 1.2 there exists N *so that for all LCCN,oo) , I t 6 Let z [oI,N] If 0 x, 2 O, we have that J,: -_ j MJXnl itnX- ZXhj ) + | t E, n.7 E (+ A ; Is Ego Ia '- T n I n ,- 1 + I | I .t- <;( because both ',- o and OL-Qo are contained in 4,oo0) Therefore the series converges to a Moore-Smith limit. Conversely, assume the series does not converge unconditionally, so there exists a subsequence I XKI whose sum does not exist. There exist E)o and sequences {PI iV j where P and for each i I .K- > E Let 'o be any finite subset of the natural numbers, and choose i so that [nPL,-"nv nV O Letting T : oU CPi, *" Vi V we have that (T (To and | n ec nrco and it follows that the series does not have a Moore-Smith limit. This completes the proof. Q.E.D. CHAPTER 2 STRONGLY BOUNDED SET FUNCTIONS Many theorems that are true for countably additive measures do not have extensions when the measures are finitely additive and bounded. However, as will be seen in Chapter 4, some of these results do carry over to the finitely additive case when the hypothesis of "strongly bounded" is assumed. In this chapter "strongly bounded" is defined and theorems are proved which will be used in Chapter 4. Definition: Let /.Z-~X be a set function. we say ,4 is strongly bounded (s-bounded) if &7iu(E,)'O whenever IEKJ is a sequence of disjoint sets in . This definition was first given by Rickart (17). Theorem 2.1. Let /4:' be s-bounded and finitely additive. Then /u is bounded. Proof: Assume/L is not bounded. Then there exists E such that C) >1 Suppose there exists a monotone decreasing sequence of sets FKCE such that (FKt+I) )I> (FK))+ for all K Then 1( -FK:,)-1K) ,and (- )IJ > /Ia) - S1(X .(). Therefore f KF, is a pairwise disjoint sequence of sets, whereas i does not converge to zero, which contradicts the fact that is s-bounded. Consequently there exists G, E such that I/A( 1 and for all LC G, ( ) /-I())I4J There exists such that > 1 (G)>| +2, so \ -N\- n Crl\>l If there exists a monotone decreasing sequence of sets F iR C , such that /pl(F+i)f/ >/ (F1)I/1 for all K we obtain a contra- diction as before. Therefore there exists G6C R-G, such that /1( )I> and for all L C&; (L1) (z )l+i i ( J D There exists P such that ()I| )l )/I3 so 0j(P- ) UG~I 4 ( P)O- (l(Pi)\ t rf] )|> > and there exists G3C P-(G UG&) such that \ >63) 1 and for all L C (3 ) j(L)L< |lG3)1+ 1 G3f(1 u G,)=O Continue in this manner to define a sequence of disjoint sets ~Gi such that for all L f/(Gi)( > This clearly contradicts the fact that A is s-bounded, hence the theorem is proved. Q.E.D. Remark: The converse to the above theorem is false. Let Z be the Borel sets of the real line, and let 6- 4or(R)RZ , the class of essentially bounded measurable functions, where is Lebesgue measure (9). Define /uE) ~= E the characteristic function of the set E /u is bounded because for all E in 2 , I E oL 1 However, letting E '=(, l) we see that E,) 0= 1 for all 7Z so / is not s-bounded. On the other hand, if k is the real numbers the converse holds, so the concepts of s-bounded and bounded coincide in the case of scalar measures. Theorem 2.2: Let /,:. )R be bounded and finitely additive. Then / is s-bounded. Proof: Assume the conclusion is false. Then there exists a sequence of disjoint sets EKJ where J / ; does not converge to zero. Therefore there exists ( )> and an infinite subcollection EKJ C I[EtJ such that for all L /(EK') Let AC be such that for all 6 (EK) > 6 and for all Ei -' /( .) 4- Either A or i is infinite. Without loss of generality, assume L is infinite. Let /1>O. Let r be a finite subset of A with cardinality greater than . Then ( ) ( \ Since was chosen arbitrarily, this proves/ is not bounded a contradiction. Q.E.D. Definition: Let/U: -- Define u(E) : .\f )'E E f uis called the semi-variation of/ . Theorem 2.3: Let/ Z. X- ThenA is s-bounded if and only if/ A is s-bounded. Proof: Assume is not s-bounded. Then there exists a /4 disjoint to zero. sequence EK such that I does not converge That is, there exists 6 >0 and a subsequence IEijC E4 such that for all i I(E >} According to the defin- ition of I for all i there exists F C E. such that I K.. Therefore does not converge to zero, and is not s-bounded. This proves that/i being s-bounded implies that/ is s-bounded. Conversely, since I/L(E) 4 (E) for every E in 2, 6m (E =o0 implies that J ([k) 0 Q.E.D. Theorem 2.1: Let/ -- be finitely additive and s-bounded. If IEl is a sequence of disjoint sets in Z then o0 / (EK) converges unconditionally. Krl Proof: Assume the conclusion is false. Then by Theorem 1.1 there exists a subsequence such that / 'I does not converge and is therefore not Cauchy. Hence there exists > 0 such that for all N there exists p > 1 where Obtain / such that I and / ( > irF, and then choose p, Q such that 6 P Z Y and )> / . In this manner we define infinite collections P and I such that for all j, qP < ( Pi j l/(E)J >E Let F = EK Then Fj is a sequence of disjoint sets t L) F i Then u and *)- (EK )|>6 so IF; does not J LJ converge to zero and/ is not s-bounded. This proves the theorem. Q.E.D. CHAPTER 3 EXTENDING THE SCHUR THEOREM In 1969 Brooks (3) proved the Nikodym theorem by using a result of Schur (20). In this chapter the Schur theorem is ex- tended to the vector case, and the result is then used to obtain a very short proof of the Nikodym theorem for countably additive, Banach valued measures. The proof of the first the -em is a good example of the "sliding hump" technique. Theorem 3.1: Let in .Il, Assume that for all i the series x is unconditionally convergent, 4=1 and also that: (*) for all A A- rX roO . i neA Then the limit in (*) is uniform with respect to A That is, for all 6 there exists (depending on 6 ) such that for all (1 and for all A in 6 . 20 Remark: If ) is the real numbers, then Ziin converges absolutely for every L and the conclusion is that i n-i Proof: Assume the limit in (*) does not exist uniformly with respect to Then there exist 6 >o and sequences i IuKj such that I A > To simplify notation, let I= K so I XKn> 6 for all K Let K, Since n r" there exists a finite set FCA such that MiAj '17 -- ; due to the unconditional convergence of , there exists R, such that for all ACFR, oo) I n ) Let T, U i(u{RT() If Ac T+l1, ) we have r, ( D ,and 7 I > U -L -X I For every n sK=O so there exists IJ such that T o for all i AI<5>T Then Zin 7fT if LI C In thl,]} Let K: 1 then t t -+ X ad T l and X 1 I rK -,TJ -CT,- >nr-l | xI6KgnI;T (- A/Tn > Therefore there exists a finite set cAKC IT s t set 62,CAg-C/,Tj such that K- . Since 1 XKj converges unconditionally, there exists K ?t= I such that for all A CCR,oo) n Kh Let T= nsa" { P2 (} ) so that for all ACT++i,o3) we have that /PA= A = o Note that (, 0 = 0 hence = PtPf 7 -' p A Kn when ACT+f,o) Also Ix nX because C [I,T;] Therefore Xi+ X-+4 1<: P uVP uL 1 a @ I l, - for AC T+ioo) Also, since P U a C cT +, oo) we have that l r ,> . In the same manner as before, obtain I such that for all 17 Iinl' and let K3= Then we have that IfA 3- ,T > and there exists a finite set C C 3f-D,T] such that I K >P Choose R3 such that for all A C [R3,.) 'A 3 A and let T3 In4t(3 u {R3) If c [T3 t,o0) we then have h1r, UpAUP 3 >-j Since G uA C T +1,oo) we have that t P, g u.r, uAI>- and u3 UA C T +1,oo) implies that acv uP, U r3 v 3 Continue to define an Infinite number of Pj and K. and u J let Then for all j | > and i> J Oi does not converge to zero, which contradicts'the hypothesis. Q.E.D. Remark: Theorem 3.1 may be used to prove the equivalence of weak and strong convergence in. iF the space of absolutely convergent series. Let fi f E( Lr l,l,." Each element of 11 is a sequence or real numbers, and we write fi as i injgl and f as x 00 The sequence {fJ converges strongly to f if i Mf i. I-, X 0 and fI converges weakly to f if converges to F) for every 9* in We know that the conjugate space of 1, is , the space of all bounded sequences, and that if : i then Theorem 3.2: Let Zf. f I ':=I,A,." Then the fL converge strongly to f if, and only if, they converge weakly to f. Proof: Strong convergence always implies weak convergence, so I will prove the converse. Assume the fi converge weakly to f Writing f# as Xi0: and f as 4 let i Xi -X .* For any A define y* in oo by: 01 if K E ' Then z because the fi converge to f weakly. Therefore, by Theorem 3.1 0o we have that A -/X ih-XhI:= and the f( converge h'l strongly to Note that we only needed to consider those elements of joo which take on the value O or I Q.E.D. Theorem 3.3: Let X. ,7:1, ... Assume that for all in converges unconditionally, and that: (M) for all A A, Z 'in exists. i nEa Then the limit in (*) is uniform with respect to A ; that is, for every >O there exists I such that for all J I and for all Jn 7 t L ( Proof: This proof will be divided into six parts. Part I: If {CiK is a subsequence, then j (XiL i~~r)- o uniformly with respect to A K ntL Proof of I: Let o( = XK XK+1 Then for all K 00oo OdK 7 converges unconditionally and for all / J Kn=O . 7L= K TI0A By Theorem 3.1, Y/A 2Ek n O uniformly with respect to , and the conclusion, of I follows. Part II: For all 6> 0 there exists N such that for P oo all integers L and for all p N in ~ E L 6 7n1 i 7t:( I Proof of II: Suppose that we deny the conclusion. Then there exists 6)> such that for all N there exist I and P2N such that: (1) oo E 26 S Choose 6 as indicated in the above statement. Let No be chosen arbitrarily. Then I claim that for all I there exist P o00 L>I and P No such that Z Xin- lIn 2 If this were not the case we could find Io such that for all > o , P o o00 P No ) 2in ZDi7 For all i Xn converges, n j4: i I T sP oo so there exists NL such that for all P2 #; X l TXit < . I lh~I ? i Let R = ty NorJ i : L' / ' o] Then for all i and all P 00 p> R Xin i (in and this contradicts the assump- tion (1). Therefore, since No was picked arbitrarily, we have that for all N and I there exist L >I P N such that: (2) i ,- n i 00 f (00 Since 6n F- in exists, IZXi7n is a Cauchy sequence, so there exists K such that for all L, K , ~ J Let ',:K Since -Yi,71 converges, z I &7t: I P o we obtain NI such that for all P2 N, ,I -n According to (2), there exist V>i2 V oN such that v o0 | V V X n Y > Therefore, since -iZ )n, + / l' f 7I=1 ^ '-1 1.' have that I -i t Lin II-- -n 00,l 00 6-e_-4=_ Let A, :[,V E I t 13 Now choose ~ such that for all 5- In xn i ^ T . According to (2), obtain i3 > i A 2 42 such that I n3 =3 x 6 and since ij and i, are both greater Sa h Combining than K we also have that z Combining S= I l f nI A these inequalities, we obtain 13 > Let Continuing this process, define { } and 7,) ,?f:,2,-"- such that 7nAmXn %E k^ h > for all n Then one (X 'X +) does not converge to zero uniformly with respect to A which contradicts Part I; hence, the con- clusion of Part II is proved. S S Part III: If Se ) then in X where 1:I 7:I X),.