ON THE BIDUAL OF JC(X)
AND
THE RIESZ REPRESENTATION THEOREM
by
Hermine Engels
A Dissertation Submitted to the Faculty of
The College of Science in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
Florida Atlantic University
Boca Raton, Florida
August 1992 ON THE BIDUAL OF K(X)
AND
THE RIESZ REPRESENTATION THEOREM
by
Hermine Engels lis dissertation was prepared under the direction of the candidate's thesis advisor, Dr. Helmut Schaefer, Department of Mathematics, and has been approved by the members of his pervisory committee. It was submitted to the faculty of the College of Science and was cepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in athematics.
SUPERVISORY COMMITTEE
Chairperson, Thesis Adviso
11 ABSTRACT
Author: Hermine Engels
Title: On the Bidual of /C(X) and the Riesz Representation Theorem
Institution: Florida Atlantic University
Dissertation Advisor: Dr. Helmut H. Schaefer
Degree: Doctor of Philosophy
Year: 1992
Locally compact spaces having property (B) are introduced. (B) is the property that each
bounded subset of the topological vector space IC(X) is order bounded. It is shown that there exist spaces X exhibiting (B) that are not paracompact. Also discussed are the consequences of this property for the dual and bidual of IC(X) . Both show features resembling those from the Banach space case (X compact). Under (B), for example, the bidual of /C(X) is Riesz isomorphic to IC(Y) for a suitable locally compact space Y.
We also introduce doubly bounded Borel functions Bb(X) which are bounded and vanish outside some compact set. Without using the Riesz represe ntation theorem they are imbedded in the bidual of IC(X) , extending the approach of H. H. Schaefer (19][20] for compact spaces to locally co mpact spaces. Further it is shown that the regular Borel measures form a band in the Riesz dual of
Bb (X). These results permit to give a topological characteriz ation of regular Borel measures on
X, which yields the Riesz represe ntation theorem as well as a distinguishing property of regular
Borel measures as fairly immediate conseq uences. Finally, some relations between Baire and Borel measures are cl iscussed.
Ill ACKNOWLEDGEMENTS
I would like to thank my advisor, Dr. H. H. Schaefer, for the many stimulating discussions we shared on the topics of this dissertation. His patience, enthusiasm, and encouragement were most
beneficial to me in completing this work.
I am further indebted to Dr. J. Brewer for his help in making my transfer to Florida Atlantic
University possible. I am grateful to Dr. S. Locke for his kind help in resolving many administrative difficulties. I would also like to thank the rest of my co mmittee, Dr. C. Lin and Dr. A. Mandell for their time and interest.
Special thanks go to my family. Without their understanding, support and love this dissertation would not have been possible.
IV TABLE OF CONTENTS
ABSTRACT ...... Ill
ACKNOWLEDGEMENTS IV
INTRODUCTION
TERMINOLOGY AND NOTATION 4
CHAPTER
THE BIDUAL OF K(X) 5
A. The Space K(X) . 5
l. Preliminaries 5
2. Dedekind Completeness 6
3. Bounded Sets . . . . . 8
4. Spaces Having Property (B) 10
B. The Space 9J1(X) 13
1. The Dual of K(X) . 13
2. Lebesgue Property of the Strong Topology 13
3. Weak Compactness of Order Intervals 14
C. The Bidual of K(X) . . . . . 15
1. Basic Properties 15
2. Order Continuous Linear Forms in K(X) 16
3. Representation of the Ideal Generated by A:.( X) 18
4. Order Continuous Dual of 9J1(X ) 21
II RADON , BOREL AND BAIRE MEASURES ON LO C ALLY COMPACT SPACES 24
A. Borel and Baire Fun ctions 24
1. Borel Functions 24
2. Baire Functions 26
v 3. Relations between Borel (Baire) Functions on a Locally Compact Space
and its One-Point Compactification 28
B. Borel and Baire Measures. Regularity 30
1. Identification of the Duals of C0 (/'() and ry)Lb(X) 30
2. Imbedding of the Borel and Baire Functions into the Bidual of K(X) 32
3. Regularity ...... 35
C. The Riesz Representation Theorem 39
1. Reg ular Borel Measures and the Riesz Dual of Bb(X) 39
2. Topological Characterization of Regular Borel Measures and the Riesz
Representation Theorem 41
3. Baire Measures 43
4. Supplements 46
LITERATURE 49
VITA .... 51
VI INTRODUCTION
In his papers on Radon, Baire , and Borel measures on co mpact spaces I and II ([19] [20]), H.H.
Schaefer chooses a new approach to illustrate the connections and relations between the set theoretic and fun ctional analytic aspects of the tllPory of intPg ration on compact spaces. The subject of this thesis arose from the id ea to extend this approach to locally compact spaces. The connection between the locally compact case and the compact case is established through the one-point compactifi cation of X ; however, the details of this progra m turned out to be considerably more complicated than one would anticipate. In solvin g this proble m we noticed that for the space K(X) of all continuous, real-valued fun ctions on X with compact support a nd es pecia lly its I idu a l (he re de noted by K(X)) , some signifi cant results ca n be obtain ed . These as well as the preparatory work for C hapter II are contained in C hapter I.
In Section A of C hapter l we explore certain propertif' s o f the space K(X) (particularly, order completeness properties ) whi ch are indispensable for the sequel. We also introduce (Definition
A.3.1) locally compact spaces having property (B). Property (13) of a lo call y compact space X is the property that each bounded subset of the topological vec tor space K(X) is ord er bounded ; it is well known that every paracompact X has that property. For this, a new proof is given in Theorem
A.4.1 ; more importantly, it is shown in A.4.l that ther(' f'X is t. spacf's X exhibiting (D) that are not paracompact. The consequ ences of this property are numerous a nd signifi cant. for example , weak compactness of the order intervals of the dual 9J1(.X) of K( .Y) (13 .3.2) or order continuity of the strong topology ;3(9J1(X) , K(X)) (8.2.1).
Moreover, property (B) impa rts a particula rl y transparent stru cture to the bidual K(X) of
K(X) ; on the whole, both 9.11( X) and K(X) exhibit Ri esz theoretic features resembling those which are well-know n to hold, and basic for topological measu re theory, whe n X is compact. Perhaps the most striking of these is that unde r (B) , the bidual K(X) is Ri esz ison1orphi c to K(Y) for a suitable locally compact space Y (C.3.l and C.4.3). These qu estions a re investigated in Sec tions Band C.
A main result of Section C is stated in Theorem C .l .3. 1!. tnrns out that the order continu ous I in ear fo rms on the id eal J of K( .Y) generated by ,\~ (X) are precisely the e le me nts of 9J1( X).
This is a surprising res ult (Example C .l.2) whi ch leads us to ma ny other in teresting statements (I.C.3.3, I.C.4.4, II.B.1.3, !LC .l .l) and turns o ut to be essentia l even fo r the new proof of the Riesz
representation theore m (II.C.2.2).
The second ma in objective of Section l. C is to take a closer look at t he ideal J in K (X)
gene ra ted by K(X) . We come up with two results whi ch again a re very helpful in the attempt to
discuss Radon , Baire a nd Borel measures o n locall y compact spaces. F irst, J is order dense in the set of all order continuous linear forms in K(X) (C.2.1) and second , there exists a locally compact space Y such t hat J is Ri esz isomorphic with ..\:(Y) (C.3 .1 ). These results permit us to give a characteri zation of the order co ntinuous dual of 9J1(X), whi ch turns out to be Riesz isomo rphic to the ideal of a ll bounded , continuous fun ctions on Y that a re in tegrable against any p. E 9J1 (X).
Moreover , the consequences of property (B) (hinted at above) a nd , additionall y, barreled ness of
9J1(X) come into view very clearly now (Theorem C.4.3). If both assumptions ho ld, the bidua l K(Y ) of K(X) shows a ll essentia l features famili a r fr om the Ba nach space case (X compact). Unfortunately, outside of para.compactness of .X no reasonable conditions seem to be know n t hat imply 9J1 (X) to be barreled . Finally, the rather special case of X being discrete is cha racteri zed by refl exivity of either K (X) or 9J1 (X) (C.3.4 a nd C.3.5).
C hapter I I sta rts wit.h the definitions and basic properties of Borel a nd Ba ire fun ct ions; for example, equivalent definitions, relations between Borel (Baire) fun ctions o n a locall y compact space and its one-poin t compactification, and id entification of the dual s paces of Co (!\) a nd 9Jlb(X). Very important for the sequel is the introduction of t he doubly bounded Borel fun ctions Bb(X) (Definition
A.1.4).
The classical Riesz representation t heorem (represe nting each continuous linear fun ctio nal on
C(X) by a unique regul a r Borel measure on.\" wh en X is compact) is the essenti a l tool for imbedding the space B(X) of bounded Borel fun ctions in to t hr srcond dua l C( X) of C( X) , the imbedding map
J bein g an order O"- continuous Ri csz isomorphi sm. In [19] an ind ependrnt approach was developed to construct t his mappin g J (see 13 .2. 1) assuming X to be compact, and t hence to obtain the
Riesz theorem a.s a. comparative ly s imple consequence. To exte nd this procedure to locall y compact spaces, we employ the one-poin t (or Al exand roff ) compactifi cation /\. of X ; t hro ug h several somewhat arduous steps, we obtain a correspond in g imbedding 4' of Bt.( S) into K( .X) (in fact., into t he ideal
K (Y) o fK(.X) generated hy }((X), Theorf' nl 13.:2.7). T lw canoni cal duali ty of9J1 (X) a nd K(Y) now immed iately defin es the in tegra l ffdp :=< p , f > (=< fl , ~ ~ (f) > ) fo r every doubly bo unded Borel
2 fun ction. C learl y, t he id eal Bb( X) of doubly bounded Borel fun ctions is the largest subset of the space of a ll Borel functions X- !11.: that. will r ermit. such a n in1b edding.
Section B.3 begin s with a set theoreti c definitio n (3.2) of (regular) Borel measures. We also give a functional a nalytic characteri zatio n of regul a r Borel measures, id entifying them with certain order a-- continuous linear forms on Bb( X). The main result in Section B provid es us a. lattice property of the regul a r Borel measures (Theorem B.3.5). This property is needed and used to show that the regular Borel measures form a band in the Ri esz dual of B&(X) (Theorem B.3.7) ; see a lso B.3.8.