= X( Proof of III: This follows because the sums in question are finite, hence the limit may be interchanged. Part IV: For all A Xk exists. Proof of IV: If A is finite, the conclusion holds, so assume A is infinite. Let = IKj Letting K i nK we see that {Z1< satisfies the hypothesis of Theorem 3.3; consequently, we may apply the results of Parts I, II, and III to the double sequence {~CKJ To show 2 7t exists it suffices 00 to show that 1 exists, where s/- <= . K=:I . In order to prove 2-6K exists, I will show that K^ Kvy is a Cauchy sequence. Let >0 By Part II there exists an N such that for every i and v2 N .K- l-iK . Therefore, if V,S2 AJ we have 1 K; b i < for all . From Part III choose I such that for all i I , gla = K- 3 and Yf K i-. Then for all i I , EVA V 700 Since 6 was chosen arbitrarily, it follows that _ is Cauchy, and the conclusion of Part IV is proved. For all A Xn exists, and Theorem 1.1 indicates that 00 c on converges unconditionally. Part V: For all a A i (Note that the second sum is defined by the result of Part IV.) Proof of V: Since the conclusion is true for finite , we may assume 6 is infinite. Let n] i k X g We need to show that on on AwL ij- i IK and since j[Ki satisfies the hypothesis Z K=1 K: of Theorem 3.3, we may apply the results of the first four parts. Let >o According to Parts II and IV, choose N so E o0 W oa that I IK -K By Part III, choose I so that for every 1 I , N N oo K 1/K [C Then for every (2 I K K I oo oo Since E was chosen arbitrarily, n -iK = xi K . L K=I K=I Part VI: The proof of the conclusion of the theorem. 00 Proof of VI: Let c<'-= Xji- X By Part IV, D'n con- 00 verges unconditionally, so for all i in converges uncon- ditionally. Also, for all L Ai -'7 = -Zt Z( n -) Snb n 7i A X- x = O where the last inequality results from Part V. Therefore, by Theorem 3.1, for all 6>0 there exists I such that for all i21 and all A > /\ ii t / tEX ~- in t; I ,where the last equality follows from Part V. This concludes the proof of the theorem. Q.E.D. Theorem 3.3 is the desired extension of the Schur theorem, and will be used to prove Theorem 3.4, the Nikodym theorem. Definition: Let / : Z/-- 6 / is countably additive if /#(=E - (E whenever iEs} is a sequence of disjoint sets in . Remark: /L being countably additive implies / is s-bounded because if/U is not s-bounded, then there exist 6>O and disjoint sets {En such that for all n / (F I > ; hence Ku(LE|) is not a Cauchy sequence and/U is not countably additive. Definition: Let U1 : Z --' be countably additive, nrl,Z,- , and let [EiJ be a sequence of disjoint sets in Z- The /I are uniformly countably additive if for all 6>0 there exists M such that for all m n and for all n L:/ ti (E)I E . Theorem 3.4 (Nikodym): Let /: u x X be countably additive, n=: 2,-. "If/ A(E) 4~L/,( ) exists for every E then/t is countably additive and the countable additivity of the /,, is uniform. Proof: Let JEg, be a pairwise disjoint sequence of sets 31 in and let (~ =/k (Fn). The double sequence ~Xi' satisfies the hypothesis of Theorem 3.3, so by the result of Part III of the proof of that theorem, for every 5>o there exists N P 00 such that for every integer 4 and all P2N s> in- = p n.)-a (E. ) Therefore the 7L are uniformly countably additive. On the other hand, /L =I C, = A e AE) gin E- n the last equality following from Part V 00 00 of the proof of Theorem 3.3. Since X : T (E) we have that / u Eln ) = I.(E) and / is countably additive. Q.E.D. n=1 We give a proof due to Brooks and Mikusinski (5) of a result of Banach (2). Lemma 3.5: Let C be a separable Banach space, and let { be a bounded sequence in Then there exists a subsequence {iK* such that YnK,() exists for every X in Proof: This proof consists of a standard Cantor diagonal process. Let iX L be a dense subset of X and choose M>O such that | for each n so 9nix, is a bounded sequence of real numbers, and there exists a subsequence Y,l of / i so that K YK (x,) exists. Likewise, there exists a subsequence of such that m. ,K(X ) exists. Continue in this manner to define subsequences sIonI so that +(, ,K j C r i: ,KIJ KK and ,K(XL) exists for every n For each integer K let Y .KK " Then {YK is a subsequence of tnI and for all j , ' K n(xj) exists. We now show that YK( ) exists for each X k Let S(K. XE and 6O Since f(jj1 is dense in X ,pick Xj so that IXj -I< M, and choose R such that for all K, 1 R I~ K t J X) I Then for all K,~ R , 7V 7) (Xj) x)4- O (xj) n'(x) f + = Therefore the sequence JIuK) is Cauchy, and A ( () exists. Q.E.D. 00 Definition: Let X h =, I, *Z, Xn is weakly 71=1 * unconditionally convergent if for every subset A of the integers, there exists XA such that TX A
fhAThe following theorem is known as the Orlicz-Pettis theorem, (13, 14), and the proof is due to Brooks and Mikusinski (5). Theorem 3.6: If X is a separable Banach space, then a series 2 t converges unconditionally if, and only if, it con- verges weakly unconditionally. Proof: Unconditional convergence always implies weak uncon- ditional convergence. In order to prove the converse, I will assume that the series converges weakly unconditionally but not unconditionally, and then arrive at a contradiction. 00 If F-X does not converge unconditionally, there exist an 0>o and finite disjoint sets Ai such that InAj>l for every integer L Letting we have that ._' -nfZ^ E' also converges weakly unconditionally. According to the Hahn-Banach theorem, there exist Xm such that /I=1 and I X(-)I= I for every integer ) . S (-yj 00 By Lemma 3.5 there is a subsequence Xi of I such that ^-4Xi xtL(X) exists for every X in M and to simplify notation I will assume that Mi= for all i Let 0Q = X(3). 34 For any A there exists 2a so that Z L = OA) for all Therefore, m't, Tdi n, ) a X; n A) Since 2 Xin) exists for all \ and uncon- L 1t6b ditional convergence implies absolute convergence in the scalar oo field, it follows that X*ir()I,) ~o Therefore, the double ?tI sequence 4inj satisfies the hypothesis of Theorem 3.3. Letting 0d : = d we have from Part IV of the proof oo of Theorem 3.3 that Z 107 theorem we have that din n n = uniformly in . I 'iA C.6 ? Choose positive integers t, and NX such that for all n 2M da I\ and for all iNW and all A ih.-g. 1 * Let )9n>tA Then I2J |^Hn 4 jI^l St '- d | which is a contradiction. Hence the theorem is proved. Q.E.D. CHAPTER 4 EXTENSIONS OF THE VITALI-HAHN-SAKS AND NIKODYM THEOREMS In this are extended and strongly to the whole (15). chapter the Vitali-Hahn-Saks and Nikodym theorems to the case where the measures are finitely additive bounded. The first theorem to be proved is the key chapter, and is an extension of a result of Phillips Theorem 4.1: Let 11: P(7Z) be finitely additive and s-bounded. If m a = c for all A then n 2 /Un ( ): uniformly in (That is, for every 6 0 there exists N such that for all A N and for all D , Remark: Since the / are finitely additive and s-bounded, 00 it follows from Theorem 2.2 that Z// -(t) converges uncondi- -t'l tionally. Therefore (Wt) exists for every A and the tE&A conclusion makes sense. Proof: Assume the conclusion is false. Then there exist an > 0 and sequences Ii, ~ such that I / > L= I 2 ,..... To simplify notation, assume n = . Let ijI /I. f > so there exists N, such that / '(() /6 for all c N,00) Letting 7=-A lnEI N,-1 we have I I ) 1> )=--Z,) Since j) 0O for all j we can pick L such that 1/i () / L ) converges unconditionally, so we can pick N,> ,I such that for all ACI ,l ) , IAl < /6. As a result, L Z .L( + a 2v 6 2r',-o + / 2) and therefore J~ P2 ) JE/iaf) ^) Z i > 1- 6h > .. Let ?j, lLi,a,-ij Choose i3 such that /3 / 7 and pick 3 so that for all A /6 Then, as before, una0l 3 o) / ; 3 r > and we let ?'3 3- n . Continue in this manner to define an increasing sequence of positive integers JNJ such that: -JK- I WE nK (2) nLJ' 76 for all A and K,3;-* ; (3) letting K AKnld[ K- ,1)K- (No1) we have Note: If A is a finite set disjoint from LK-IJ , then it follows from (1) and (2) that A / < l . Let 6,= f.*((,tl)]t: I h)= ,,.'- For all values of ?m , Of is an infinite set. Also the sets iJ are disjoint because fnl V implies that 2 ml' -I)-12~m -) for all values of n, and Xz Since the sequence fK) is disjoint, it then follows from the preceding statement that {u(mj l, is a sequence of disjoint sets, where UB, is defined to be Ulr :TE In other words, I have divided a countably infinite collection of sets into a countable number of countably infinite collections; and then have taken the union of each collection. Since all the sets were disjoint, all the unions are disjoint from each other. Since /Ai, is s-bounded it follows from Theorem 2.3 that / l is s-bounded, so /A (UB,)- O Hence there exists an no so that ,Ai(Bmo EIA. ?" o because i,( \)> . Let PL -.m is a countable collection of the 7? , so by the same process as above there exists P ,an infinite subset of ,such that / (Up < Again ? . Continue this process to obtain a sequence of infinite sets i2 where PmiC C and i,(Ul )/
Let r consist of the first element of r for every 7t . P is an infinite subset of TKI and for each value of h , tr-f consists of a finite number of elements. If rE then /K(, r)llI)- /K ufr-(ru )~J) iK(fr nrK]) because r . u fr-(rufrKU) f is a finite set disjoint from LNK>,NK41 , so so ufp[ Kf- t j)I) / UiK ([ fl '))] Combining all three inequalities we obtain K i r) TK -- > * Therefore / UP) if ? EP There are an infinite number of ?' in P so /iK K)U I cannot converge to zero, which contradicts the hypothesis. This proves the theorem. Q.E.D. Remark: Theorem 4.1 implies Theorem 3.1. Corollary 4.2: Let /J,: -4~ be finitely additive and s-bounded, n=lI,Z,-' Assume .& /',(E) exists for all E Then if EK) is a sequence of disjoint sets, . ~I(/ at, "-/ K) 0 uniformly in A . Proof: Define Vj .(7)-4 X as follows: A/a)= A( -i Ej( The V satisfy the hypothesis of Theorem .4, so J, (,a /n)(E) -- 1 0o uniformly A K C6 K E6 in Q.E.D. Corollary I.3: Let /IV: be finitely additive and s-bounded, ?=Il,Z,. Assume A/ (E) exists for all E 7?1 and let /u(E) =k /n?() Then / is s-bounded. Proof: Assume /U is not s-bounded. Then there exist 6>0 and a sequence of disjoint sets fEg} such that for all K , I(EK)I>E Let iI Since /mUn()= 0( there exists K, such that /1h (E) / However ,i /)(E)/ i //EK) j> so there exists z > so that ) >(E- Therefore m -~U i) ( >) I- C Because I/LrF 1(EK)O there exists KX such that K2) 3. &. |j (E )/= Ii(E )I > e so there exists 3 > > so that I)3 (F 9)I and I Z-/ S 3)( 3 )3 3 Continue this process to define sequences {n Kj where iI} is monotone increasing and for every integer , S/l ) > Therefore the sequence IZ1/ ( / x)(Hj does not converge to zero uniformly in A and this contradicts Corollary l.2. Hence the theorem is proved. Q.E.D. Definition: Let / -h be finitely additive and s-bounded, 7Z~:1,I,. We say the ~ are uniformly additive if /tm Z" X (EK 0 uniformly in A and n whenever {EK) is a pairwise disjoint sequence of sets in 2 That is, for all 6 >0 there exists J such that for all J 2 J all n and all 'i KEJO,) n A Remark: When the un are countably additive this is equi- valent to the /"u being uniformly countably additive. Theorem h.h: Let /U,.' ( be finitely additive and s-bounded, ?=/,', ," Assume u (E) = M/n('E) exists for every E Then the U, are uniformly additive. Proof: Let EK be a sequence of disjoint sets and let >O . Let V,= P(7)-4 be defined by ) )--(/ -, ( E) By Corollary 4.3 /x is s-bounded, so the X satisfy the hypothesis of Theorem h.1. Hence there exists an integer N such that: (1) fJ/-) ( < E for all A2 N and all A Since u is s-bounded, it follows from Theorem 2.4 that o0 V- I( ) converges unconditionally, so by Theorem 1.2 there exists I, such that: (2) K ,) E for all Combining inequalities (1) and (2) we have that for all t 2 N 1 ,f x(EK) M ^ 1 11E) and for all ,,o) a n f,) /(?nr,,-1-4 ) (Ex Each /L is s-bounded, so there also exists 1 such that LKEf n[r, ) I< for all 6 and all 7n N Consequently /Mh(EK) 4< (.6 for all n and A Since E was chosen KE6 n +I oo) arbitrarily, this proves that the are uniformly additive. Q.E.D. Remark: Corollary 4.3 and Theorem 4.l are the intended generalizations of the Nikodym theorem. Corollary L.5: Let /h: E-~ R be bounded and finitely additive, n =l, a,-- If /(E) ,/E ) exists for all , then the /, are uniformly additive. Proof: It follows from Theorem 2.2 that the u are s-bounded, so the conclusion follows immediately from Theorem l-.. Q.E.D. Corollary 4.6 (Nikodym): Let /An: -9 ~ be countably additive, I, 2, ** If / l E) = /n,(E) exists for every E then /u is countably additive and the countable additivity of the n is uniform. Proof: Let > 0 and let {Egi be a sequence of disjoint sets. By Corollary l.3, u is s-bounded, so i /U(EK) exists, / K=I hence there is an I, such that for all i 1 , 1 j (EK- 00X(E:) /X- By Theorem h.-, the u are uniformly countably additive, so there exists 1L such that for all n and all 2 i , I /E2) which means ni <)/ < because each ,u is countably additive. Therefore: (2) U EK) = j /A( p 0 for all 1i I . Let I- 7 IJ11J Obviously # is finitely additive, so: (3) Combining inequalities (1), (2), and (3) we have K) - = I ,ewr )h h S.E Since E was arbitrary, this implies that :- / .Q.E.D. EI)= (K ) Definition: Let /~: X and V:W '- R+. We say /I is absolutely continuous with respect to V if for every 6 > 0 there exists 6 >0 so that v(E) < implies IJE) < for all E in i1 If / : we say the /I are uniformly absolutely continuous with respect to d if for all 6>O there is a 6>o such that VE) 6 implies I/r(E)k6 for all E and n . Remark: The next theorem is a generalization of the Vitali- Hahn-Saks theorem, and is the most important result of this dissertation. Theorem h.7: Let/U : .-4- X be finitely additive and s-boun- ded, n:L, ." Assume that Am/u.(E) exists for every E Let ) be a non-negative (possibly infinite) finitely additive set function defined on If Uh is absolutely continuous with respect to ) for each L then the /n are uniformly absolutely continuous with respect to . Remark: If ) were assumed to be bounded, then the /f would automatically be s-bounded. Proof: Deny the conclusion. Then there exists an >0O 45 such that for every 6 > o there exists an E, such that )(E)4 and J1/n(E)I ) for some 7 Let E, /h, be such that vE,)
and /II(E, )I Let that /in, (E) v(E)) and /In;(E) >) Then for all Fc Ez , Now assume that E, EK , l- nfK have been chosen. Let SK+1 be such that for all : I ..K V(E) < CKI implies /hE)< 3 Now choose K+l and l+ so that v(EKti) SK+1 )K+i K and I/ (E'+,) > E Then for all FC K+ and for all Z .***i , 1/ i(F) j . Resubscripting for simplification, let /UI "-'/i We then have that: (1) for all I I (E)> and for all J : l*-l- and all FcE , JF)< Let F= : = I and assume there exists an L, >2 such that / II E Let F2 I E L- Assume FI'.FK have been chosen in this manner, along with i" iK * 46 Assume there exists it( > iK such that J/iK4I(FKnELK,) Then define FK+1I F El X+ If this process continues we obtain a countable collection of sets FK such that VKIFKF K1)1- / ,(F)nE and because of (1) and the fact that FK n E- zl ELK+I Therefore |I(,, -1:)/ -(FK - -f = f- Note that IFK-FK+I} is a sequence of disjoint sets, and S (/i4 / f j) (FgK-.+) I} does not converge to zero uniformly in 6 which contradicts Corollary 4.2. Therefore the process of choosing the FK and lK has to stop, so there exists FK lK 2., such that for all J > K, J(K E) Let f:= K H, = F /, ( , and E =p, HI Then the following three results obtain. i. (H,) Since H, cEi it follows from (1) that Jt(H1) I < E E I, /6 i. >JL HI1t Y We have that E = F, (,-F )U(F F3)U ..U(FK_,-F) U F< | 1> 6 and / (FiO )= // (F n El ) I / since F, n Ez C Ej and 2n< II( -FJ)h I/'F E)\ < --< /6 because 13 > i >2 hence 3 >' Thus for xJi -1( F +-Fj)j I+'jEi +) and e I, ( ) I |/',-F,-F ,. .-+ I,"(F-,- FK)I+ /t i (, )| ---- +.* + i2J(H I since H,:FK Therefore 16 3Z-L +. =C III. 'E ) -- Note that Ep +i (E+i- H,) U(Ep + n H,) H1 was chosen so that for all J> P /Ij(EjnH,) I Hence E ,< I 0(E, )) i(E, H,) + |it-(Ep)1,pn H,)|< V (Et Thus E-~ IFP' Note: From I and II we have that )(H ~ ~E ' Let FE= Assume that there exists Z > 1 such Lt /r.^ I F' ['t r^ ') that l/ui i 2 and let F E If we could continue to choose i indefinitely, then we would ob n a contradiction the same as before. So there exists obtain a contradiction the same as before. So there exists r and I ?.2 so that for all J > ti IF ('F n )j ) AP2 =1A / I+ 1) 8 r + p, + .- Ep,+pt (H, uH.) Then H, n H, = D because Hn C l I) (and and E (H nH =E (Recall the construction of then obtained. r'. ,1 H) 3 Since H~C EF1 C EP..l 1V (Ha) The following results are and P,+4 >2- we have that 2 +132 SI E- ( F2- ) (K )- F3) U'. U " < 1 ) l -j F - [lal"iH) , since HI. r and tL > I we have U =F=' E"') Since F(') Sso () c (C)C P1 E; L Ei,1C Ep,,l that 6 (IF (F 1( E < 1/ 1 1 1 2 S+ E P, + +Lz ( 6 .2 Likewise I/ ( j - E3 F (I) - because i > J4i J for all J:1,,..K-I Hence < +- + Let Let Hz = F ( and and K = ^-^') W . P;L+ (1) E, P.,+ L + 1 .1 J+1 '+ n F + )( F/; ) ECI < n J+, " 49 E - 8$ 1r+ ^"^)| and it follows that I/aN I III'. ffor all . Since HH the sets (Ep,+p n H,) , (nEp pl nH ) and (EP +Pi+ (H I u ) are disjoint. Therefore 6 / (EpP+ ) I I2)(Ep, p Lin iH') + /2) YEP, +P ,in i E J'(2)E (f, H .1) Note that E PE Pi", H) and |)(Ep +P+L because of the construction of ,l and H2 Also, by definition )I/ ,(EP +Pl- HHl) jI (i (E) Therefore Note: From I' and II' we obtain J i >(-1 f 31 We now proceed to the general induction step. Let K 2) /A (0)= -, E El Assume that the following statements are true. For each J = K Hj and Pji2 are defined and () = Ep.--Hj .' =/ EP,+i )i'EU)> E( ( HpnHj0O if P=I---K )j) 0J-,) IJ-> S>- j- f , All of the above statements are true when K= I will now show that their being true for K implies they are true for K+ I (K)= (K) Let F, I E 1: Assume there exists lZ>i-l such that I( K)f E<) K+X and let F ) F I E . If there exists 3 > so that (/ ( n 1Ei3 ) , S(K) i (K) let F3 = F 3 If this process of choosing ij and Fj should go on indefinitely we obtain a contradiction because for all j Ijl -,- j J / ( n fl (-0) (K) (K) E( whereas -j _, E C, C p +.. +- PK + ij and Ij > imply I (P F E (K))' zP, +-t" .+ P,+j '4 that L +Z ' Combining these two inequalities we see that for all J , S 1(K) II >K) K))3 P( -/4jl, (J ji -~ > + Therefore l K' i F jj does not converge to zero uniformly in This contradicts Corollary 4.2. Hence there exist E(K) S(K.) n() -- -3 J (K EJ )\I 3 2-K+ and i, a such that for all J > L , . Let PK+, = L. (K) , I-K+=, Ft p(K') (K) L P:Ep . In order to show the induction step is valid, I must prove the following four statements. I". For all j : I\' K Hj 1HK+I= + (K) (K) (K) HK+1 C E = FK) by construction, and E, \j 1 SHence H K+l r (J, = 0: 11 1 Since HK+1 c E +... +PK+ 2P, + + + K P -. 3Z because P > 2 . I /.a ...(HK Z'l- Note that E) FK)- ( ,(K) U(K) U (K) I 3 l Also I I(- () I fK))F : )I' n+ E ( K++) b e F(Ei.+ P K)+ . ZK+-+I because F L CE p and I Li HK '- PK + PK+1 i H+1 P +""* + Pg + Pg+, +1 FK) UF* (K 'P) (K) " 7-14pt(+r+ I /'"Pj +--' + P;(+ 1P+,+ i K+1 j H E,+...+PKt + , I- H )/ 1 l + - PK, -IK+H Likewise K) (KF ) 3 I ( K)-t 1:I r3)i3 +." K1+; because F n E E + ...+P + 3 and 2 3 . S ) (K) (K) _ In the same manner, for j :...- we have I ( rj Ij IjP ++PK+jl.+ < + because J j+1 Therefore E- K (E I I++K + -+++z '"* + + )/ (HK)l ; the first inequality follows from the induction hypothesis and the second inequality results from the definition of HK+I and the inequalities in the preceding sentences. Therefore, S- ..-:+--^iT-(++i +*+ Z} <. Ir, ( 1 (K) 6 e Since 2.+l4 +I + K + we have 6 1_5_ 6 (K) (K+X H+1 IV''. For all ( / I (E ) > 6 +- --E (K) K) I. (K( (K) (K) \ ^..^EK~ -H,,)t+ H pK+ +PK+K++.Jf 11 H K 1^ )j > because of the induction hypothesis. As a result of the construction of HK+I and because PK+I+ >P +I we have J/p+, +i( p + H2) 2 I Thus c, PK+,^ :,-HKJ|> PO I+-^ Note: It follows from II' and III" that J(1A (KK-0) (H I ( CE- -- C- 3-6 Therefore, by induction we may define infinite collections [H and {/) such that for each K2 L (K-) (K-) i 31 > 32 {HK is also a sequence of disjoint sets. Letting % ,) =V - O -- 1 then | +,- ) J /6- for all 72 So (V n+iJ M+ ) K tI(H 10 does not converge to zero uniformly in L which contradicts Corollary l.2. This concludes the proof of the theorem. Q.E.D. Definition: A linear topological space is said to be locally convex if it has a base consisting of convex sets. Let U :{UL dE AJ be such a base. Let = : U Ua:U oI For each U, in define I( X) = a: a >O, iU and ( [)- : EII() .. Then is a pseudo-norm for the space 6 and has the following properties. (a) 6)0 o0 (b) + +oo00 (d) if X)Ur then a( )_ I That is, / has all the properties of a norm except that it may be possible to find an X such that >XO and (Y)=0 In the theorems of this chapter the fact that I=1 0 implies X=0 was never used, and with appropriate modifications the theorems are true if the set functions take values in a locally convex linear topological space. First of all, one has to define s-bounded and absolute continuity differently. Definition: Let X be a locally convex linear topological space, and let :({A3 be a family of pseudo-norms that determine the topology of E Let / U-X and 'L: -- Rt be set functions. We say / is strongly bounded (s-bounded) if A ()F:O 0 whenever {E} is a sequence of disjoint sets. // is absolutely continuous with respect to V if for every o( and 6 there exists 6>o such that (E6)/ implies ( )) The topology on 6 determined by a single pseudo-norm o( is not necessarily complete. We then let K be the completion of , and if /L: Z- Z is s-bounded, we consider /4 to have values in Ej and it is then true that converges unconditionally in )c if [Exn is a sequence of disjoint sets. The statements of Theorem 4.1 and Corollary 4.2 do not change if 9) is locally convex, except that An(t) is considered to be an element of for a 'tEA fixed o Taking into consideration the new definition of absolute continuity, the extensions of the Vitali-Hahn-Saks and Nikodym theorems for finitely additive vector measures are valid when the range of the measures is a locally convex linear topological space. CHAPTER 5 A COUNTEREXAMPLE In this chapter an example is given to show that weak con- vergence does not imply strong convergence in the space of countably additive set functions. Let /t: 0(01)-9 R be countably additive, and assume that for all A t (A): 0 Then from Theorem 4.1, we have that V/u, (A)= 0 uniformly in Z which in turn implies that D/-- )j/Un = O where lI is the total variation of/L . Since Theorem 4.1 was very important in proving theorems about set functions from a O0-algebra to a Banach space, it is reasonable to pose the following question: if -> R and if j/ T (E)=O for every E does it follow that - I/, nl= 0 ? The answer to this question is, "No," as the following counterexample shows. 