In Section C we reach our final goal: Under t he duality of Bb(X) and 9J1(X) established above, we prove that the regul ar Borel measures on X a re precisely the linear functiona.ls in 9Jl( X) (C.l.3), thus showing these measures to be t he continuous fun ctionals on JJh( .X) for a very natural Lebesgue topology (namely, o(Bb(X),9J1(X))). Sin ce K(X) is dense in Bb(X) for this topology, t he Ri esz representation theorem now emerges as a. simple appli cation o f a well kn own basic result on the extension of uniformly continuous mappin gs. fn addit ion, we obtain another characteriz ation of regul ar Borel measures (Theorem C.2.4). All of these results appear to be new. The approach also yields a very transparent illustration of the relations between Baire and Borel measures (in particular, of the intrinsic regularity of Ba.ire measures) .
Finall y, the last section (C.4) discusses the topologization of the Riesz space Bb(X) , which places t his space into a. certain para ll el t. o the topologization of A.: ( X) that served as the starting poin t to define the space 9Jl(X) of a ll Radon measures on X .
3 TERMINOLOGY AND NOTATION
The general fi eld of this work is Jnt egration o n Locall y Compact Spaces in the spirit o f Bourbaki
(5] (6]. In addition, for t he theory of Ri esz Spaces (Vect.or Lattices) a nd Topological Vector Spaces (in
particul a r, the duality t heory of locall y convex spaces) o ur mos t. im portant refe rences are Schaefer
(17] (1 8] and , to some extent, Fremlin (9]. Genera ll y we foll ow t he termino logy and notation of (17]
and (18] which is wid ely used elsewhere as well. For example, t he notation CJ( E, F) a nd f](E, F)
are e mployed for the weak and strong topologies, respectively, on E with respect to a given dua.lity
< E,F >.
For t he benefit of the reader we li st a. few terms perh a ps not so commonly used , or specially
introduced for t he purpose of t his t hesis, wit h indicati o n of t he page wh e re t hey first occur.
Ba.ire (function) 26
Baire measure 43
Borel (fun ction) 24
Bore l measure 35
Doubly bounded real fun ction 25 o(E,F) 13
Proper (conti nuous) map 18
Properties (B),(B0 ) of a locally co mpact space 8
Quasi-Stonian space 6
Radon measure 13
Regul a r Baire measure 43
Regul ar Borel measure 35
Sto ni an space 6
4 CHAPTER I
THE BIDUAL OF X::(X)
A. The Space K(X)
1. Preliminaries
Let X be a loca ll y compact space a nd le t. A.:( X) he the vector space of a ll continuo us , real valued functions on X with compact support. For a.ny fix ed compact subset I\- C X , denote by
(KK(X) , 'I A-) the Banach space of fun ctions in K(X ) t hat a re supported by I\- , with the uniform norm ge reratin g 'I.g .
The space K(X) is the union of the lin ear subs paces KK(X ), wh ere I\ runs through the compact subsets of X . We eq uip K(X) with the fin est locally convex topology 'I for which a ll the canoni cal injections ig: KK(X)-+ K(X)are continuous , that is , the inductiv e topology with respect to the famil y {(KK(X), 'IK ,iF\) : /\- C X , I\ compact.}. The spaces (Kg(X) ,'Ig) a re 13 a nach spaces and therefore barreled [17 , 11.7 .1 Coroll ary]. Sin ce the property of bein g barreled is preserved under the formation of inductive topologies, K(.Y) is barreled [17 , 11 .7.'2].
'I is fin er than the topology 'Iu of uniform convergence on compact subsets of X, hence Haus dorff. Thus by [17 , II.6.3] (K(X) , 'I) is the inductive limit of t he subspa.ces (KK(X), 'IK ); i.e.,
(K(X), 'I) is isomorphic with the locally convex space (f} A.:K(X)/N , where N is a. closed subspace of ffi K 1,(X).
The topology on KK(X) indu ced by 'I is coarser than 'I1,· . On the othe r hand s in ce 'I is fin er than 'Iu the topology on Kg(X) induced by 'I is fin e r than 'If\. Thus 'I induces on each KK(X) its own Banach space topology.
Finally, let us note that 'I is the order topology 'Io. 'Io is the fin est lo cally convex topology for which every order interva l is bounded. By [17 , V.7.3] K(X) is a. locally convex vector lattice; hence K(X) possesses a 0-neighbourhood base of convex solid sets. A.:(.X) is , in genera l, not order complete.
5 2. Dedekind Completeness
In this section we characteri ze o rder co mpleteness pro perties of the space K(.X) in terms of
topological properties of )1... We recall that a topological s pace X is call ed Stonian ( quasi-Stonian)
if each open subset (F0 -subset) of X has open clos ure.
2.1 PROPOSITION. Let X l1e loca lly compact. Thc11 A.:(X) is /J edekind com.plete if and only if
X is Stonia 11 .
PROOF. If X is Stoni a n the space Cb (X) of all bo unded , continuous, real-valued fun ctions is
order complete by [18 , 11.7 .7) . K(X) is a n id eal of Cb (.X); hence o rder complete .
Suppose now K(X) is order complete. First. we look a t a compact s ubset ;~.· of X which is the
closure of its in terior . There exists a n open relatively contpact s ubset W of .X such that ;~. · C W .
Let J\ 1 denote t be closure of W. 1~. ·, is compact a nd we have A.:g(X) C KK,(.X ). Furthe r let
KK(X).L =: {f E /Cg,(X ): II I A lrJI = 0 Vg E A.:,,-(X)} be t[l(' s ubset. ort hogona l to K.1<( X ). Now
we cl a im that /Cg , (X)/K.g(X).L and C (!l.') a rr lattice isorn orphic.
To this end , first observe that. each I E A.~ g(X).l C K.g, (X) must va nish on ;...· ; indeed , if
f(to) # 0 for some to E 1~.· then If( t) I 2: 15 > 0, say, i 11 some open subset U # 0 of 1~.· , by virtue
of the denseness of f...· in ;~.· . Then by the le mma of Urysohn t.lwrP ex ists a. fun ction g E KK(X )
(g( UC) = {0} and g(t 0 ) = l ) s uch that. 1/IA IYI # 0. On the othe r ha nd , if f(I~.·) = {0} then
obvio usly I E /Cg(X).L. It fo ll ows from t his that a ll fun ctions in a give n equivalence class [f) E
K.A·,(X)/K.1.;(X).L have identical res trictions h to 1~. ·, a nd therefore that [f)~ I 1,- defin es a lattice
isomo rphism of K.K, (X )/!Cg(X ).L into C( l\') , which turns o ut to be s urjective.
To show that the restricti on map is surjective let. I E C (I\) . Since 1~. · , is compact, by Tietze's
extension t heorem f can be extended to a. continuous real function f 1 o n 1~.· 1· Further by the lemma. of Urysohn t here exists a. continuous fun ction g, 0 S g S l , s uch t hat g(I~.·) = {1} and g(WC) = {0} .
Thus gft E K.K, (X) is an extension of I E C(J\) .
A:,,., (X) is an id eal in K.(X) a nd therefore orde r complete by assumption. Hence K.K(X).l is a. projection band and (the complementa ry id eal) A:K ,( X)/ Kg(X).l is ord er complete as a band of
K.K,(X) . It fo llows that C (l... . ) is order complete, whi ch by [1 8, 11 .7.7 Coro llary) implies that f{ is
Stonian.
6 Now let G be an open, relatively compact subset of X. Again we can find an open relative ly
compact subset W in X such that G C W C W. 13 y the preceding part of the proof W is St.onian.
Hence G' is open in W. lt fo ll ows that C: is open in IV a nd , sin ce W is open in X , G' is open in X .
Finally, let G be a n open subset of X a nd let x be a. bounda ry point of G'. We show that x is an
interior point of G'. Let U be an open , relatively co mpact neighbourhood of x. The n U n G' is open
a nd relatively compact. Now X E u n G' a nd by (6 , 1.1 .6 Propos ition 5] X E u n G' c u n G' c G.
But by the second part of the proof U n G is open in X. Thus x is an interio r po int of G, which
implies that G is open .
There is a n ana logous res ult conce rning Decl ekind a--completeness of K(X ). But the proof is
slightly more delicate and based on the following lemma.
2.2 LEMMA. Let E be a Dedekind a-- complete Riesz spa ce, and denote by J a a--id eal of E. Th en
th e quotient map q : E --+ E I J is sequ entially order continuous, and E I J is Dedekind a-- co mplete.
PROOF. To show that q is sequentially ord er co ntinuous it suffi ces to prove that Xn ! 0 in E
implies q( :en ) j 0 in E I J . For this, it is enough to sec that 0 ::; y ::; q( x,} (all n E ~) implies fj = 0.
There exists y E E+ such that y = q(y) . The id entities :I:, + y = .c , V y + Xn 1\ y (n E ~ )imply that
y = (x, V y- x,) + x, 1\ y.
But Zn := x, V y- Xn is in J for a ll n , sin ce q( xn V y) = q(x,) V y = q( x,). Since E is Dedekind a-- complete and J is a a-- ideal, it follows that :: :=sup,::, exists a nd belongs to J . On the other hand , inf,(x, 1\ y) = 0. From y = ::, + .c, 1\ y it now fo ll ows that y =sup,::,+infn(x, 1\ y) = z E J .
Therefore, y = q(y) = 0.
Let us now show that EIJ is Dedekind a--complete. Suppose (x, ) is a n in creasin g sequence in
EIJ such that 0::; Xn::; z for some z E EIJ. By induction on n, we can find an in creasing sequence
(xn) in E + such that X11 = q(x 11 ), for a ll n . Also, there exists z E E+: q( :: ) = i. The sequence (!In) ,
defined by y,. := X 11 1\::, is in creasing, bounded above by:: and satisfies q(yn) = q(x, l\ z ) = Xn i\ Z =
x,. On the other hand, sin ce E is assumed Dedekincl a--complete, y :=s up,y11 exists in E. Since we have just shown that q is sequentia ll y ord er co ntinuous , it foll ows tha t q(y) = sup, q(y, ) =sup,x,.