56 Counterexample: This counterexample will show that if Z/ : Z-R are countably additive and if u /-(E) =0 for all E then it does not follow that A u-= 0 . Construction: Let 1 be the Lebesgue measurable subsets of (0, I) and let A be Lebesgue measure restricted to . Let -f be defined: \ if X is in ( ) o o < Z a, even foO -1 if X is in ,( ) o0 <2 a. odd 0 otherwise Let ? (E) a d A Then for all >0 A(E)<& implies I fd)<$ which is equivalent to IL,(E)1< Note that if K >n then /a :O First it will be shown that if is an open interval, then / (6) =0 Let 6>o and choose n so that 6 . Let = -fru[i":^ 3i<9 c-frrp .f.7z: ;i f3 Then f ('-U ) (:a ) L( C and this inequality combined with the statements in the last two sentences of the preceding paragraph give us that I/4(6}< 2 6 for all K > . Since E was chosen arbitrarily, the conclusion follows. Now let V be an open set in F and let > 0 There exists a countable collection of disjoint open intervals fl 00 such that V = U( (I am considering the null set to be an o00o open interval.) Then A(V) = A( and, since A(V) oo , there exists N such that A(V- ,)Z 6 ; hence for all 2 , I/x / that for all K>T( //K k)I< Then letting T= 7 A L: I, N1 we have that for every K >T, 1-K(V) I IIK(v/IM -A)/ + K() ( I + N(): Therefore we have that & /U, (V):O for all open sets V Let E be a set in 1 and let O There exists an open set V so that ECV and \(V-E) /Yt(V-E ) E for all n By the result of the last paragraph there exists T such that for all > T //kiV)I E hence /(IE)I J /(+ V ( th(V-E)l 6 This implies that A/6un (E)=0 for every E in . The /- are countably additive, so they satisfy the hypothesis 59 of the counterexample. However, / f(o,) ifnl X= I for all n, so j[jnlI does not converge to zero. REFERENCES 1. Ando, T., "Convergent sequences of finitely additive measures", Pac. J. of Math., 11, 395-h04 (1961). * / / / 2. Banach, S., Theorie des operations lineaires, Monografje Mate- matyczne, Warsaw (1932). 3. Brooks, J. K., "On the Vitali-Hahn-Saks and Nikodym theorems", Proc. Nat. Acad. Sci. U.S.A.;' 6, 468-471 (1969). 4. Brooks, J. K., "Representations of weak and strong integrals in Banach spaces", Proc. Nat. Acad. Sci. U.S.A., 64, 266-270 (1969). 5. Brooks, J. K. and J. Mikusinski, "On some theorems in functional analysis", Bull. Acad. Pol. Sci., Math., Astron., Phys., 18, 151-155 (19707. 6. Darst, R. B., "A direct proof of Porcelli's condition for weak convergence", Proc. Amer. Math. Soc., 17, 1094-1096 (1966). 7. Dunford, N. and J. Schwartz, Linear Operators, Part I: General Theory, Interscience, New York (1958). 8. Dvoretzky, A. and C. A. Rogers, "Absolute and unconditional con- vergence in normed linear spaces", Proc. Nat. Acad. Sci. U.S.A., 36, 192-197 (1950). 9. Hewitt, E. and K. Stromberg, Real and Abstract Analysis, Springer Verlag, New York (1965). 10. Hahn, H., "Uber Folgen linearer Operationen", Monatsh. Math. Physik., 32, 3 (1922). 11. Hilderbrandt, T. H., "On unconditional convergence in normed vector spaces", Bull. Amer. Math. Soc., 46, 959-962 (1940). 12. Nikodym, O. M., "Sur ler suites convergentes de functions parfaitment additives d'ensemble abstrait", Monatsh. Math. Physik., 40, 427-432 (1933). 13. Orlicz, W., "Uber unbedingte Konvergenz in Funktionemaumen", Studia Math., 1, 83-85 (1930). 14. Pettis, B. Math. J., "On integration vector spaces", Soc., Wh, 277-304 (1938). Trans. Amer. 15. Phillips, R. S., "On linear transformations", Math. Soc., 48, 516-541 (1940). Trans. Amer. 16. Phillips, R. S., "Integration in a convex linear topological space", Trans. Amer. Math. Soc., 47, 114-115 (1940). 17. Rickart, C. E., "Decomposition of additive set functions", Duke Math. J., 10, 653-665 (1943). 18. Rickart, C. E., "Integration in space", Trans. Amer. Math. a convex linear topological Soc., 52, L98-521 (1942). 19. Saks, S., "Addition to the note on some functionals", Trans. Amer. Math. Soc., 35, 967-974 (1933). 20. Schur, M. J., "Uber lineare Transformationen in der Theorie der unendlichen Reihen", J. reine u. angew. Math., 151, 79-111 (1921). 21. Vitali, G., "Sull'integrazione per series Rend. Circolo Palermo, 23, 137-155 (1907). BIOGRAPHICAL SKETCH Robert Jewett was born in Portsmouth, Ohio, on August 20, 1945. When he was eleven years old his family moved to Fort Myers, Florida, where he graduated from high school in 1963. In the Fall of the same year he went to the University of Florida on a golf scholarship, and played on the golf team for two years. In 1967 he received his Bachelor of Science degree in Math, and from that time on has been working toward his doctor's degree. He is a member of the American Mathematical Society. He was married to Suzanne Strobak on July 3, 1971. / I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ames K. Brooks, 'Chairman associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David A. Drake Assistant Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Zoran Pop-Stojanovic / Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Gene W. Hemp Associate Professor of Engineering Science and Mechanics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 41' - Stephen A. Saxon Assistant Professor of Mathematics This dissertation was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council, and was accepted as partial ful- fillment of the requirements for the degree of Doctor of Philosophy. August, 1971 Dean, Graduate School L2_/^ |

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PAGE 1 Generalizations of the Vitali-Hahn-Saks and Nikodym Theorems By Robert Stanley Jewett A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1971 PAGE 2 ACKNOaTLEDGMENTS The author would like to thank his advisor Dr. James K. Brooks and the members of his supervisory committee, Dr. David Drake, Dr. Gene Hemp, Dr. Steve Saxon, and Dr. Zoran Pop-Stojanovic for their help in writing this dissertation. 11 PAGE 3 TABLE OF CONTENTS Abstract iv Introduction 1 * Chapter 1: Unconditional Convergence 6 Chapter 2: Strongly Bounded Set Functions lit Chapter 3: Extending the Schur Theorem 20 Chapter h' Extensions of the Vitali-Hahn-Saks and Nikodym Theorems 35 Chapter 5: A Counterexample $6 References 60 Biographical Sketch 62 111 PAGE 4 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy GENERALIZATIONS OF THE VITALI-HAHN-SAKS AND NIKODYM THEOREMS By Robert Stanley Jewett August, 1971 Chairman: Dr. James K. Brooks Major Department: Mathematics The Vitali-Hahn-Saks and Nikodym theorems are two very closely related results concerning sequences of countably additive, real valued measures. The purpose of this dissertation is to improve on these theorems both in the statements and the proofs. That is, stronger theorems will be shown to be true, and the proofs will be easier that those usually given. The author and J. K. Brooks extend the Vitali-Hahn-Saks and Nikodym theorems to the finitely additive vector case. In 1969 J. K. Brooks used a result of Schur concerning uniform convergence of a double sequence of real numbers to derive simple, direct proofs of the Vitali-Hahn-Saks and Nikodym theorems. In Chapter 3 of this dissertation the above mentioned theorem of Schur is extended to the case where the double sequence is contained in a Banach space, and then this new result is used to obtain a short proof of the Nikodym theorem with the measures taking their values iv PAGE 5 in a Banach space. However, the Vitali-Hahn-Saks and Nikodym theorems do not generalize directly to the case where the measiires are finitely additive, bounded, and Banach valued, and the additional hypothesis of "strongly bounded" must be assiimed. With this assumption, the theorems may be extended, and the proofs follow the same general line as those of Brooks' for the countably additive, scalar case. Instead of the Schur theorem, a generalized form of a resiiLt of Phillips is used, and the Nikodym theorem follows, but the proof of the Vitali-Hahn-Saks theorem requires a difficult construction with measurable sets. At the end of Chapter ij a discussion is given to indicate how the theorems may be extended even further to the case where the measures take values in a locally convex linear topological space instead of a Banach space. PAGE 6 INTRODUCTION This dissertation concerns the Vital i-Hahn-Saks and Nikodym theorems (7). In 1907 Vitali (2l) proved the following theorem: If Tiv' tÂ°/ 'J Â— ^ ^ ^^6 Lebesgue integrable functions converging almost verywhere to T , then .^cim. I'f^ciS and )TciS exist and are qual if and only if the indefinite integrals of the Th are uniformly absolutely continuous with respect to Lebesgue measure. Hahn (lO) then proved in 1922 that if -p^: [o,^-^ R are Lebesgue integrable functions and if .,Â£/rYv jTn<^5 exists for every measur able set Â£ , then the indefinite integrals of the T^ are uniformly absolutely continuous with respect to Lebesgue measiire, and they converge to a set function which is also absolutely contir.aous with respect to Lebesgue measure. In 1933 Nikodym (12) generalized these two results when he proved that if fS, T.,/^) is a measure space PAGE 7 and il A^ is a sequence of u -continuous measures such that ^Si/nv X-^(Â£) exists for all Â£ in ^ , then the A^ are uni71 f ormly absolutely continuous with respect to /x . A little later Saks (19) gave another proof of Nikodym's theorem, and it became known as the Vitali-Hahn-Saks theorem. Using the countable additivity of the control measure, Saks defined a complete metric on the measurable sets, and then applied the Baire Category theorem. If the control measure had been only finitely additive, the metric would not necessarily have been complete, and this technique could not have been used. Nikodym (12 ) proved tht if JUyi is a sequence of measures defined on a (T -algebra such that jj.