Therefore, E I J is Dedekind a--complete.
7 2.3 PROPOSITION. Let X be lo ca lly co mpa ct. Then K( X) is Dedekind IJ- complete if and only
if X is quasi-Stontan.
PROOF. If X is qu asi-St.oni a n the sp<~C<' C'b(,\) of a ll bounded , continuo us, real-valu ed fun ctions
is countably orde r complete [18 , IT .7. 7). Thus A.:( X) is co u nta bly ord er complete sin ce it is an id eal
Let now K:(X) be countably order complete. First we show that a compact subset J( C X is a
quasi-Stoni a n s pace whenever I\ is tlw closure of its interi or in X. To this end , we use the following
result fr om the proof of Proposition 2. 1: If r,· is a compact s ubset of X which is the closure of its
interior and 1\. 1 is a compact subset of X such that. I,· C k 1 , then A.:1,·, (X)/K:K (X)l- a nd C'(J\.) are
lattice isomorphic.
Now K:K, (X) is an id eal in K:(X) and therefore DedekinJ IJ- complete by assumption. Since
K:K (X)l- is a band in K:K, (X) (especially a IJ-id eal) it foll ows fr om Lemma 2.2 that C(I\. ) is Dedekind
IJ-complete. Thus by [18 , 11.7 .7) I\ is qu asi-Stoni an.
It remains to show t hat X itself is quasi-Stonian. Let G = U ::;-o= 1 Fn be a n open Fa-subset of
X (Fn closed, n E fl). Let .c0 be a boundary point of G, let U be a ny open, relatively compact
neighbo urhood of x0 , and let V := U. By the preceding, V is co mpact quasi-Stonian. Lettin g
Gt := G n v t hen , sin ce c, = u ::;-o= l(F, n V), Gt is an open Fa-subset of V , and Xo is in t he
closure of G 1 in V, sin ce x0 is interior t.o U. V being quasi-Stonian , 0 1 is open in V; thus there
exists an open subset W of X such that. x0 E V n W C 0 1. Since xo E U we have, a fortiori ,
Xo E v n (W n U) c GJ. But w n u c v a nd Gt c G wh encf' it fo ll ows t hat Xo E w n u c G;
hence .1: 0 is in terior toG in X , which implies t hat G is ope n.
3. Bounded Sets
3.1 DEFINITIONS . Let X be a lo cally compact spa ce.
(a) X is said to have property (B) if every bounded su bset of K: (X) is order bounded.
(b) .X is said to ha ve property ( B0 ) if every open G C X such lh al G is not co mpact contains a
sequence without duster po ·in t in X .
8 3.2 PROPOSITION. Let X be a locally compact space satisfying co ndition (Bo). A subset
B C K(X) is bounded in ( K( X), 'I) if au d only if for som c co mpact /\. C .X , B is a bounded su bset
of (KK(X) , 'IK ).
PROOF. Let B be a bounded s ubset of(K( .Y) , 'I). Sin ce (K(X) , 'I) is locally solid , the solid
hull of a bounded subset is bounded. Therefore we ca n assume that B is solid . Since 'I induces on
each KK(X) its own topology, it is enough to show that B is conta in ed in some KK(X) .
Let Sf := [Iff > 0] and suppose UJEB Sf is not compact. Then, since X has property (B0 ),
UJEB 81 conta in s a sequence T := (t n ),. EN without. acc umula tion po in t.. By constru ction for each
ln E T there exists a fun ction f n E B s uch t hat f n(t 11 ) f. 0. Further s in ce X is completely regul ar
there exists a. continuous real-valu ed fun cti on Yn, O :S [} 11 :S I, s uch tha t Yn(tn) = 1 a nd g,(tm) = 0
for a ll n f. m . Since B is solid , it fo ll ows fr om [ y ,,f~~.[ :S If., [ t hat f/ 11 / ., E B. SincP Sr is compact for f E JC(X ) a nd sin ce T has no accumulation po in t., T n 51 is fini te. Therefo re we can defin e
J..l E K'(X) by J.t := L ~=l akbt•, tk E T, Cl'k := k/ h(tA: ) , and we have l'·(g,Jn) = n. Thus B is not weakly bounded which is equivalent to B not bein g bounded a nd t hi s is a contradiction to our assumption.
3.3 COROLLARY. If X is a loca lly compact space with prop erly ( Bo). the family of all intervals
[-f , f] in K(X)form a fundam ental syslem of bounded se ts of K(X) . In particular, if X has property
(Bo) th en X sa tisfies co ndition (B).
PROOF. Let h. be a compact. s ubse t. of X . Since X is completely regular there exists a continuous fun ction h · E A::(.X) , 0 :S fg :S 1, s uch that fl.; (I\.)= {I} . If U1,· denotes the unit ball of KK(.X) , we have UK C [- JK , h ·]. Now if B is bounded in K(.X) , by Proposition 3.2 B C >. UK for some 1\· C X a nd .A E lit Hence {m[-h·,h ·]: mE f' l,/\. C .\ compact } is fundamental with respect to the fa mily of bounded subsets.
3.4 C OROLLARY. Lei X be a lom//y co mpact spa ce with properly (B). K(X ) is refl exive if and only if eve ry interval[- f , f] in K(X) is iT(K(S) , A:''(X))-compacl.
PROOF. Since K(X) is barreled , it. is re fl exive if a nd o nly if eve ry bounded set in K(X) is relatively iT (JC(X) , K'(X))-compact [17 , [\1 .5 .6] . The assertion fo ll ows now fr om Coroll ary 3.3.
g In gene r· a l, t he statement of Proposition 3.2 is not true, and it. can even occur tha t (K(X), 'I)
is a. Banach space as t he foll owing example shows.
3.5 EXAMPLE. Le t. w 1 be t he fir s !. un co un table ordinal and let. X be t he open o rdinal space
(0 , wt). On X we cons id er t he o rd er topology which has as a. base the set. of all intervals (a , ,8] =
{x: a< a;:::; ,B }. The assertion is now that (A.:(X) ,'I) is a Ba nach space.
In o rde r to s how t his let C 0 ( X ) be t he set of continuo us fun ctions on X t hat van is h at infinity.
C0 (X) , with t he uniform no rm, is a Ba nach space a nd A.:( X) is dense in C'0 (X). Iff E C'0 (X), then for all n E N the set A, := {;r E X : lf(x) l 2: 1/n} is compact . Thus A, C (0 , O'n] for some
O'n E (O ,wt). Let a be the least uppe r boun d of (o,),EN· Then t he suppo rt o f J is contained in
(0 , a) and f( x ) = 0 for a ll x E (c~ , w 1 ) , whi ch implies that f E A.:( X ). li enee K(X) = C'0 ( ..X) a nd
(K(X), 'Iu) is a Banach space.
In general, 'I is stri ctly fin er than 'Iu . But fo r X = (0 , wt) these topologies agree: Let U be a closed , absolu tely co nvex 0-neighbourhood in (K:( ,\' ), 'I) a nd let Un),E N be a seque nce in U which
'Iu-converges to f E K(X). T hen t here exists a n n , rt < w 1 , s uch t.hat f n(t ) = 0 for l 2: o· and a ll n EN. Tf !\ := {,B < w 1 : (3:::; o} , t hen J, converges to J in K1\(X) . llu t U n Kf(( .X ) is closed in
KJ<(X) by definition of the inductive topo logy. This implies that f EU; hence U is 'Iu-closed, and so a. barrel in (K(X) , 'Iu ), i. e ., a. 0-neighbourhood for the topology gene ra ted by the uniform norm.
Thus (K(X), 'I) is a Banach space but there does not exist a. compact s ubset !\ of X such that the unit ball of K (X) is contain ed in (K 1\(X), 'I1,· ).
4. Spaces Hav in g Property (B)
4.1 THEOREM. A sufficient con dition f or th e loca lly co mpact space,\' to hav e property (Eo) is that X be paracompact. This conditwn is not necess ary.
PROOF. To s how that. the condition is s ufTi cient let. C be a n open s ubse t. of X such that C is not compact.. A locall y compact space X is para.compact if and onl y if ,\- is the sum of a family
of locall y compact, u-compact spaces (X0 ) oE A [6 , 1. 9. 10 T heorem 5). T hus , sin ce C is open in
X , Co := G n X a is open in X a for each n EA. If Ga f 0 for infinitely rn a. ny n then there exists a. sequence consisting of elements contain ed in di stin ct open sets Ca. This seque nce has no accumulation poin t sin ce a ny union of subsets X a is both open a nd closed in X . On the other hand if
10 Ga =/= 0 for only finitely many a there exists an a E A such that Gz;- is not relatively compact. Since
Xa- is a--compact., Ga- is not contain ed in any .Xa,n. wh ere Xa = U ~= 1 Xa,n, Xa,n c X a- compact.
Any sequence consisting of elements .tn E Ga,n +l \ Ga,n, n E l\1, has no accumulation point. Now the assertion follows by Corollary 3.:3.
In order to show that the condition is not necessary let X be the space [0, lJ(O,lJ, endowed with the product topology, and let E := {IE X :either f(t) = 0 or f(l) = 1, t E [0 , 1]}. Further let
X a be the subspace X \ E of X. -Yo is locally con1pa.ct since X is compact and X 0 is open in
X. Tf X a were pa.racompact it would be the s um of a. family of locally compact a-- compact spaces
[6 , 1.9.10 Theorem 5]. Now X a = {!EX : :3t E [0 , 1] such that. 0 < f(t) < 1} is convex; hence connected. Thus the sum consists of one element only, namely X 0 . But. Xa is not countable at infinity, sin ce the filter of neighbourhoods of E in X does not have a countable base. To s how this suppose (U,)n?.I to be a. co untable base. There exist. finite s ubsets H,, 11 E R!, of [0, 1] such that U, :J IT(al]\H .. [0, 1] X fltEH .. [0 + c1 , l- ct], 0 < c1 < 1/2. II := UnEN Hn is countable. Let s E [0 , 1] \Hand U = IT(a,l)\{ s} [0 , 1] x [0 + C8 , I - c,]. Then there does not exist a neighbourhood
Un such that Un C U . Hence Xo is not pa.racompa.ct..
It remains to prove that X a has property (Ba), hence also property (B) (Corollary 3.3). Let
G C X 0 be open, G not compact and let ~ be the filter of subsets of X 0 with relatively compact complement. For any finite H C [0 , 1] a nd kE N, we consid er the set. C(/1 , /.;) = {f E X o: t E H =? f(t) < 1/kor f(t) > 1-1/k}. Then (G(H , k))fl,!: is a base o f~ and c;c rf. ~- Hence GnG(H, k) =!= 0 for a ll H and a ll kE N.
We a re going to show that each set G( /-1 , 111) n G contains a. n element f such that f(t) = 1 outside a suitable finite se t. F0 . Let h E G(H, m) n G. Since G n G(If, m) is open there exists (gn)nE N in G without cluster point in Xo. Let H 0 be any finite subset. of [0 , 1]. Then as shown above there exists ft E G(H 0 , 1) n G and a. finite set Hb C [0, 1] such that. ft(t) = 1 wh enever I rf. Hb. Suppose now tlH' fun ctions h, ... , f n have been constructed, a.s we ll as a. finite s ubset 1-r:,_ , and an in creasing sequence Ho C !11 C · · · C Hn_ 1 , with these properties: fm E G(Hm-t,m)nC:(m = l , .. . ,n) a nd fm(t) = 1 whenever t rf. H~_ 1 . Now we define H n := H n-1 uH:,_, and select fn+ 1 E G( Hn , n+ 1)nG s uch that fn+ 1 (t) = 1 whenever 11 t f/:_ H~. , where H;, is a suitable finite s ubset of [0 , 1) . Now let A = U::"= l Hn ; A is countable, and Vn E f.!: f n(t) = 1 if l E [0 , 1] \A. Moreover, for each tEA , the sequence {fn(t)} has at most the two cluste r points 0 a nd 1. Thus {/n} has a s ubscq 11 ence {!I n} s uch that the limit of {!ln(t)} exists for each t and is 0 or l. Consid ered as a sequence in X, {!In} converges pointwise to an element of E; it the refore has exactly one cluster point in X. As a sequence in X 0 , {!In} is contained in G and has no cluster point. 12 B. The Space 9J1(X) 1. The Dual of fi:(X) Let ~l(X) := (K:(X) , 'I)'. By [17, V.G.l] ~l(X) is the orde r dual A::(X)* of A::( X), i.e., 9J1(X) is the set of all linear forms on A::( X) that a re bounded on each o rd e r interval. 9J1(X ) is called the space of all real Radon measures on X . On 9J1(X) we will consider the strong topology ,8(9J1(X) ,K(X)) , i.e., the topology of uniform convergence on the bounded subsets of K(X) . By [17 , V.7.4 ] (9J1(X), ,8(9J1(X) , K(X))) is an order complete locally convex vector lattice; moreove r, sin ce A::( X) is ba rreled , (9J1(X) , ,8(9J1(X) , K(X))) is a. complete lo cally convex space. In this section we consider only those properties of the spact' 9J1(X) which a re indispensable for the following study of the bidua.l of A::( X). Many aspects of the space ~1(X) have been studied; in particular numerous results (such as consequ ences of the weak-star completeness of the cone of positive measures) can be found in [5 , C hapter II] , however for the greater part in the exercises. 2. Lebesgue Property of ,8(9J1(X) ,A::(X)) Let o(~l(X) , A::( X)) be the topo logy of uniform converge nce on t he o rd er intervals o f A::( X) [9 , 2.24]. Then we have the following propos ition. 2.1 PROPOSITION. Let X be a locally co mpact spa ce will! properly (B). Then th e strong topology ,8 (~1(X) , A:: (.X)) coin cides with o( ~1(.X) , A."(X)) ; in pa.rlicula7'1 it is order continuous. PROOF. Since by A.3 .3 the bounded sets of A.:( X) have a. fundamenta l system consisting of order inte rvals of A::( ..-Y) , the topologies coin cid e. In orde r to show that the topologies are order continuous let Po E 9J1( X) , Pa l 0, a nd let c > 0 a nd f E ,q X l+. Then there exists a. p 00 such that J-t (f) :::; L Hence for J.l E A ,"' := {J.t a : J.l :::; J.l } we have J.L a (f) :::; L Thus 00 0 1 0 0 00 A "' C {J.t E 9J1(X) :< 1~-ti,J > :::; c}, which proves the assertion since the latter sets form a 11 0 0-neighborhood base for o( 9J1(X ), A::( .X)). 13 3. Weak Compactness of Order Intervals 3.1 PROPOSITION. if ,\ z.s a locally compact space satzsfying conditzon (B), th en th e strong topology ;3 (9J1(.X) , K(X )) is th e topology induced on 9:11( ,\) by O( flK K~,(X) , EBK KK(X)) . PROOF. We consider the dual system < f1 1,. K~,· (X), ffiK Kg( X) >. Since K(X) is isomor phic with the locally convex space EBK KK(X)/N , wh ere N is a. closed subspace of EBK KK(X) , 9J1(X) is a.lgebra.ically isomorphic with the polar N° C flK IC~<(.X). The topology ;3(9.H(X) , .K(X)) is that of uniform convergence on the O'(ffi"· K1dX)jN , N°)-bounded subsets of ffiK !Cg(X)/N and O(flK KK(X) , ffiK KK(X)) is the topology of uniform converge nce on the O'(ffiK KK(X), flK KK(.X))-bounded subsets of ffiK K1..:(X ). By [17, IV .4.1] ;3 (flg K~· (.X), ffiK KJ<(X)) induces ;3(N°, K(X)) on N° if every bounded subset of K(X) is the canonical image (under¢: ffiJ< KF<(X) -+ ffiJ< KK(X)/N) of a bounded subset offfig !Cg(X). But this is true since by A.3.3 every bounded subset of K( X) is contained in some i nterva.l [-f , f] and if f E K J< (X) then [-/,/] is bounded in EBJ( KK(X). 3.2 PROPOSITION. Let X be a locally compact space with properly (B). Then each order interval of9J1(X) is weakly (i.e. , 0'(9J1(X) , K"(X))-) compac/. PROOF. As mentioned in the proof of Proposition 3.1 9J1( .X) is algebraically isomorphic with the subspace N° of flK K~,(X). Now for each K, J\. C X compact, KK(X) is an AM-space. Thus by [18, II.9.1] K~· (X) is anAL-space and by [18 , 11.8.3 and II.5.10] each order interval of KK(X) is O'(KK(X),K~(X))-compact. Let [-p,p] be an order interval in N° Then (-p, J.t] is the intersection of a. product of order intervals in K~d X),!<..' C X compact, with N° Further by (17 , 1V .4.3 Corollary 1] and since the product topology on fl}( K'g(X) coin cides with the strong topology on flJ< K~<(X) we have O'(flgK~.(X), ffii K"(X)) is the topology induced on 9J1(X) by O'(f1 1,. K~,· (X), ffiK K~<(X)) ([17, IV.4.1 Corollary 1] and Proposition 3.1). Hence [-lt,p] is 0'(9Jl(X), K"(X))-compact. 14 C. The Bidual of K(X) 1. Basic Properties By K(X) we will denote the bidua l of A::(.\"). From [17 , V 7.4] it fo ll ows that (K(X),f3(K(X) , 9J!(X))) is an order complete locally convex vec tor lattice unde r its canoni cal order . 1.1 PROPOSITION. Iff E K( ..-Y)+ th en the bipolar· of[-f, f] q x) is th e interval [-f,fh:cx) in K(X) . In particular, the ideal generat ed by K(X) equals UJEK(X)+ [- f , fh::cx)· Moreover, if X satisfies condition (B) , th en K(X) = UJE A.:(XJ + [- f,Jh:c xr PROOF. K(.X)+ is convex and closed [17, Y.7.2]; hence by [17 , !V.3 .3] cr(K(X) , 9J!(X))-closed. Therefore by [17 , TY.l.5 Coro ll a ry 2] the polar of [-f, f)A.:(>; l( = (f- K(X)+) n (-/ + K(X)+)) is t he cr (9J1(X ), K(,Y))-c losed convex hull of (f- K(X)+) 0 U (-/ + K(X)+) 0 Thus the bipolar of 0 00 [-f , fkcx l equals (f- K(X )+ )" n(f + K(X )+ ) . Since K(X )+ is cr(K(X ), 9J!(X))-closed it follows by [17 , IV .1. 3] that K(X)+ is the cr(K(X ), 9J1 (X))-clos ure of K(X)+ · Thus by the bipolar theorem [17 , IV.1.5] we have [-/ , /]~0( X ) = (!- K(X)+) n (-! + K(X)+) which equa ls [-f,Jh::cxJ· The second assertion follows fr om [17 , IV .5.4] a nd A.3.3 . 1.2 E XAMPLE. Let E = U(O, l ) a nd let f E E be the fun ction f( x) = .r. f is a weak order unit of E [1 8, Il .6 Example 1] ; thus the principal id eal E.r is ord er dense in E but Er is not equal toE since E does not contain a n order unit. Let M be the set of real valued , measurable and a .e. fini te fun ctions on (0, 1). Then by [21 , 1.7 .5] the order continuous dual of E1 is g iven by (EJ)~ 0 = {gEM : Jto ifgld,\ < oo }. Let g(x) := 1/:r. Then g E (E.rJOo but g rf. L00 (0, 1). This example s hows that an ord er continuo us lin ear f'o rr11 de fin ed on an order dense id eal does not necessaril y have a n ord er co ntinuous ex tension t.o the entire space. This explains the signifi cance of the following t heorem . 