(B) MaÂ«v jU^(E) ^::ists for all F , then il is countably additive and the coiintable additivity of the JUL y^ Is uniform. This is known as the Nikodym theorem, and in most books it is proved as a consequence of the Vitali-Hahn-Saks theorem. Rickart (18) and Phillips (I6) extended these theorems to the case where the measures are Banach space valued. The proofs of these results rely heavily on the fact that the set functions in PAGE 8 question are countably additive, and there seems to be no way to generalize their techniques in order to prove Vitali-Hahn-Saks and Nikodyiti theorems for finitely additive, vector valued set functions. However, in I969 Brooks (3) gave a short proof of the Nikodym theorem and then used that result to prove the Vitali-Hahn-Saks theorem. His proof used a difficult construction with measurable sets along with a theorem of Schur (20) concerning the equivalence of weak and strong convergence in /C / . In the same paper he gave a short proof that the truth of the two theorems in the scalar case would imply their truth in the vector case. An extension of the Vitali-Hahn-Saks theorem to the finitely additive, scalar case has been proved by Ando (l), but his theorem was not extended to the case where the measures are vecf^'P valued. In 1970 Brooks and the author proved Vitali-Hahn-Saks ana i\likodym theorems for finitely additive, strongly bounded, vector valued set functions, where strongly bounded is defined as in (17). The strongly bounded hypothesis is especially needed in the Nikodym theorem, because the statement would not even make sense if the PAGE 9 measiires were only bounded ^ the strongly bounded hypothesis may be dropped in the Vitali-Hahn-Saks theorem if the control measure is assumed to be finite. The proofs of the theorems are built around a generalization of a theorem of Phillips (15) and an extension of a technique of Darst (6). The first two chapters provide the background material for the results proved later in the dissertation. However, Theorems 1.3 and l.li are stated only for completeness, and are not used in subsequent proofs. Although most of the theorems in these , apters may be found in Hilderbrandt (11 ) or Rickart (17), new proof c ire given by the author. The purpose of Chapter 3 is to generalize a theorem of Schur (20) and to use the new result to construct a direct proof of the vector case of the Nikodym theorem. Since all of the measures are assumed to be countably additive, the theorems rely only on the results of Chapter 1, and not of Chapter 2. Chapters 3 and l, -.re independent, and no theorem of Chapter 3 is used in a proof in PAGE 10 Chapter 1|. The most important theorems are proved in Chapter kBoth the Vitali-Hahn-Saks and Nikodym theorems are extended to the finitely additive case, and it is shovm how the newer version of the Nikodym theorem implies the usual version when the measures are countably additive. The concept of "strongly bounded" is of utmost importance here, but Corollary I4.5 and the remark following the statement of Theorem i^.7 indicate when the condition may be dropped. PAGE 11 CHAPTER I UNCONDITIOML CONVERGENCE Throughout this dissertation the following notation will be used. L is a cr-algebra of subsets of a set S . (P(??) is the power set of the natural numbers ^ , and Â£ and A are generic notations for sets belonging to Z. and P(7l) , respectively. R denotes .the real numbers. 3^ is a Banach space over the real or complex numbers, and M is the norm of )< Â£" 3^ . ^ is the conjugate space of dC , with the norm of an element in ^* defined in the usual way. Let yi,ynÂ€n with n 6 m . Define [f^/mji {yi^n+l,,>Â».} ^ If^ is a real valued measure defined on a 0" -algebra Z then J^l is the total variation of u , which is defined in the PAGE 12 standard way. In this chapter we define the term "unconditional convergence", and establish other conditions equivalent to it. Definition : Let ^y^e^ ,n-l,Z," Â• ^ TV ^ -> 71 is a one-to-one, onto mapping, then { n(n) \yizi is called a rearrangement of /^n l-^-i Â• The series ZK is said to be unconditionally convergent if the sum of every rearrangement of its terms converges. Remark; A series that converges absolutely will converge unconditionally, but the converse is not necessarily true. In fact, Dvoretzky and Rogers (8) proved in 1950 that the two conditions are equivalent if and only if the Banach space in question is finite dimensional. The difference between absolute and unconditional convergence was further demonstrated by Brooivs ([() in 1969 when he showed that weakly and strongly integrable functions with range in a Banach space correspond to unconditionally and absolutely convergent series in the space. 00 Theorem 1.1: Let X>, f 5^ j^"',X. "' . The series Z-^H 1 71 = 1 PAGE 13 converges -unconditionally Â±Â£ and only if every subsequential s\m converges, that is, if i ^^'f is a subsequence, then .4^i converges. Proof ; Let { ''kJ tie a subsequence and assume the sum does not converge. Then there exist i>0 and sequences {Pij and j^tl > 6 . Let [l/^}~=, = {^^}Z.l'unl^''K'''^''^'>"''''4 Â• Then form the foil ing rearrangement of {^ ji] : ^np_ ; Â• Â• Â• , ^n y, > i/i ; ^ np >"'^ where for each i , /'^ <^i PAGE 14 as they appear in {^nl , and call this rearrangement l^n, )'' >^ft^ I . Let Piji>P, . Again rearrange \^K' ^^m"'>^a] appropriately, and label this arrangement l^ri^^^,)"->^n^J , Since y^^!^5 ii" f f r>Â«., ^,] , SfC^/i^riA!] , we have that {^n ^i 'Â• >'^^nuJ^'^^ui^^l^ '" ^^u>^] has the same order as I'^nj . In this manner, define j^/< Â• K= ^i/-'| and {u)^] so that Â•^if. for every i , Yi^^K /* ^ Â• Then I >*/<[ is a subsequence / K-utj^ + i I ' whose sum does not exist; thus the converse is proved. Q.E.D. 22. Theorem 1.2 ; Let X^ Â£ 3c j>l=',Zy''' . If 2_^W converges n=( unconditionally, then for all 6 > , there exists an N such that for all A C [hl,oo) j Zl/x I < ^ . Proof ; Assume the conclusion is false. Then there exists Â€>0 such that for all N , there exists AcCnI/Oo) such that Let N,= I . Then there exists ti^c[^^^oo) g^^h that Ifr^ ''^ I > 6 . Also, there exists /) , a finite subset of Z\, , such that I^Yp ^ P X * ^^"^ ^3. ' ^''**^ ^ * There exists PAGE 15 10 t^jC[N^+ljOo) such that LVa '^ / ^ ' ^Â° ^^^^^ exists a finite set Px^: ^z such that L^n^^ I > -f" Â• Note that P, HP^z O , where D denotes the null set. Also, for all SÂ£ P, , tcP^ , 5<.i: . In a similar way define an infinite number of finite sets /j_ such that for all L , \^n.'^h4' and if t PAGE 16 11 and let (7 . By Theorem 1.2 there exists N so that L^a ^ P X for every A^ L^/Â°Â°) . Let X be chosen so that |X*U I and define A,r(7if[N,oo):X*(X^)>o} Then A, C A, "^^',Â«') , and A, (1^,-0 . Hence H K Km) = Note that for all -^ > N , /ynjA nfry PAGE 17 12 For every N there is a value of L so that I^l'Â„->''V,\C(N.C.) . therefore Â£,l"r(' PAGE 18 13 nfA'^P ^ Â• ^Â®^ ^o-tl.f^l . li" a, ^(r,>ro , we have that Into-(To P**-^ > because both (T, (To and Ci-O^ are contained in [rV,oo) . Therefore the series converges to a Moore-Smith limit. Conversely, assume the series does not converge unconditionally, so there exists a subsequence I^^KJ whose sum does not exist. There exist 6>0 and sequences [Pc] , [v^<;j where Pi<^i ^ Pi-hi and for each L , I I_^'*^k I > ^ . Let (To be any finite subset lK--Pi I of the natural numbers, and choose c so that {^P; >' Â•Â•;^v^ ?n 0^= D . Letting CT = (To U f ^P^ , Â• " > "" V^J we have that (X > (To and \nÂ£(r riKTo I ' ^^^ ^* follows that the series does not have a Moore-Smith limit. This completes the proof. Q.E.D. PAGE 19 11; CHAPTER 2 STRONGLY BOUMDED SET FUNCTIONS Many theorems that are true for countably additive measures do not have extensions when the measures are finitely additive and bounded. However, as will be seen in Chapter l^, some of these results do carry over to the finitely additive case when the hypothesis of "strongly bounded" is assumed. In this chapter "strongly bounded" is defined and theorems are proved which will be used in Chapter i;. Definition ; Let lL'.Y.-^^ be a set function, we say n, is strongly bounded (s-bounded) if ^^T^/'lHfj-O whenever |f^| is a sequence of disjoint sets in Z. . This definition was first given by Rickart (17). Theorem 2.1 . Let ll-^~^^ be s-bounded and finitely additive. Then u. is bounded. PAGE 20 15 Proof: Assixme u. is not bounded. Then there exists u such that iM^^n -^ . Suppose there exists a monotone decreasing sequence of sets /^^E such that (A{fV+()l "^ |/^(/^)r 1 for all K . Theny.(/l^-f^Â„)y^('rK)-'-XfV+.) , and )/^ f/^'^J) > l//"/^,,)] //'('KJ/ ^ -^ Â• Therefore /rf<''V+|( is a pairwise disjoint sequence of sets, whereas ] U-yK \{-^q\ does not converge to zero, which contradicts the fact that u. is s-bounded. Consequently there exists 0. C E such that jMiOilj ^1 and for all iC G, , iM-fin -\m(Q\-^1 . There exists R f H such that )y/?)i>)yu(t,)| + 2, so \^(({G)\>\/R)\'\jii(KnG)\>l If there exists a monotone decreasing sequence of sets fj, C K~ C// such that IA'('|(+/)| '^ '/'( f^/j '''-^ ^or all K , we obtain a coi mtradiction as before. There Therefore there exists G^C R~ G, such that iMiGiJI ^1 and for all i C G;, , Im(l)\ ^ lufG;^\ i 1 . G,r\G^^ exists P such that L fr)| > l//ft)|+ Ja&)/+ 3 , so |/.fPfe^C^;?))) > Lfr)/LfP/l(;,)| L /P/0{?i)) > 1 , and there exists (J^C P-fo f GJ PAGE 21 16 such that Iywf(r3))>l and for all L C G3 , j u(i)\ < IjliO^j j + 1 . Continue in this manner to define a sequence of disjoint sets l^L) such that for all [ , \lJi{Gi)\^l . This clearly contradicts the fact that jU is s-bounded, hence the theorem is proved. Q.E.D. Remark ; The converse to the above theorem is false. Let 2be the Borel sets of the real line, and let yt oCf^iH^L,^) , the class of essentially bounded measurable functions, where A is Lebesgue measure (9). Define juiE) ~ ^E , the characteristic function of the set E . u. Is bounded because for all E In 2, / r E 1^ Â— -^ . However, letting h^-^Ti^Tt+'J , we see that /^(E-i,)) 1 ^o^ ^11 ^ , so // is not s-bounded. On the other hand, if 7c is the real numbers the converse holds, so the concepts of s-bounded and bounded coincide in the case of scalar measures. Theorem 2.2 ; Let M'^^R. be bounded and finitely additive. Then JU is s-bounded. PAGE 22 17 Proof ; Assume the conclusion is false. Then there exists a sequence of disjoint sets |C|^f where //'v '"K/fi/; i does not converge to zero. Therefore there exists Â£ >0 and an infinite subcollection I'-K J ^ i^K) such that for all i , /A(f^ )/> 6 . Let A<^fZ he such that for all i Â£ A , JU (E^ . ) > ^ ^'^ ^Â°^ ^H Lilt' A , /u(ff(.)^~^ ' Either A or 7?A is infinite. Without loss of generality, assvune A is infinite. Let A1>0 Â• Pa f^ Lst / be a finite subset of Z^ with cardinality greater than "^ . yU (}J ^K .) ^ r/^r%) > (t)^ ^ /^ Â• Since /^ was chosen arbitrarily, this proves U. is not bounded a contradiction. Q.E.D. Definition ; Let U' 2Â— ^X Â• Define u,(Ej Pof Zd . Theorem 2. 3 J Let JUL '. ^_ ^ a! Â• Then ytl is s-bounded if Then /Uf jlu[^j}E ^LjE <^E( . n is called the semi-variation a: /_ T 7^ . Then At and only Â±f //. is s-bounded. Proof; Assume ju. is not s-bounded. Then there exists T PAGE 23 18 disjoint sequence I'-K) such that j P-^'-l PAGE 24 19 and then choose p , G such that ^ ^ f\ ^ ^j and 1 /_/^i^Ki)\y ^ In this manner we define infinite collections )fjl and | fjl such that for all j , Pj < ^^. < ^.^^ , | >_y"fc^)|>6 . Let 1; ' .