1.3 THEOREM. If ~1 denotes th e ideal ofK(.X) generat ed by K( .X) th en J 00 can be identified with 9J1( X) (i.e. , the order continuous liu ear f orms on J are precisely the elem ents of 9J1( X) , and thus ha ve MYl er co ntzn'!wus extensions to K( X)). l5 PROOF. Let 0::; f E A::( X) and denote by V the interval [-f , f] in JC(X). Since the polar of [-f, fk(x) is a ;J(9J1(X) , K(X))-neighbourhood of 0, the bipola r (which equals V by Proposition 1.1) is an equicontinuous s ubset. of i"(X); hence compact for t7(I"(S) , 9J1(S)) [17 , IV.l.5 andlii.4.3 Corollary]. If ,1r is the ideal in K(X) gene rated by J then Vis a lso t7(]J,9J1(.X))-compact. Let QJ be the saturated hull of V. By the l\ilackey-Arens theorem [17 , IV .3.2] the topology 'Isn of uniform convergence on the sets of QJ is co nsiste nt. with the duality < 9J1(.X), .J.r >. In particular (9J1(.X) , 'I 0 is the gauge of [-f , !] , i. e. P(J.L) =< f , lit I >. Since p- 1(0) is an id ea.! of9J1(X) and sin ce pis an L-semi-norm, 9J1 p(X) := 9J1(X)/ p- 1 (0) is an L-normed Ri esz-space. By the above 9J1~( . \") = :11 . Let 9J1 ;:(.X) be the completion of 9J1p(X) . Then 9J1; (X) is an AL-space with dual ,1! . If v is a n order continuous I in ear form on J then by [18, Il.8.5 and 11.8.7] the restriction of I/ to .J.r is t7(,7J , 9J1;;(X))- continuous. Let I/o denote 1/l,qx); then v0 E 9J1(X) a nd1/0 has a. unique O"(f"(X) , 9J1(X))-continuous extensio n v to K(X) . Now we have v =von [-f,f]qx)· further vis O"(:T , 9J1(S))-continuous on [-J,f]J. Since [-f,fl.J is dJJ , 9J1;;(X))-compact and sin ce 0"(,1 ,9J1( .X)) is a. coarse r IJ a. usdorff topology, the two topologies agree on [-f, fl.J This implies that // is also O"(J , 9J1( X) )-continuous on [-f , Jl.J. Thus, since [-f,Jk(x) is O"(K( .X) , 9J1( X) )-dense in [- f , fLr by Proposition 1.1 , v a nd v agree on [-f , fl.J. Hence they agree on every interva l of ,7 and s in ce v E 9J1(.\") it follows that// E 9J1(X). 1.4 COROLLARY. if.\" is a lo cally compact space sa lisfyiug condition (B) then 9J1(X) can be identified with th e order· contin-uous dual of K( X). PROOF. The assertion follows with Proposition 1.1 a nd Theorem 1.3 . 2. Order Continuous Linear Forms in K(X) Let X be a ny lo call y compact space a nd let I\. be the Al exandroff" compactifi cation of X . By T: K.(X) ___.Co(!\.) C C(X) we denote the canoni cal imbeddin g. T is an interval preserving lattice homomorphism a nd by [ 18 , Exercise I I I. 24] t he second adjoint r" has the same properties. Let 1 8 0 := r"- (0). Since r" is an order continuous ho momorphis n1 8 0 is a. band , and since K(X) is order complete, B0 is a projection band [18 , 11 .2.10]; i.e. K(.,\" ) =Eo+ B ~. Thus the restri ction of r" to B~ is an injective and interva l preserving lattice homo morphis m. 16 2.1 PROPOSITION . Bi} is the set of all order co ulinuons linear f orms ·in K(X) ; i.e. Bi} K.( X) n 9J1( X no. Moreover, the ideal in }:(X) geu er-a.t cd by A.:( X) I S onleJ· deu;; e l1l Bi} . PROOF. Firs t we show tha t K(X) n9J1(X) ~ 0 C Bi}. Leth E K.(X) be o rde r continuous a nd let h0 be the compone nt of It in 8 0 . Then h0 is ze ro 0 11 the s pace 9J1 b(X) of bounded measures on X , and sin ce 9J1b(X) is order de nse in 9J1(X) , it foll ows that. h0 = 0. Thus hE Bi} . In o rde r to prove the reve rse in clusion let h 1 E K( ,\") and let h"! be a n ele me nt of the order dual 9Jl(X)* s uch tha t lh2l :S lh1l- By [17 , V.7.4] (IJJ1(X) ,,t3(9J1( .X),A.:(X))) has a. neighbourhood base of convex solid sets . For such a neighbourhood U we have lh 1(U)I :S c, c a consta nt. Since U is solid it follows that lh2(x)l :S lh 2l(lxl) :S lhll(lx l) :S c, :r E U, which shows that h2 E K(X). Thus K.(X) is an id eal in 9J1( X)*. On the other hand by [ 18, 11 .4 .3 Coro ll a ry] 9:11( X )00 is a ba nd in 9:11 ( X)*. This implies that K(X ) n 9J1(X)oo is a band in K(X) . Now let ,1 be the id eal ge nerated by K.(X)in K(X) , i. e., ,7 = U J EA:( XJ+ [-f,J]x=(X)" Since t he elements of J a re o rde r continuo us it. foll ows fr om the a bove that the ba nd ge ne rated by J is conta ined in 9J1(X)00 n K(X) a nd that J C /J r}- . ll c nct' to prove the assertion it is enough to show tha t t he ba nd gene ra ted by :J equa ls Bi}, i.e. that. ,7 is orde r de nse in Br} . Now < Bi} ,91lb(X) > is a se pa rated dua lity a nd ,7 C Bi} separa tes 9Jlb( X). 9J1 b(X) is the dual space of the AM-space C0 (X) (X the Al exa ndroff compact.ifi cat.ion of X) and thus a n AL-spa.ce [17 , V .8.4] . The refor e each orde r interva l in 9Jl b( X) is weakly compact [ 17 , V .8.6 Corollary], which implies that the topology o(Bi} , 9Jl b(X )) is consiste nt with tha t dua lity [17 , IV .3.2]. He nce by [17, IV.1.3] a nd [17 , IV.3.3] J is o(Br} , 9J1 b(X))-dense in B{ Le t now hoE Be}-, ho 2: 0. There exists a. filter ~with base in ] such tha t lim~ = h0 . Let ~' : = {F 1\ ho : FE ~}. Then lim~'= h0 since the la ttice operatio ns a re continuous fo r o(Bt , 9Jl b(X)) [9 , 22 B] Thus, if U(h o) = {h E Bi} :< ih- hoi, 11 > < f }, 11 E 9J1 b(X)+ , f > 0, is a n o( Bi}, 9J1 b(X))-neighbourhood of ho there exis ts an F' E ~'su c h that for a ll hE F' < ho- h , p > < f, i. e. < h , Jt > > < ho, J.l· > - f. Hence for h:=supF' we have < h, J-l > 2: < h0 , 11 >. On the other hand < it., Jl > :S < ho, Jl > since J-l 2: 0. - - Therefore< h,p > = < ho, Jt > fo r any I' E 9J1 b( ,\")+ , which implies tha t h0 = h. He nce h0 =supF', where F' C ,1. 17 3. Representation of the Ideal Generated by K.( X) Let X be locally compact and let }\. = XU {w} be the A lexand rofr cornpacti fi cation of X. Since C(I\) is an AM-space with unit, by (18 , Il.9.1) C(I\T' is an AM-space with unit. Thus by (18, II.7.4) 1 11 there exists an isometric Riesz isomorphism of C(!\")' onto C( L) , wh ere L is the 1J( C (I\")' , C(I\ )") compact set of extreme points of P := {k E C(J\)'" : k 2: 0, JJkJJ = 1}. Let r : K.(X)--> C(I\) be the canonical imbedding and let r" : K(X)- C( L) be its second adjoint (up to isomorphism). Let us recall that a co ntinuous map betwee n locally compact spaces is proper if the inverse image of a compact. se t. is compact. 3.1 PROPOSITION. Let X be any lo cally co mpa ct space. Ther·e exists a locally co mpact space Y and a proper, con tinuous map K(X) generated by K.(X), th e eva luation map K.(X) - ,7 corresponding to the mapping f f-> f o NOTE. We do not. claim that a Ri esz isomorphism J- K.(}' ) determines Y uniquely to within homeomorphism. However, the space Y co nstru cted in the subsequ ent proof is the unique open (hence, locally compact) subset of L which is the union of the supports (in L) of a ll fun ctions r" ( z) , z E J. Therefore, in the sequel we wi II often id entify the id eal J with K. (Y) C C ( L) and consider K(Y) to be a lattice id ea l in K(X). In particul a r, if .X satisfies (B) (A.3.1) then J = K(X) and hence, K.(Y) can be viewed as the bidual of K.(X) (C.l.l). PROOF. Let <1> 0 be the composition of the isometric Ri esz isomorph ism of C(I\ )" onto C( L) and the evaluation map C(I\.) --> C (I\. )''. Then <1> 0 is an isometri c Ri es;~ isomorph ism of C (I\) into C( L) . Further let OePL and 8ePK denote the set of extreme points of PL := {It E C(L)': J.1 2: 0, JJpJJ = 1} and Pg := {v E C(I\)' : v 2: 0, JJvJJ = 1} respectively. We claim that the adjoint Tis iJ(C(L)' , C(L))-compact since it is closed in PL (extreme point ofT and let P. e = Af-1-1 + (1-).. lJ-12 , J-11, lt2 E PL. Since Tis a face of PL it follows that p 1, p 2 E T , 18 a nd sin ce fJ e is an extreme point , we have f./·1 = fl'2 · Thus fJ e E OePL . Now by [18 , ll.7.4] L is homeomorphic to OePL a nd /\. is ho meomorphic to 8e PI< . T hus ~ defines a surjective cont inuous (hence proper) map L - ]\ .. Let Z := <1> - 1 {w} (w here J{ = XU {w} is the o ne-poin t compact ifi ca.t ion of X) and let Y := L \ Z. T he n Z is closed, hence Y open. If X is not compact, Y is locall y compact, but not compact, s in ce is not compact. It foll ows from the above t hat : V __.X is cont inu ous, s urjective and proper [6, I.10.3 Proposition 7] . It rema ins to show that .J and K( Y ) are Ri esz isomorphic. From C.2 it follows that the 11 11 restriction of T to .J is injective. Further by [18, Exercise 111.24] T is an in terval preserving lattice homomorphism. Hence r 11 (.J) is a n id eal in C' ( L) . On the other hand r" (.J) C K(Y): let f E K(X) , 7 := f o E C'(L) and let S1 