^4 i-K; Â• Then ] 'j \ is a sequence of disjoint sets ^""^ ]m(Fj)J'\xMEk)\>^ , so [/x(Fj)j._. I does not converge to zero and u, is not s-bounded. This proves the theorem, Q.E.D. PAGE 25 CHAPTER 3 EXTENDING THE SCHUR THEOREM In 1969 Brooks (3) proved the Nikodym theorein by using a result of Schur (20). In this chapter the Schur theorein is extended to the vector case, and the result is then used to obtain a very short proof of the Nikodym theorein for countably additive, Banach valued measures. The proof of the first the ^em is a good example of the "sliding hump" technique. Theorem 3.I ; Let y^Â„ f ^ > L 71l,X,--Â• Assume that for all L , the series 2_ in is unconditionally convergent, and also that: (-X-) for all A , Xrv Zr>^tn-0 Â• Then the limit in (Â«Â•) is uniform with respect to A . That is. If (depending on Â£ ) such that for all i>I and for all A , lH^in ^^ . InfA 20 PAGE 26 21 Remark: If 7C is the real numbers, then ^^in converges absolutely for every i , and the conclusion is that Proof ; Assume the limit in (Â«Â•) does not exist uniformly with respect to A . Then there exist 6>o and sequences t^Kj ) l^K) such that j^eAu^ ^^ Â• To simplify notation, let i'^= K , so l^^*^"|>^ for all K . Let K,/ . Since \ f,^^\^^ > there exists a finite set T, C A , such that yTTn '^ I ) -7=" ; due to the unconditional convergence of ^ '^ , there exists (^^ such that for all A(^[R,,'^) , I V A ' ^ P ?" * Let T, = nm^t /^, ^[R,j) . If Ac[T^-^\,oo) , we have For every ji , Jutrt ^/<>, =0 j so there exists I^ such that 6 T, ^ for all L>Jy, , /\^|^3^ . Then IIl^inkTif t > I >Â«<^[^n->tffi,7;j] . Let K.--1 , then I >< ^ ^ ^^a^'^H rl^ Â£ 2. / '^a'^l > r _i_ Vi. , Therefore there exists a finite set /^cAK^-f/JJ such that \^p^^^l>^ Â• PAGE 27 22 Since X. K,)i converges unconditionally, there exists Hj, such that for all Ac[Rj^^oo) , J(i^^^^\'^T Â• ^^^ "^1 = ^'*^<^Crx^(^x]) > so that for all /l^ cfl^ + Z^oo) Â„e have that P,f)A--Q , p, nA=D . Note that Pr\r--Q , hence ^^Â„^^ = Sr'i)^UA 'Sr''^'' '5^^'^ ^ "hen ACrT, + .,-) . Also iSn'^'-'^l^r^^''^^^ S , hecanse n C [l Jj . Therefore Inf /; Jr;,uz^ I ^ )n^/i ' / M M |nÂ£A M ^ ^ ^ ? ifor A ^i\-^^,Â°Â°) Â• Also, since /I A CfTj + l^oo) , we have that )I^^^ J>f . In the same manner as before, obtain X such that for all i>I , L-l^iyil^y , and let Kj = I . Then we have that Iwf Ai/ -fi Tll^^'T'^ ^ ^^'^ there exists a finite set r^cAn^.[lJ^'] such that \}jr*^^^ \'^f Â• Choose R^ such that for all A C [^3,00) , \ijÂ£^^^ \^T ^ ^^^ 1Â®^ "^3 ' yÂ»<>^(Pj ^{f^i]) . If A <^rTj +/;<Â») . we then have ^Y ' Since Pj U A C [l^+\^'>o) Â„e have that ^/r.l)\ur3UA\>S ' ^-^ /l^r^McfTj.^oo) implies that E^"^.^ J si: PAGE 28 23 Continue to define an infinite number of Pj and Kj and let r^UPj . ^enforall J , |Â„^;^"(>f and {J/'"]" does not converge to zero, which contradicts^ the hypothesis. Q.E.D. Remark ; Theorem 3.I may be used to prove the equivalence of weak and strong convergence In . J^ , , the space of absolutely convergent series. Let f4 i j^, , L-I.l,--' Each element of X\ is a sequence of real niimbers, and we write 11 as /^tnjn-i and f as V'y^ly^zi Â• The sequence jfij converges strongly to f if Matxj Z_/^iÂ„-X^| -O j and ifj_] converges weakly to f if 2? *Â• J converges to PAGE 29 21; so I will prove the converse. Assume the Tl converge weakly to -f . Writing f^ as f iri)n-i and T as i njjir/ , let ^4 X ^tn " ^n Â• For any ^ , define y* in i^oo by: V f/ if K f A (_0 if K ^ Zi Then trn. Z'^cn = ^ Z(^Ln-)(n) ^ Jun. U*Cfi'i)-0 because the Ti converge to T weakly. Therefore, by Theorem 3'1 oo we have that Â£C^yt^ L. l^in' ^-k] O , and the TJ; converge strongly to T . Note that we only needed to consider those elements of x^g which take on the value O or / . Q.E.D. Theorem 3.3 : Let Y;^ f^ , i,7i-l,X,"Â• Assume that oo for all i , )L^Ln converges unconditionally, and that: (-;f) for all A , Jim, Z^tn exists. Then the limit in (-"-) is uniform with respect to A ; that is, for every Â€ > o , there exists I such that for all j > I and for all A , jEp-I^f/Uj(e . t Proof : This proof will be divided into six parts. Part I ; If {tj^] is a subsequence, then PAGE 30 25 Si/m. T-i^^K^ ' ^i-H+i'^^' Â° uniformly with respect to A . Proof of l ! Let ^^^ ^ 5^^ ^ Xi n Â• '^'^Â®" ^Â°^ ^^1 *< * 00 Z_ K n converges unconditionally and for all A , Â£,/n\. ZI*^/^n-0 By Theorem 3Â«l3 ^W/**** (k^^ ' ^ uniformly with respect to A , and the conclusionof I follows. Part II ; For all 8 > O , there exists N such that for P CO I all integers L and for all P>M , H^ltxTl^iy\.\<-S . Proof of II : Suppose that we deny the conclusion. Then there exists oyo such that for all N there exist I and P>N such tnat: (1) ibjy^-ZXjJlS . Choose as indicated in the above statement. Let No be Â• chosen arbitrarily. Then I claim that for all T , there exist L>1 and P>No such that \L.^Ln-T.^in\lS . If this 'n:( 71=1 I were not the case we could find Iq such that for all t > IÂ© , Plf^o , IZ-^tnH^inUS . For all L , Lin converges, '71=1 7t=/ I ^=' so there exists AJ; such that for all P> hi, , I H^tn" H^tTi < . PAGE 31 26 assiimpLet R = >Â«>^{A/o^y\/t : t = '/ "Â•/loj Â• Then for all i and all P> Q , lYl^i-n-jyinl^S and this contradicts the tion (l). Therefore, since A/q was picked arbitrarily, we have that for all N and I , there exist t >I t PZ t^ such that: Since -tt PAGE 32 27 f 3 . Let Continuing this process, define {'m} and \^rÂ»\ t'^^'l/^,'" such that \h^^^ ' nc A^r^'^ r T ^or all m . Then -'Cwn/ 2( 'm.^ ~ ^M.+i>i ) does not converge to zero uniformly Tie A / with respect to A , which contradicts Part I; hence, the conclusion of Part II is proved. S S Part III ; If Sf ?Z , then ^ L^inT^n , where ^X= 'W^ ^i-n. . L Proof of III ; This follows because the sxms in question are finite, hence the limit may be interchanged. Part IV ; For all /\ , ZI ^>t exists. Proof of IV ; If A is finite, the conclusion holds, so assume A is infinite. Let A = V^k\ . Letting j^l^i^ ^c rif, we see that {^inS satisfies the hypothesis of Theorem 3.3; consequently, we may apply the results of Parts l", II, and III to the double sequence l^cK) Â• To show 2_^?t exists it suffices ' 7lÂ£A PAGE 33 28 oo to show that 2_ ^X exists, where Ou-: '^^ ^LK oo C ^ ^ "joo In order to prove Â£Â—^ii exists, I will show that \^ <^\s/z\ is a Cauchy sequence. Let Â£>0 . By Part II there exists an N such that for every l and v>N , j^^tK " Il'^tK K "^ . Therefore, if y.SlH we have \i.B^^^^ L.^iA^f ^Â°^ ^^^ ^ Â• From Part III choose I such that for all Ll ~L , ^hi^'-^ '%,^'^\^3' and /?/<.'< Â£^/< l^l" . Then for all i>I , Since 6 was chosen arbitrarily, it follows that j tl ^ni is Cauchy, and the conclusion of Part IV is proved. For all A , 2-.^n exists, and Theorem 1.1 indicates that 00 ^j 'n converges unconditionally. Part V : For all A , Jt>m. T^ltiZ\ . (Note i,hat the second sum is defined by the result of Part IV. ) Proof of V ; Since the conclusion is true for finite ^ , we may assume A is infinite. t Let A = [^k} ) ^LK'^i-T^ii ' We need to show that oo oo MnrA. Z!^<;Â«zl^K , and since I^lk] satisfies the hypothesis PAGE 34 29 of Tneorem 3-3 , we may apply the results of the first four parts. Let 6yo . According to Parts II and IV, choose N so that l-'^K -T.^K < f and for every i , H^lk " Z/^,v <Â£ By Part III, choose 1 so that for every t > I , r^iÂ« -Z^k\<Â€ . Then for every i>l , iL^tK T ^h OO Â°Â° yj Since Â€ was chosen arbitrarily, Â£i/m, zl^iK ' ^^K Â• Part VI ; The proof of the conclusion of tre theorem. Proof of VI ; Let oi'^^= ^in' ^n ' ^^ ^^^"^ ^> X-^^ ^Â°^~ oo verges unconditionally, so for all i , ^ t-n converges unconn-l ditionally. Also, for all A , JunvH'^Lyi Jc^r^x, Li^in'^yi) i nit I nÂ£^ i^^-nTA''^ TU^^ ' ^ ' where the last inequality results from Part V. Therefore, by Theorem 3Â«1, for all 6>0 , there exists 1 such that for all ill and all A , Â€>\ Z'^i. n I r |h?/'^ "J?/'' j ' I^a'"" '^|?a''' I ' "here the last equality follows from Part V. This concludes the proof of the theorem. Q.E.D. PAGE 35 30 Theorem 3Â«3 is the desired extension of the Schur theorem, and will be used to prove Theorem 3Â«U> the Nikodym theorem. Definition ; Let ^'.L. Â— '3^ . ^ is countably additive if jU-Itizi^) ~ ^-./'('^n) whenever l^n\ is a sequence of disjoint sets in 2. Â• Remark : jU being countably additive implies u is s-bounded because il JUL is not s-bounded, then there exist Â£>0 and disjoint sets {En] such that for all n , /"frn)|>^ ; hence l^, "^'Ik-., is not a Cauchy sequence and ic is not countably additive. Definition ; Let ia^;Z.~^7c be countably additive, n-l.^.j--' , and let jEij be a sequence of disjoint sets in H . The A>t are uniformly countably additive if for all 6>0 there exists M such that for all w>A\ and for all n , L/">ift) ' lL^-n(^i)}<^ . Theorem 3.k (Nikodym); Let y/^; ^Â— ^ 3c be countably additive, fl-1,2,-. If n.(B) ' ^^*^ /^yi(^) exists for every E , then /z is countably additive and the countable additivity of the U^i is uniform. Proof ; Let lEyii be a pairwise disjoint sequence of sets PAGE 36 31 in 2and let ^iy^'/^ci^n) . The double sequence {>^iÂ«] satisfies the hypothesis of Theorem 3 '3, so by the result of Part III of the proof of that theorem, for every 5>o , there exists N P n = i such that for every integer c and all P2 N , S ^ lH^LnY.^'-'Â»U-i n = i 7i:/ 7l=i additive . Therefore the ^i are uniformly countably On the other hand, /^ fe^n) " ^^^//.(u^fxj = X^n, E A^^n) = OO oo Â£^ Y-^in L.^n , the last equality following from Part V of the proof of Theorem 3.3. Since ^^n -Ji/^i^fU , we have n-i n-i that m(^ ^>ij= Z^^i^h) , andyU is countably additive. Q.E.D. We give a proof due to Brooks and Mikusinski (5) of a result of Banach (2). Lemma 3.5 : Let x be a separable Banach space, and let /^nj be a bounded sequence in 3t . Then there exists a subsequence /i/n^c such that -^T^'J/n (><) exists for every X in X . Proof ; This proof consists of a standard Cantor diagonal process. Let i^l\l:i be a dense subset of 3^ , and cnoose A1>0 such that ly*j PAGE 37 32 for each n , soj|^n(^<)j is a bounded sequence of real numbers, and there exists a subsequence )yi,K] of )y>,r so that M^ij^ )(i^\) exists. Likewise, there exists a subsequence fyi^KJ of jy, i^f such that Â£um, 'j^, K^^^ exists. Continue in tnis manner to define subsequences (\i(\ so that ^^* +i^ kJ^,, ^ [yn,KJÂ»(=/ ^^d Sn 'j^^kM exists for every )i . For each integer K , let y^ ' ^u u ' Then |y>if^{ is a subsequence of ly'Ti [ , and for all J , '%*'!^n^^^j) exists. â€¢e now show that m^ 'Jxu^^^ exists for each )/.Â£ 'X . Let Xe )^ , and 6>0 . Since [^jjjr/ is dense in 3Â€ , pick Xj so that /Xj -)^j< fj^ , and choose R such that for all K^^ 1 R , Ifyn^yn^)ft)|< f . Then for all K,i> R , I y'>Â«^^^j^ ~ y*'^^^''^ l-^'^^'^i'^ ' '^^Â®^Â®^Â°^Â® ^^Â® sequence JMn ^'^M is Cauchy, and Arn' 'i^l^^(%) exists. Q.E.D. oo Definition ; Let X^f ^ , nl,Z ' " Â• 2_^n is weakly ' n-i unconditionally convergent if for every subset A of the integers, there exists X* such that H >< ^^x) 5^ ('>^a) for all X* in 3c . PAGE 38 33> The following theorem is known as the Orlic z-Pettis theorem, (13, 111) J and the proof is due to Brooks and Mikusinski (5). Theorem 3.6 ; If ^ is a separable Banach space, then a 00 series 2n. converges unconditionally If, and only if, it converges weakly unconditionally. Proof ; Unconditional convergence always implies weak unconditional convergence. In order to prove the converse, I will assume that the series converges weakly unconditionally but not unconditionally, and then arrive at a contradiction. If 2_ n, does not converge unconditionally, there exist an Â€>o and finite disjoint sets ^i such that LfA-^i '^^ ^Â°^ every integer L . Letting ?. r Z_ >i we have that > ?