and ST denote the support off a nd 7, respectively. Then 1 since 51 is compact in X, ST = <1> - (51) is co mpact in Y sin ce is proper. Thus 7 E K(Y). Now by C. l.l r" ( ~1) = r"(U JEK(X)+ [-f,f]K(X)) = UJ EK(X l+ [- 7,7] C K()") T herefore r"(.J) is an 11 ideal in K(Y ). ln order to show that T ma ps .J onto K(Y ) le t. g E K(Y ); g has compact s upport ) 59 in Y. Let 3.2 COROLLARY. K(X) is strongly closed m K(Y) . PROOF. Let h be a n element of the f]( /C(X) , A.' (X )')-closure of K(X) in .J. T hen there exists a filter J on K(X)such that his the limit of J . Let DyE C(L)' , y E Y , be defin ed by by(g) = g(y) 11 for every g E C'(L). By [17 , IV.7.4] a nd sin ce C'( L) is a Banach space t he map by o T is strongly continuous. Thus t he filter (by o r")(J) co nverges in JR( t. o (by o r")(h) , for every y E Y . If R denotes the equivalence relation 1 equivalent class. Let h0 (.r) := r"(h)(y) , x EX, y E <1>- ( {:r } ). We cla im t hat h0 E K(X). Let W be an open set in IIR. Then ( r" h) -l ( W) is open in Y and saturated wi t h respect to R ; i. e . the canoni cal 1 image of ( r" h) - ( W) is open in Y/ Rand t herefor<' open in X ( L/ R and f{ a re homeomorphic sin ce L/ R is Hausdorff [6, I. 8.3 Proposition 9] and therefore compact [6, 1.10.4 Proposition 8]). 19 1 3.3 LEMMA . Let X , Y , be as in Pro posilio u 3.1. Ea ch class <1> - ({ x }) x}, x E X, contai11s precis ely one isolated ]Win / y~,. of Y . PROOF. T he poin t fun ctiona ls li.,. a nd liYr defin e sca la r la ttice homo morphis ms on K (X) and K (Y ) respectively, a nd each li y,. is a n extens ion of li_,. sin ce < f o <1> , li y,. > = < f , b,f> (Yx) > = < f , lix > for all f E K (.X). Sin ce A:( X ) is u (A.:(Y) , 9J1 (.X ))-de nse in K(Y) , t he re ex ists ex act ly one linear functional by,. s uch that by x is u (,\; (Y ), 9J1 (X))-continuous. Since t he intervals in K (Y) a re u( A::(Y), 9Jl (X) )-compact , by-" is order cont inuous a nd , as a. Radon measu re on Y, has therefore open an d closed support. In orde r to prove t he latte r let U C Y be the la rgest ope n se t s uch t hat by) U) = 0. T he n [i \ U is a. closed now here dense subset of r' a nd by (8] o r (1 8, Exe rcise 11. 24] we have byx (iJ \ U ) = 0, which implies bv./ U) = 0. Since U is maxima l and since U is ope n (Y Stoni a n , s in ce K (Y) order com plete , A.2. 1) it foll ows that U = 7J . T he refore the com plem ent of U is open a nd closed. He nce {iix} is open a nd closed a nd t hus Y.c isolated. Conve rsely, let y be an isola ted point of Y . T hen by is orde r continuo us on K (Y ): let A l 0, A C C(L) , and assu me th at lim Aiiy (f) = lim Af (y) =: u > 0. T he n i.l'\ y ~ A whi ch is a contrad iction . T hus by C .l.3 a nd C .3. 1 li y is a n element of 9J1 (X ) a nd it fo ll ows t. hat. by is u(K( Y) , 9J1 (X))- cont inuous. 3.4 PRO P OSIT ION. Let X be aug locally compact spa ce. 'l'heu th e following are equivalent: (a) K( X) is refl exive. (b) K (.X ) is semi-r·efl exive . (c) X is discrete. PROOF. (a) => (b) is immediate from t he defini t io n o f re fl exivity. (b)=> (c ): By Lem ma 3.3 a nd si nce A::(X) is semi -re fl exiv<' <1> - 1 ({ x }) consists of exacLl y one 1 isolated poin t of Y. But y = u l· EX { (c) => (a.): Since X is discrete K(X ) is t he loca ll y co nvex d irect sum of t he famil y {K {x}(X) : x EX} and t hu s re fl ex ive by (17, !V.5.8] . T he discreteness of X ca n a lso be cha racteri zed by sem i- reflexivity of 9J1 (X ) , a.s fo ll ows. 20 3.5 PROPOSITION . For any locally compact spa ce X , the f ollo wing are equivalent: (a) l)J((X) is re.fl exzvc (under ,B( l)J((X), A::( X))}. (b) 9J1 (X ) is semz-re.flexive (under ,8(9J1 (X ), ,\:(X))}. (c) X is discrete. PROOF. (a)=> (b) is immediate fr o n1 t hP definitio n of re fl exivity. (b) => (c): ff 9J1(X) is semi- re fl exive then , by definition , e very continuo us lin ea r form o n the strong bidual K( ...-Y) is contained in 9J1 (X ). Hence ,B (x=-(X ), 9J1 (X)) is consiste nt with the duality < K(X) ,9Jl(X ) >. Since K(X) separa tes 9J1(X) , K( ~\") is strongly dense in K(X) (17 , IV.1.3] a nd, in particular, in K(Y). Thus by 3.2, K(X) (mo re prt>cisely, r"(A.:(X))) equals K(Y) 1 1 This means t hat every f E ,q}"") is co nstant on each equiva lence class 1 contains an isolated point Y:r of Y , it follows t. hat <1>- ( {.!:}) = {iix} a nd , furthe rmore, t hat Y is discrete . As in t he proof of Proposition 3.4 , this im p li es that X is discre te. (c) => (a): From Proposition 3.4 we co nclude that. K (X) is re fl ex iw'; hence 9J1 (X) is re fl exive unde r ,8( 9J1 (X) ,K(X)) by (17 , lV.5.6 Coro ll a ry 1] . The preceding proposition contains the well -kn own result that for a compact space .X , the Banach space 9J1 (X) is re fl exive if a nd on ly if it is fini te dime nsional (equivalently, X finite). 4. Orde r Cont inuous Dual of 9J1(X) 4.1 T HEOREM. Every ord er continuous linear form h on !)]( (X) is gi ve n by a nnique bonnded function h. E C(Y) such th at h(p.) = Jy hdp., Jl. E 9J1(X ). PROOF. Sin ce K (Y) is oHler complete (3. 1), Y is St.oni a n (A.2. 1). T herefore , a nd sin ce Y is locally compact, Y has a base of compact, open a nd closed sets. Let A be compact, ope n a nd closed in Y a nd let PA: K(Y)- K (}' ) lw t he strongly continuo us map h f-' h y_A , \'A denoting the characteristic fun ction of A. Furtlwr let P;1 lw the adj oint of PA · Sin ce< PAh , J.t > =< hxA,J1. >= J hxAdJ.l = J hd(XA · J.t ) = < h, P~J.l > , t he adjoin t is given by Jt ...... , \A · J.t . By (18, II .2.9] PA and P~ are band projections. Let J. LA denote the lla.do n rn easu re \A · Jl. on }" . Let now Jl E 9J1 (X)+. T hen P·A E 9J1 (.X)+. In fact, by Theorem C.l .:-1 we have 9J1 (X) = X:(Y)B 0 ; hence l)J((X) is a n id eal in K(Y)* (1 8, U.4.3 Coroll ary] and , sin ce 0 ~ P. A ~ p. , it fo ll ows t hat 21 flA E 9J1(X) . Moreover , flA is a bo unJed measure on X , for each compact., open and closed set A C Y. In order to show this let f E A."(X). Then p 11 (f) = J, 1 {f o IJ.tA(f)l :Ssup {IJ(t)lp(A): t E Now if h 2: 0 is an order continuous linear form on 9J1( .X) , de note by ho its restriction to 9J1b(X). Since 9J1 6 (X) is an ideal in 9J1(X), h0 is order continuous, i. e . h0 E (9J1 6 (X)no C C(L). Therefore h0 defin es a bounded, continuous fun ction f1. : Y ---> lit such that h(p.A) = ho( P.A) = JA hdp. = jy i1Adp. , where hA := il'AA E K(Y) . Since s up11 p 11 = p. in 9J1(X) (in fact, }' is the union of its compact, open a nd closed subsets) and since his order continuous, we have h(lt) =sup 11 h(p.A) =sup11 J}' i1Adp.. Finally, since i1 =sup AhA and since his continuous on }··, it fo ll ows fr o m the definition of the integral [5 , C ha pter 4) that h(p.) = J}. hdlt. 4.2 C OROLLARY. For any lo cally co mpact space X , th e order con tinuous dual of 9J1(X) zs Ries:: isomorphic to th e ideal of all h E Cb(Y) /hal are integmble against any p. E 9J1(X). PROOF. Each hE Cb (} ') which is int.egra bl e against. a ny I'· E 9J1(X) is order continuous, since i1(p.) = J}, il.CIJ1 =sup.4 J hxAdJl =supr~il. A(Jt) and hA is an cle me nt o f the band 9J1(X)00 in 9J1(X)*. Hence ·lj; : h ~--+ h is a linear isomorph ism with 1/• ((9J1( .\no)+) C C6( Y )+. The assertion follows now from [18 , TI.2.6). 4.3 THEOREM. Let X sat1.sjy cou rlilio11 (B). Theu one ha ~; K(X) = K(Y) . Moreover, K(X) and K (X) are co mplete lo ca lly co nvex spaces and K(X) = K(Y) equals th e band of all order continu ous lin ear f or·ms on 9J1(X) wh eneuer 9J1(.X) is barTelerl or bon10logi ca /. In particular, this is true wh enever· .X is paracompact. PROOF. The first assertion res ults fr om I .1 and 3. l. If 9J1(X) is barreled it follows from [17 , V.7.4) that K(X) is a complete locally convex space . A.:(X) is completf' as it is strong ly closed in K(X) (3.2). By 2.1 and 3.1 K(X) is orde r dense in 9J1(X n 0 a nd by [17 , V.7.4) a nd [18 , £1.4.3 Corollary) II:( X) is a band in 9J1(X) 00 . He nce I'( X) = K(Y) = 9J1( .\')00 . Since K(X) is bornological the strong dual (9J1(X) ,f)( 9J1(X) ,K(X)) is complete [17 , IV. 6.1). By [17, Il.8.4) eve ry complete bornological space is barreled. Finally, if X is paracompact, then by (A .4. 1) X has prope rty (B) a nd 9J1(X) is barreled, since it is a product ofF-spaces [6 , 1.9.10 Theore m 5). To cla rify the relationship between the s paces occurring in Theore m 4.3 let us add the foll owing. 22 4.4 PROPOSITION. Let X be a lo cally compact space aud assume (B). Consider thes e asser tions: (a) 9J1(X) is barreled. (b) Every strongly bounded subset of K(Y) is onier bounded. (In particular, Y satisfies (B).) (c) K(Y) is a perfect Ries:: spa ce. (d) Every contin·uous, 9J1( ,\)-unive rsally mlegrable fun ction on Y has compact suppo1·t. Then (a){::} (b)=> (c)? (d). PROOF. By Theorem 4.3 the hypothesi s implies 9J1(.X)' = A.:(Y) . (a) => (b) : Since 9J1(X) is barreled , every strongly bounded subset of K(Y) is equicontinu ous [17 , IV.5.2], hence order bounded, since the inte rvals in K(Y) form a fundamental system of equicontinuous sets. (b)=> (a): (b) implies that 9J?(X) is infrabarreled [17 , IV .5], hence that 9Jl( .X) is barreled , sin ce 9J1(.X) is complete [17, IV.5.3 Corollary]. (a)=> (c): By Theorem 1.3 and Proposition 3.1 we have K(} ' ) ~ 0 = 9J1(X). He nce (K(Y)Q 0 )'~ 0 = 9Jl(X) ~ 0 . Thus by Theorem 4.3 K(}') is perfect wh eneve r 9J1(X) is barre led. (c){::} (d): By Theorem 4.1 , (d) is equivalent to the equality K(Y) = 9J1(.X)t 0 , hence to the perfec tness of K(Y) by the preceding. 23 CHAPTER II RADON,BOREL AND BAIRE MEASURES ON LOCALLY COMPACT SPACES A. Borel and Baire Func Uons 1. Borel Functions 1.1 DEFINITIONS. Let X be a topologi cal space and let B0 (X) denote th e lin ear span of {xA A E 2l} where 2l is th e alg ebra of subsets of X genera ted by all open se ts. (a) The (bounded) Borel fun ctions on X are th e smalles t sel of 1-eal-valued functions on X which contains B0 (X) and is closed under· pointwise co nverge nce of (bounded) se qu ences. Th e family of all bounded Borel funct io ns will be denoted by iJ(X) . (b) A set A C X is a Borel set iff its charactenstic fun ctwu ts a. fJ01d fun ction. The family of Borel sets of X will be denoted by 23( X) . 1.2 PROPOSITION. Let S( PROOF. Let £ := {L : L linear subspace of IW. x s uch t hat 1> C L C S'( )} . In (£,C) every chain has a n upper bou nd (union ). Thus by Zo rn 's le mllla t he re exists a maximal lin ear s ubspace L,a~.. We claim that. Lma.•· = S'( ). Suppose t hat L,wJ· is not equal to .5'( 1>) . The n the re exists an f E S'() \ LmaJ· such that. limn-= f n = f , f n E L,w1· for a ll n E N. Now we define LJ := {af + g: a E Ill:., g E Lm a_, }. Obviously Lf is a linear s ubspace of JW. X a nd we ha ve C LJ C .5'(1>) (the second in clu s ion foll ows fr om of+ g = lim,_= (af, +g), (af, +g) E L m.ax for a ll n EN) a nd Lma.c C Lr ( Lma.v -:f Lj), whi ch is a contradiction to t he maxima li ty of the subspace Lmax· Hence Lma:r = 5(<1>). If 1> is a. subalgebra (or a subla.tt.icc) t.he proof is similar, except. for t he last part. If 1> is a suba.lgebra we define A := { A C ,\ A suha lge hra s uch t hat C A C .5'( <1>)} and Ar := 24 {L~=O av !" , av E A ma x }. Now it is easy to see tha t Ar is an a.lgeb ra; si nee C Ar C S ( ) and Ama.v CAr (A ma.r f:- A.r) we a rrive at a contradiction. If is a s ublattice we set V := {V C .\ : V sublat.t ice s uvh that C V C S( define V.r to be the set of a ll lattice polyn omials whi ch have as variables e le me nts from Vmax or f.(We recall that a lattice polynomial is a fun ction of variables .r 1, ... , J: 71 which is either one of the x;, or (recursively) a join or meet of other lattice polyn omia ls (4] .) V.r is a lattice and we have C V:r C S( 1.3 PROPOSlTION.The boun dedn ess co ndition on E(X) in Definition 1.1 is not essential; if the Borel class Ca(O' < Wj) is defin ed by Co= Eo and c,J th e set of all pointwise limits of sequences in U a<,6 C0 , th en f E Ca i/] (! 1\ nl) V ( -nl) E Ba fo1 · each n E 1\l. PROOF. [tis read il y seen by transfinite induction that. each of the spaces Ea a nd C 0 (a < wl) is a lattice (Riesz space) of real functions. Thus it suffi ces to show that, fo r each a a fun ction f 2 0 is in Ca iff f 1\ nl E E a for a ll n E f\1. We prove both directions by transfinite induction. The condition is necessary. Let f E Co. Since Eo = Co we have f E B0 a. nd f 1\ n I E J3 0 for each n E f\1 . We now assume that the statement is true for a ll ordinals ;3 Wj. Let f E Ca. Then there exists a. positive sequence (fA.) in u ,6 n there exists a ko such that fk(t) > n for all k 2 ko. Hen ce fork 2 k0 we have (h 1\ nl)(t) = n = (f 1\ nl)(t). Similar for f(t) < n. If f(t) = n then fk(l) :'::: n implies (h 1\ 11.l)(t) = fk(t) and fk(l) > n implies (h 1\ nl)(l) = n. The co ne! it ion is sufficient. Let. (f 1\nl) E 8 0 for alln E I\!. Then f = li mn.-oo (! 1\nl) E B0 = Co. We assume that t he assumption is t rue for all ordin a ls ;3 < o , a < w 1 . Iff 1\ nl E Ba there exists a pos itive sequence h E Ui3 lim71 _ 00 (f 1\ nl) =lim71 _ 00 (limk-oo h 1\ nl) = limk - ('X) h· Therefo re f E Ca. 1.4 DEFINITION. A real fun ction on .\ is ca lled doubly bounded if it is bounded and vanishes outside some co mpact set. 25 2. Baire Functions 2.1 DEFINITIONS. Lei X be a topologica l spa ce and let Cb(-\) denote the space of all bounded, continuous, rea l- val-ued functions X - Itt (a) Th e (bounded) Bair·e fun ctions are th e s 111 alles / se t of real-valu ed fun ctions on X which co ntains Cb(X) and is closed under pointwise conve rgence of (bounded) sequences. T he family of bounded Baire fun ctions on X will be denoted by B(X). (b) A set A C X is a Baire se t ijf its characteristic function is a Baire functwn. T he family of Baire sets of X will be denoted by ~ (X). Propositions 1.2 a nd 1.3 of the previous section have prec ise a na logs for Baire functions. ln the foll ow in g lemma suppose X is a topologica l space a nd ~(X) is the a--algebra generated by a ll open Fa sets. Further let 2l denote the algebra ge nerated by a ll open Fa-sets; we defi ne :=:(2l) := {xA : A E 2l}. As in t he previous section by 5(<1> ) we understand the sequent ia l closure of a set C IPI. x , with respect to the product. topology. 2.2 LEMMA. ~(X) coincuies with th e set ~ o (X) {A XA E S(:=:(2l ))} and 2l equals {G1L.G2L. .. . L.G,. ; n E I~ , G; open Fa-s ets}. PROOF. Since ~ 0 (.X) C ~(X) it is enough to show that. ~o(-\) is an a lgebra; because then it is a a--algebra by defi ni t ion (seq uential closure). We defin e the set {2l' : 2l' a lgebra such t hat 2l C 2l' C ~ o(X) }. By Zorn 's lemma there exists a maximal a lgebra 2lrna.r s uch that 2l C 2lma:r C J o(X). Now we claim 2lmax = Jo(X). Suppose 2lmM #- Jo(X). T hen there exists a n element Ao E Jo(X) such that X Ao =limn-ooXAn, An E 2lma.r, n 2 1. But then if the set 21 := { ( Ao n 8)L.C; 8 , c E 2lmax} is an algebra, we would have 2lmM C 21 C ~o(.\) (2lm a"· -:f. 21) whi ch is a contradiction. It remains to show that 2l is an algebra. If 8 = 0 and C = X t hen ( A 0 n 0)L.X = X ; thus X E 21. Further let B 1 , 8 2 , C1 , C2 E 2lmax· Then we have [(Ao n B 1 )L.CJ] n [(Ao n B2)L.C'2 ) = [( A0 n 8I)n((A0 nB2 )L.C2 ))L.[C, n((Aon 82)L.C2)] = [AanB, nAonB2)L.[Ao nB, nc2]L.[C1 nAon B 2)L.[C'1 nC'2) = A0 n[(B 1nB2)L.(B 1 nC2)L.(ll2 nC', ))L.[C'1 nC'2). Since t he elements in the brackets are from 2lmax. 21 is stable under finite inte rsections. Final ly [( A0 n B 1 )L.C'1)L.[(A0 n B 2 )L.C'2) = (Ao n BJ)L.(Ao n B 2 )L.C'1 L.C'2 =An (B 1 L.B2 )L.C1 L.C'2 which shows that 21 is a n a lgebra. The second assertion foll ows immed iately fr om the definition of an algebra. 