/ also converges weakly unconditionally. According to the Hahn-Banach theorem, there exist X-^ such that IXrt^hl and j^MiiytJ]' \^m\ for every integer >U . By Lemma 3.5 there is a subsequence \^^i{ of { rn-]f^, such that ZCftrv X-niibO exists for every X in 3c , and to simplify notation I will assume that W^-^ for all L . Let ^tn'^c^^Ti)' PAGE 39 3h For any A , there exists i^ so that J^^ nJ )^i( a) for all c . Therefore, Jt^nt. H'^irt r ^^^ T-^L^^-n.)yU^ )^-(^a) Â• Since Z. ^i^^n) exists for all ^ and unconditional convergence implies absolute convergence in the scalar field, it follows that ZriX;('?n)M^ Â• Therefore, the double n-i sequence i*^4nj satisfies the hypothesis of Theorem 3.3Â« Letting <^ yy, ^iorj^'^i yi > we have from Part IV of the proof L oo of Theorem 3Â«3 that 2_ \'(yi\^<^ , and from the conclusion of the theorem we have that j&jyn. I Z t n ~ 2_ n = o uniformly in A . i InfA nfA / Choose positive integers t^i and hi^ such that for all nl N^ , Let yK>rm^{t^i,r^x] . Then 6 < j 2^| ^^(^Ol = Kr^>n | I w. >n. " Â•^m, I + !*'>Â«. I ^ "3" } which is a contradiction. Hence the theorem is proved. Q.E.D. PAGE 40 CHAPTER U EXTENSIONS OF THE VITALI-HAHN-SAXS AND NIKODYM THEOREMS In this chapter the Vitali-Hahn-Saks and Nikodym theorems are extended to tho case where the measures are finitely additive and strongly bounded. The first theorem to be proved is tne key to the whole chapter, and is an extension of a result of Phillips (15). Theorem i^.l : Let Xiy^: P{/lJ-~^ 3t be finitely additive and s-bo\anded. If ,Â£i/n\. a (^)O for all A , then -"^ 2rTxi'^l O uniformly in A . (That is, for every 6 > <^ there exists /V such that for all 7i > /S/ and for all A , Remark ; Since the ij. are finitely additive and s-bounded. oo it follows from Theorem 2.2 that / /^>^ L W converges unconditionally. Therefore ^ ryo'^' exists for every A and the 35 PAGE 41 36 conclusion makes sense. Proof ; Assiime the conclusion is false. Then there exist an Â£ > O and sequences (^tj , |^i,j such that L'^ i | > ^ ^ LI , ffi , To simplify notation, assume 71~ L . Let i-i' I . i-ieÂ£^ I ^ ^ } ^o there exists A/, such that iJtA^'^^^ I ^ ^ for all A C [M ,Â«=) . Letting 7^=A, /OEl, ^"0 wehave JL/^>(t)\ = JZAM I> I^^,^^)| Since Jci/yyu LL ('t) fÂ°^ ^H t^ , we can pick 6j_ such that 2_ l/^i ( w) ^ JZ ' ^ / ''i converges unconditionally, so we can pick /\l^ > Ki, such that for all A^L^(2,Â°^) , |W)|< ^ . As a result, E A^,W . I ^''^^'^ + 51 /^^:i ^^^ + ^ ^^ ' ^^'^ therefore > '] Choose L^ such that /_ // ^3'^'/ ^ 7^ and pick that for all A , \YLf^^^^^^ < fl . Then, /V, so as before PAGE 42 37 / EA^^^^ I >A , and we let % = ^i, ff^.^/^j-lj . Continue in this manner to define an increasing sequence of positive integers l^ij such that: (3) letting Tf^^K^[l^K-i,f^K''l > (^o' ^) . we have Note : If A is a finite set disjoint from lNf(.\,^n'U , then it follows from (l) and (2) that \/^i-J^)\ ^ f . I^t ^m= i^Y^>Â«-')i^--| ' M = /,a/" Â• For all values of 7k , U-fri. is an infinite set. Also the sets { i3^j are disjoint because 7w, ^ "w^ implies that 2 '(2^,-1)^ 2 U?Â«^-lj for all values of 71, and Tlj^ . Since the sequence I^k) is disjoint, it then follows from the preceding statement that l^^Tnlyn:! i^ a sequence of disjoint sets, where U Uy^ is defined to be L/ I T*' f l^TnC Â• In other words, I have divided a countably infinite collection of sets into a countable number of countably infinite collections^ and then have taken the union of each PAGE 43 38 collection. Since all the sets were disjoint, all the unions are disjoint from each other. Since y^i, is s-bounded it follows from Theorem 2.3 that /^<-/ is s-bounded, so >^ /Ui:,(UBÂ„)-0 . Hence there exists an Wo so that /^i,i^Â°ynJ ^"T Â• Therefore for all AcUB^ , o |A,W/ PAGE 44 -}>3 js we lyU/ ((j(/^/l/^)jJ < "^ . Combining all three inequalities Obtain KM|>/AÂ„('rJj-|r-f >-|:-f-|-=f . Therefore \/^L^(Ur)l>^ ^ '^ f T Â• There are an infinite mimber of ?J{ in P , so j/^i^'l'^/^/l i/cannot converge to zero, which contradicts the hypothesis. This proves the theorem. Q.E.D. Remark : Theorem i^.l implies Theorem 3.I. Corollary I4.2 ; Let Xi^:^--^^ be finitely additive and s-boimded, yi-l.Z/" . Assiime Â£1^ /^-^fEj exists for all E . Then if /tKj is a sequence of disjoint sets, Xom, 2~ '/Â«Â•*; "/ hJ(E^kJ ~ ^ uniformly in A . Proof ; Define '^y^' (P(7l)-^ ^ as follows: )4.^A) = (/^n+/ ~^'N^l({t^J Â• The v'^ satisfy the hypothesis of Theorem li.l, so Ji^nv 2[rn-H '/^nji^Hl ~ -^'^ Z_ Hi^/^^ = O uniformly in A . Q.E.D. ana Corollary 1^.3 ; Let yUy^' T.~^^ be finitely additive 5-bounded, 71= I, X,'-. Assume /M/w, y" ( Â£ j exists for all Â£ , PAGE 45 ilO and let /^(^) ~J^J^n(^) Â• Then yU is s -bounded. Proof ; Assvune u is not s-bounded. Then there exist 6>0 and a sequence of disjoint sets jExj such that for all K , j/"^^KM^^ Â• Let >t,r| . Since iim./^n,(^i^)= O , there exists K, such that //^>Â», fc./KT Â• However ^^j/^Â„ (Â£Â«,)/ = ' l/'l^K,)))^ so there exists n^> W, so that \/J.y^ {Â£^^^\y^ . Therefore |6^n^-/"?t,)(fK,) j ^-^^ ' J " -J ' Because Jliyrro U.-H (j^\-0 , there exists K;^ such that I'^^j.lfKj/l ^ 3" * so that |/*n3 (f K, ) I > ^ and I ^/n,-/^ Â« J(Â£/,J | > f ' f = f Â• Continue this process to define sequences \^\\ , \^j\ where j^ij is monotone increasing and for every integer l , (Wn" /'W' )( ^K' )l ^T Â• Therefore the sequence )LY*'i^i~/^L)\^^j}(. does not converge to zero uniformly in A , and this contradicts Corollary lj.2. Hence the theorem is proved. Q.E.D, Definition ; Let M. ' l_''^ JC be finitely additive and s-bounded, 71 = I, X,"' Â• We say the yU^ are uniformly additive PAGE 46 Ul if Zi/m. 5^r; **1 A ' ^ uniformly in A and n whenever (EkI is a pairwise disjoint sequence of sets in Z_ . That is, for all Â£ > O , there exists J" such that for all J 1 J , all n , and all ^ ' kfrJ,Â«Â«)nA / ^ '^ Â• Remark ; When the Uy^ are countably additive this is equivalent to the Ufi being uniformly countably additive. Theorem U-k ' I^t ^^: L~^ ^ be finitely additive and s-bounded, 11 = 1,1/ ' . Assume u(E) ~ X^ /^-niE) exists for every E . Then the U-n are uniformly additive. Proof ; Let |Â£k{ be a sequence of disjoint sets and let dfO . I^t V^=(P(n)-^^ be defined by \^-n(t^) ^ {^n-/^)(tiL^l) Â• By Corollary \x.3> fJi. is s-bounded, so the v'71 satisfy the hypothesis of Theorem l^.l. Hence there exists an integer H such that: (1) \!uL^^~^^(^^\^ ^ , for all 71 >N and all A . Since u. is s-bounded, it follows from TheÂ£)rem 2.\\ that ^j '' '^ ' converges unconditionally, so by Theorem 1.2 there exists I , such that: PAGE 47 and for all 142 Combining inequalities (l) and (2) we have that for all n 1 M iKeAntl.,"] )<'Â«Â• Each Uy^ is s-bounded, so there also exists J.^, such that iKf^nfr 00) I ^Â°^ ^11 ^ ^i^i^ ^11 n< ri . Consequently l^'r ^ Ki6 for all n and A . Since Â€ was chosen arbitrarily, this proves that the Uy^^ are uniformly additive. Q.E.D. Remark ; Corollary i|.3 and Theorem I4.i1 are the intended generalizations of the Nikodym theorem. Corollary U.^ i Let /^^Â•' ^"^R be bounded and finitely additive, n=/,i,--. If /^(b) -= /o*rx./^y^(B) exists for all Â£ , then the Uy^ are uniformly additive. Proof : It follows from Theorem 2.2 that the Uy^ are s-bounded, so the conclusion follows immediately from Theorem I4.i1. Q.E.D. Corollary ii.6 (Nikodym) ; Let M-nT. "^ ^ be countably PAGE 48 h3 additive, ?i = /,2,'-. If /J-(t) ^^^^^^^/^yJiB) exists for every E , then JU is countably additive and the countable additivity of the JJ-n is uniform. Proof ; Let Â€>0 and let {En] be a sequence of disjoint sets. By Corollary 14.3, yU is s-boimded, so TL /^i'^Ki exists, . / K=l hence there is an I, such that for all til, : By Theorem ii.Ii, the Uy^ are uniformly countably additive, so there exists I^ such that for all n and all / 2 I^ , \II..^^(^Kn^~X' which means \/^niK=L Kjl ^ i" tiecause each yU;i is countably additive. Therefore 00 for all c > Ij^ . Let I rÂ»Â»Â«i'' |I, ,1^5 . Obviously // is finitely additive, so; Combining inequalities (l), (2), and (3) we have j'' 1^=1 ^1 Â•< + -jjl" = Â€ . Since 6 was arbitrary, this implies that PAGE 49 Definition ; LetM.L~^^ and V Â• Z""^ R . We say ii is absolutely continuous with respect to v* if for every 6 > O , there exists 0>O so that ^(B) PAGE 50 xes such that for every S>o , there exists an Â£ , such that v'^E) PAGE 51 1^6 Assume there exists i ^^, > i^ such that '^'"K+il Â« l^^,) | Tf Then define rK+l ' 'K ~ ^^'k+I Â• '^ this process continues we obtain a countable collection of sets Fk such that and 6 because of (l) and the fact that Fk H Â£"; ,, C E; Therarore ) f/'c,,, "/. J^Fk" F,Â„) | > X . X = X . Note that |'K''k+0 is a sequence of disjoint sets, and iKiiS ^*^"^' J)\^~^*\) |j=/ does not converge to zero uniformly in A , which contradicts Corollary [j.2. Therefore the process of choosing the FÂ« and L^(^ has to stop, so there exists fy^ , i-K I ^ such that for all ]><-[(, , K^KnÂ£j)|^f . Let ^=.K ,H, ^Fk ./xf^//.,, , and t ^ -'^P^-ti n, . Then the following three results obtain. I. Ia('h,)| PAGE 52 hi we have that F, = F. = (F. -F, ) U (F^^F,) U Â• U (F^.F^) U F^ . \/^,(Ql > 6 and l/^i(F, -FJ I -/A^F, n f, J I ^ ^if^ < f , since F.nEi.C^i^ and 2 ^ . ^ . /a(/^ "Fj/r | A^F, D E J| < ^Â•Â•a+i /6 because Z , > i^ > 2 , hence tj > V . Thus for I"-*y^ -+ 3j + '"+ |/l( F/J I , Since H,-Fk . Therefore III. KYff)l>f-f . Note that Cp + i (tp^^j HJ L/(Â£p ^.^ D H,j . n, was chosen so that for all J > P, , JAj( S ^ ') | ^ 7/~ Â• Hence Note ; From I and II we have that |(/^a.'A'')( Hj | > ^ ~ -^' J^ . Let r, ti . Assume that there exists tj_ > I such that A'i;,^ C^-t^ y I Z y, and let h^^ = p, b.If we could continue to choose c 'K j indefinitely, then we would obtain a contradiction the same as before. So there exists rj^ PAGE 53 1^8 and i-K ^^ so that for all J > i-K. , (i) ^ (1) 4 and (I) (>) Then H,nH^=D because Hx^E; and Â£ , Yl H ,= D . 6) (Recall the construction of E, .) The following results are then obtained. JX'YHjj.e-f-f I'-. )/'A)k^Since Hi^ t , ^ Â£p|tl and P,-K >2 , we have that /'i'YHJI .Since Hj=Fk"' Â• Since F> E/"c E/"c Ep , and C,>, ,Â„e have that j/lY^ '" f^,')^ KVpf'^ <) | < Tpri^ < ^ . Likewise j/^1' (f/'^Fj^^'^) | i kfY^/o Ec^',) I < /T^vi^ ^ ^ because i^^^ >j for all J=|^i,...K-l . Hence ^ i< -1+ X+ . . . + + PAGE 54 h9 X'^rHj),f.)^>J| ,,,,,, ,,,,,,3 ,,^, )X'>(H,)| > III.. )/'f'(Ef')l>6-f-|, for ail.. Since H,nHi = U , the sets (f p, + p^+i H H,) , (^d^fl^i " H.) , and ('f,;.P,.r^H, OH,)) ^^^ ,.3.Â„.^^^ Therefore Â« ^ |/*f' (fp,tP^n ) ) i (/"f (^P, + P;,. i '^ H,) | + Note that l/'fY^P,-''.^."^^.))