26 2.3 PROPOSITION. Le t .X be a Hausdorff topologica l spa ce with pro perly T4 (i.e ., disjoint closed subsets possess disjoint neighb our'lwods), or let X be locally co mpac t and countable at infinity. The Bair·e se ts l.B (X ) co m cide wi th th e cr- alg ebra J (X ) gcnemted by all open Fa-sets. PROOF. Let X be a topolog ical space with property T 1. We first prove the in clu s ion j"(X ) C l.B ( X ). Let G be a n open Fa -set, say G = u.. > I t~,. Without restriction of gene ra li ty we can choose t he sequence of closed sets (Fn) in creasin g. For every n E H Fn a nd G0 a re disj o in t closed sets. T hus by [1 6, 7.2] t here exist continuous fun ctions J,, O ::; J,::; 1, such th at f n(x ) = 1 for all x E Fn a nd f n(x) = 0 for all x E G 0. Hence Xa = li mn - oo f n, which shows t hat Xa is the pointwise limit of a sequence fr om Cb( X ); hence G E l.B (X). Sin ce B( X ) is a lliesz space ( Proposit ion 1. 2 for Ba ir e fun ctions) a nd sin ce XG , 6G, = l:x.c;,- ,\ c;, I,G; (i = 1.2)Fa-sets, it fo ll ows fr om Lemma 2.2 tha t 21 C l.B (X). F in all y, again fr om Lemma 2.2 it. fo ll ows th at. ;t(X ) = i"o( X ) C l.B (X). In order to prove t he in clus ion l.B (X ) C J (X) we :; how t hat ll(X ) is contained in t he set F(X) of a ll j" (X )-measura ble fun ctions. F(X ) is sequ ent ia ll y closed sin ce the po intwise limit of measura ble fun ctions is measura bl e [3, IJ. 9. 7]. On t he other hand F(X ) contain s t he cont inuous fun ctions: Let f E C(X ). T hen for a E ~ [! Now if y_. is a Ba ire fun ction then A= [:x.A 2 1] E J (X ). Second, let X be locall y compact. a nd coun tabl e at infini ty lt. is evident fr om t he p revious proof t ha t we o nl y have to show t ha t. t he characteristi c f11n ctio n of an open F,-set is a Ba ire function . Let G = U ~ 1 F1 be an open F,-set in X . Furt her let [,' = XU {w} be t he one-poin t com pactifi cation of X . Sin ce X is locall y compact a nd co un table a t. infinity {w} is a G6 -set inK, say {w} = n:=l Cm,Gm open in/,· _ T hus X = u := , c ~,,C ~, compact for a ll rn 2 1. Now we have G = Gn X = ( U ~ 1 Ft)n( U := 1 G ~,) = U ~n= 1 (F/nG ~,) a nd fo r a ll/, m 2 1, FtnG ~, is compact in X . For all n E Pl let us defin e t he n (Kn ) is a n in creasin g seque nce of compact s ubsets of X , each disj oin t fr om G0 . By Urysohn 's lemma, t here exist fun ctions fa E C( X ) such t hat. 0::; fn ::; 1, f n(x ) = 1 if x E f( n a nd f n(x) = 0 if 0 X E G (n E !':!).Sin ce ! n ixE Cb( X ) and sin ce Xa =limn-co f n, t he assert ion is p roved. 27 2.4 COROLLARY. The bounded Baire fun ctions B(X) coincide with th e se t F(X) of all bounded j (X)-measumble fun ctions. PROOF. The proof of the inclusion B(X) C F(X) is contain ed in the proof o f Proposition 2.3. Further it foll ows from the previous proposition that :=:(j(X)) C B(X). This proves the reverse inclusion sin ce every bounded j(X)- measurable fun ction can be uniformly a pproximated by linear combinations of characteristic fun ctions of Baire sets [3, 11.11.6). 3. Relations between Borel (Baire) Functions on a Locall y Compact. Space and its One- Point Compactifi cation Let X be locally compact. a nd le t. ]\. = X U {w } be tlw A lexandroff com pa.ct ifi cation of X . 3.1 PROPOSITION. 23(1\) comcults with th e set co nsisting of 23 ( X ) and sets B U {w} , B E 23(X) . Equivalently, th e res tr·iclion of a JJ orel fun ctio n f E B(l\.) to X ts a Bor·el function on X , and every real-valu ed extension of a Bore l function f E !J(X) to 1\· is a Borel fun ction on !\ . PROOF. The first assertio n foll ows by applying the second to cha racteri stic fun ctio ns. Therefore we only prove the second assertion. We prove both directions by transfinite induction. Let B 0 (!\) denote the linear span of {\A :A E 2l(J\') ) where 2l(A' ) is t he algebra. of subsets of I\' generated by all open sets of r\·. By [19 , 13 .2.2] 2l(J\. ) is the set of a ll finite unions C = U;'= 1 A; (for some n E N dependent on C) where A; = F; n G; for suitable closed subsets F; and open subsets G; = , n) of Hence iff E then = LjEHajXc finite, aj E ~,o r (i l , ... X . Bo(J\.) f 1 , H f = L j EH ajXCj Xx + LjEH ajXCjA(w). Now cj n X = u ;·;,, (I~; n X) n (Gji n X) ,j E H. (Since H is fmite, we can take n to be the same in teger for a ll Cj ,j E H) . For each j E H and all i = 1, ... n(Fj; n X) is closed in X a nd (Gj; n X ) is open in .X which s hows t hat fix is a n element of the linear s pa n of {x A :A E 2l(X)}. Thus fixE Bo( X) . We now assume that t he assertion is true for all ordin als f] < a, where a de notes any ordinal less than the smallest un countable o rdinal w1 . Let f E B cr ( /\.). Then there exists a sequence U n) C U i3 Ba(X). It rema in s to show that eve ry real-valued extension of a Borel fun ction f E B(X) to J\. is a 28 Borel fun ctio n on!\ . Let ift E .\' f (t) := { f(t) , T , if t = w be such a n exte nsion off E B (X). Iff E IJ 0 (X) then as 111 the first pa rt of t he proof f is of the fo rm J = L j EH O']XCJ,o E IR, Cj = u :·= l F ji n Gji Now j = L jE II OjXCJ \ x + T X{w )" Let Cj := u :'= l F jinc:j i wh ere F j i is the closure of Fj; in/\·. Since Gji is o pen in X (and in/\.) w rf_ C:ji and we have } = L jE H O')A c \ _, . + TX{ wl. Hence by th e preceding, j E Bo(J(). 1 Now we assume tha t the assertion is true for a ll ordin a ls (J < n, o < w 1. Let f E Ba(X ). Then there exi sts a sequence (in ) c U~ fn E B~ (I\· ) , (J < o , n E f':! . Since ]n(l) converges to f(t ) for a ll t E [\. it foll ows t hat j E B(X). 3.2 PROPOSITION. Let X be locally co mpact and cou ntable at znfinity. Let f( = XU {w} be th e on e point-co rnpa ctifi cation of X . Th e (}"-alg e bra~(!\ . ) co in cides with th e set consisting of~(X) and the sets B U {w }, B E ~ (.X ). E qu ivaleutly, th e restrictio n of a Ba zre fun ctio n f E B(I\. ) to X i.s a B aire fu nctio n and every real- valued exlensta 11 of a Baa·e fuu clzo n f E B(X ) to f{ zs a Baire fun ctzon on J\ .. PROOF. We only prove the first assert.i on.The seco nd t hen foll ows fr om Coroll ary 2.4. Since the complement of an open F(J-set is a cl osed Gb-set it fo ll ows from Propositio n 2.3 tha t the Baire sets coincid e wi t h t he (J- algebra ge nerated by a ll cl osed G o~ -s e t s . LetS(/() denote the set ~ (X) U {BU{w} : BE~(X)}. If E is a. cl osed G b-set Ill X then F = En X is a. closed G b-set in .X ; either E = F or E = F U {w}. On the other hand S(J\.) is a (}" -alge bra sin ce it is a. Dynkin system and st a bl e under finite intersections [3 , L2 .3]. Thus we h a v e ~(!\ . ) C S( [\.). Let F be a. cl osed G 6-set in X. We cl a im that FU {w} is a. closed G b-set in I\ . Since X is locally compact and countabl e at infinity, {w} is a (c losed) (;b-sct in /\· . It is clea r that F U {w} is closed . To s how that F U {w} is a G'J-set , let F = n~= l G, a nd {w} = n : = l Gm , G, open in .X (hence in F) , Gm open in/\·, n , m 2 1. Now F u {w} = (n ~=l Gn ) u (n : = l Gm ) = n~m =l (Gn u Cm ) Thus, sin ce Gn U Cm is open in /\. , F U {w} is a. G o~ -s e t of l\·. Since {w} is a G' b-set in /\. we have {F U{w}} \ {w} = F E ~(/\ · ) , h e n c e ~( ,\") C 23 ( !\). But this implies BU {w} E 23(1\") for a ll B E ~ ( X). Hence S(J\.) C ~ ( A.) , whi ch proves the assert io n. Thus ~ ( K) = ~ ( X) EB 23( {w} ) is a Boolean direct sum. 29 B. Borel and Haire Measures. Regularity 1. ldentincation of the Duals of C 0 (1\.) and ~1Jl 6 (X) In this section X is locally compact and r\· :=XU {w} is the one-point compactincation of X. Let us recall the definition of the support of a measure It E C(K)' [5]. , 1.1 DEFINITION. Let p E C( X)' . The S'li.JlJIOrt 5'1 of p is th e complem ent of th e largest open set G with th e J11'0 perty: iff E C(J\) and th e support off is co ntain ed in G th en p(f) = 0. In the following let Co (I\) :={IE C(I\.): f(w) = 0} a. nJ let e be the function e(x) = 1 for all x E K. Since f( x ) = [f( x)- f(w) e] + f(w) e, f E C (f\) , a nd since Co( f{) is closed in C(I\.) we have C(I\) = Co(K) EB [e) [17 , 1.3 .5]. The projection Q is denned by QJ := f(w)e, f E C(K). 1.2 PROPOSITION. Co(K)' is Banach. lattice isomorphic to Mb(X). 0 0 PROOF. We first show that Co(I() = [liw]· Let f.l E Co (/\.) . Since the polar of an id eal in C(K) is an ideal in 9J1(K) [18 , [1.4 .7] we can choose J1 > 0. Let. 5'1, de note the support of Jl· If 5 1, C {w} then the assertion follows immediately, s in ce p. - fl.(e)liu., has void s upport and he nce must be the ze ro measure. Assume now that 51