<|an. )/-?Y^p,,p^,,nH,)j PAGE 55 so All of the above statements are true when K=Z . I will now show that their being true for K implies they are true for K+ I . r M. c ^K) . ^ . . Let Tj ~ 'I J i/ = Z Â• Assume there exists ^j^/'-i such that fK) |/^ti ( r, D t^^ ; I i ^k+;l , and let r;^ = F, -t^ If there exists t^ > ^^ so that jAtj C f^ ''t^^ jj r. ^k + z , ^(K) (K) p(K) let lo ~ >x ~ ^1 ' -^ this process of choosing Lj and 'j should go on indefinitely we obtain a contradiction because for all j , )A,(Fj., -h, jll/'^j (F,., HE,^ Jl > -^K+1 whereas rj.| '' tj^^ ^ ^r ^ ^i + ' ' "'' f^K ''' "^ J ^^'^ ^J^^ iJ^plj Combining these two inequalities we see that for all J , zero uniformly in A . This contradicts Corollary i;.2. Therefore does not converge to PAGE 56 51 Hence there exist r^ and 'i.>Z such that for all j y i^ , \^0 [Ft ^^j )\<^3 ' I-et pK^r-'-t > HK+i= ^Â± > In order to show the induction step is valid, I must prove the following four statements. I". For all J = (..-K , Hjr)H,^+, = D Â• "k + 1 ^1 'I by construction, and t " '-P + Â•Â•Â•Â• Rz + i (jV.'^j) . Hence HÂ„t,n(uHj)=D . Since H^tiC Ep^+,.. +p^ + l , j/"| i^Ktlllbecause F, ^ 2 Â• iiiM. (^Â«'(H^J)>Â«-i.-|.-...--l^ . Note that Â£,'^WFrFnufFflFf)^-^fFt^'-Ff>Fi'' Â• Also (/^[Â«(PÂ«' -Ffjl^ l/^r^OE/f )1^ ^^rrr^niTi < ^K-^-i + l because p^ ^^^i. ^^Pi^ '^Ph.'^^X ^^'^ ' *^ '"i Â• PAGE 57 .+J 52 Likewise )/^ , (fx'h )i~l^l (fx ^ E^j j| < /,+ Â• tP^ + ij-HX < K+X-i-X because r^ '' '=13 ^ ^ Pf -i'Â• + Pk + ^3 ^^^ ^^*3 Â• 1 (K)/ (K) p(K)\ In the same manner, for jz \---t-t we have jr 1 ^"j "'j-ti / because J ^ij+, . Therefore Â€ -|" ^ " " ^/ ^ l/'f'^Yf f*^^]! Jk+XTT + ^K + X + Z + Â• Â•Â• -^ ^K+2+Ct-O "'' JA| ('^K+/)) ; the first inequality follows from the induction hypothesis and the second inequality results from the definition of n;^+; and the inequalities in the preceding sentences. Therefore, Since -^*^ + ;i + i +Â•Â•Â•+ ^K + i + (t-i) ^ ^'^"'"'^ ' we have IV... For all C , l/rUeT")\> ^-f-'-jfe Â£ -^ Â• Â• ' -jTT 3 because of the induction hypothesis. As a result of the construction of Hk+z and because Pk+i"*" *Â• -^ '^K + l ' we have |/'pÂ«+, +i ( &p^ + , +t '^ Hj^^,) j < ^"l^TX . Thus Kt (Â£1 ilKpKV^t^p.Â„+i H^+,)|>^-f ix^' i'<+^ Â• PAGE 58 S3 Note: It follows from II" and III" that 6) I Therefore, by induction we may define infinite collections [Hk] and [j^f] such that for each ^>X , [(/^f^''^-/ ^^0(^H^)| > ^ â€¢ 8 2*^+;l ix '' ^ " aT" 32 ^TT Â• irlKi is also a sequence of disjoint sets. Letting V a ^ V = >u' VH-/'f'''^^ . then \(K^^^^^LK)]^ h f or all 7: > Z . So //aA*> ^^' ^^ ' )h:l '^Â°Â®^ ^Â°^ converge to zero uniformly in A , which contradicts Corollary \^.2. This concludes the proof of the theorem. Q.E.D. Definition ; A linear topological space is said to be locally convex if it has a base consisting of convex sets. Let Â•^:(t'^:./f A] be such a base. Let '}'[^^iU \ Oi^^^ . For each D^ xn ^ , define I^ fx) = f^: ^>^j a ^ ^Â•'J and P^ (t)z Mil /a dil^M j . Then A is a pseudo-norm for the space ^ , and has the following properties. (a) /^MIO (b) ^()()< +00 PAGE 59 (c) fl^Cd^)-' a^/i) (d) if )(.Â£Uu , then aM^ I (e) /i^(^^i)t^^U)-^/^u(^) That is, A has all the properties of a norm except that it may be possible to find an X such that yC^O and A/^Li) O . In the theorems of this chapter the fact that \%\ ~0 implies %-0 was never used, and with appropriate modifications the theorems are true if the set functions take values in a locally convex linear topological space. First of all, one has to define s-bounded and absolute continuity differently. Definition ; Let 3^ be a locally convex linear topological space, and let //^Â•Â•^f/^j be a family of pseudo-norms that determine the topology of ^ . Let y/ ; 21Â—^3^ and v'; Â£^ Â— => R."*" be set fiUiCtions. We say u is strongly bounded (s-bounded) if jMnjU.(^K)'0 whenever {Â£^1 is a sequence of disjoint sets. Lc is absolutely continuous with respect to v' if for every oi and 6 , there exists <5 > o such that v(f)^<$ implies /}[u{^))<.C . The topology on 3t determined by a single pseudo-norm o< is PAGE 60 not necessarily complete. We then let Tt^ be the completion of 7L , and if jJ.'2-.~^^K. is s-bounded, we consider /^ to have values in oo X^ i si^d xt is then true that 2_/ (^>Â«) converges unconditionally in t i. fÂ£.l i= a sequence of disjoint sets. The statements or Theorem i^.l and Corollary 1^.2 do not change if ?i is locally convex, c~ ^ except that Z^/'nW is considered to be an element of Kj for a fixed <^ . Taking into consideration the new definition of absolute continuity, the extensions of the Vitali-Hahn-Saks and Nikodym theorems for finitely additive vector measures are valid when the range of the measures is a locally convex linear topological space. I PAGE 61 CHAPTER 5 A COUNTEREXAMPLE In this chapter an example is given to show that weak convergence does not imply strong convergence in the space of countably additive set functions. Let y/A.-n: (P(n)-^ R be countably additive, and assume that. for all A , Ay^^^(A)O . Then from Theorem i|.l, we have that .Uyrrv^^{/\)z o uniformly in A , which in turn implies that x^nj/t-f^l O , where //in I is the total variation ofy/^ . Since Theorem 1^.1 was very important in proving theorems about set functions from a (T -algebra to a Banach space, it is reasonable to pose the following question: if u : Y. "^ R and if >^5^^Â„^Â£-)=0 for every E^H , does it follow that ^^'>>t' //^n/ O ? The answer to this question is, "No," as the following counterexample shows. 56 PAGE 62 57 C ounterexample ; This counterexample will show that if /t^; Z'^R are countably additive and if J^xm. iU^(E)-0 for all Â£" , then it does not follow that j&yry\j \jj.y^\~ O . Construction ; Let YL be the Lebesgue measurable subsets of \S> t I ) i and let A be Lebesgue measure restricted to L. Let -f^ be defined: {I if X is in (^n. J pT ) > O ia < i^ , a even I if X isin(^^,|Â±i) , o<(i<^ , CL odd O otherwise Let /c^(E)-J'fjidA Â• Then for all S>0 , ^(b)<6 implies )iy^dy\ PAGE 63 58 Since Â€ was chosen arbitrarily, the conclusion follows. Now let V be an open set in Land let 6 > O . There exists a countable collection of disjoint open intervals i^i) oo such that V U u(^ . (l am considering the null set to be an open interval.) Then AtW = .1A((9[;j ^ and, since X(\/) ^ oo , there exists M such that XiV-QuAc Â£ ', hence for all yi , |/"nW-/^K(V; (S[^)|< 6 . For all t-/^..-,N/ , let Tt be such that for all K>Tt. , JMk((^uI< IT Â• Then letting T-VW'i'^L-^ = fj'-->^j } we have that for every ii>T , //^kM/lM'^)-M^,^<^)l-^\t^K((^i)l<^ ^^(fr)= l^ Â• Therefore we have that y^^ My,\^)-^ ^Â°^ ^^^ Â°PÂ®^ ^^^^ ^ Â• n /" Let Â£" be a set in Z , and let ^ >0 . There exists an open set V so that ECV and X(v-^)<Â€ , and it follows that M'riv^'^)^^ for all n . By the result of the last paragraph there exists T such that for all >^ > T > \/^yJy)\^^ ' hence |y"7i(Â£)| |a W \ ^ I/^hC^-E) j < XÂ€ . This ijnplies that Ziyrro jU y,(B] = O for every Â£^ in Z. . The yW^ are countably additive, so they satisfy the hypothesis PAGE 64 59 of the counterexample. However, \/^Tt\(o,l) -ilfn\
PAGE 65 REFERENCES 1. Ando, T., "Convergent sequences of finitely additive measiires", Pac. J. of Math ., 11, 39^-hOh (l96l). / / / 2. Banach, S., Theorie des operations lineaires , Monografje Matema tyczne, Warsaw {1932). 3. Brooks, J. K., "On the Vitali-Hahn-Saks and Nikodym theorems", Proc . Nat. Acad . Sci . U.S.A. ;'6l4, ii68-lj71 (196?). k' Brooks, J. K. , "Representations of weak and strong integrals in Banach spaces", Proc . Nat. Acad . Sci. U.S.A., 61^, 266-270 (1969). 5. Brooks, J. K. and J. Mikusinski, "On some theorems in functional analysis", Bull . Acad. Pol. Sci., Math., Astron . , Phys., 18, 151-155 (197077 6. Darst, R. B., "A direct proof of Porcelli's condition for weak convergence", Proc . Amer . Math . Soc, 17, 1091^-1096 (I966). 7. Dunford, N. and J. Schwartz, Linear Operators , Part I: General Theory , Interscience, New York (l9Sti). ~ 8. Dvoretzky, A. and C. A. Rogers, "Absolute and unconditional convergence in normed linear spaces", Proc . Nat. Acad. Sci. U.S.A., 36, 192-197 (1950). 9. Hewitt, E. and K. Stromberg, Real and Abstract Analysis , Springer Verlag, New York (1965). ' 10. Hahn, H., "Uber Folgen linearer Operationen", Monatsh . Math . Physik. , 32, 3 (1922). 60 PAGE 66 61 11. Hilderbrandt, T. H. , "On unconditional convergence in normed vector spaces". Bull . Amer. Math . Soc . , 1|6, 959-962 (19I4O). 12. Nikodym, 0. M. , "Sur ler suites convergentes de functions parfaitment additives d' ensemble abstrait", Monatsh. Math. Fhysik. , 1^0, k27-h32 (1933). 13. Orlicz, W., "Uber unbedingte Konvergenz in Funktionemaumen", Studia Math., 1, 83-85 (1930). lli. Pettis, B. J., "On integration vector spaces". Trans . Ainer . Math. Soc, hh, 277-30i^ (1938). 15. Phillips, R. S., "On linear transformations", Trans . Amer . Math. Soc, I48, 516-5I1I (19U0). 16. Phillips, R. S., "Integration in a convex linear topological space". Trans . Amer. Math. Soc, hi, III4-II5 (I9I4O). 17. Rickart, C. E., "Decomposition of additive set functions", Duke Math. J., 10, 653-665 (19^3). 18. Rickart, C. E., "Integration in a convex linear topological space". Trans. Amer. Math. Soc, 52, 1+98-521 (19142). 19. Saks, S., "Addition to the note on some functionals", Trans . Amer. Math. Soc, 35, 967-971^ (1933). 20. Schur, M. J., "Uber lineare Transformationen in der Theorie der unendlichen Reihen", J. reine u. angew. Math . , l5lj 79-111 (1921). 21. Vitali, G., "Sull' Integra zione per serie". Rend. Circolo Palermo, 23, 137-155 (1907). t PAGE 67 BIOGRAPHICAL SKETCH Robert Jewett was born in Portsmouth, Ohio, on August 20, 19li5Â» When he was eleven years old his family moved to Fort l^ers, Florida, where he graduated from high school in 1963In the Fall of the same year he went to the University of Florida on a golf scholarship, and played on the golf team for two years. In 196? he received his Bachelor of Science degree in Math, and from that time on has been working toward his doctor's degree. He is a member of the American Mathematical Society. He was married to Suzanne Strobak on July 3, 1971. / * 62 PAGE 68 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. C /(/nui (Y /1\-^^^<'^-? James K. Brooks,Chairman /Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David A. Drake Assistant Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Zoran Pop-Stojanovic Associate Professor of ^Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. w -cl. Gene W. Hemp Associate Professor of Engineering Science and Mechanics PAGE 69 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Stephen A. Saxon Assistant Professor of Mathematics This dissertation was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council , and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1971 M'ege of Arl/s ar-" '' Dean, College of Arlis andfSdiences Dean, Graduate School PAGE 70 GAl 1 13 5.78. xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E32BUQNPM_92WWWK INGEST_TIME 2017-07-14T23:04:39Z PACKAGE UF00097676_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES |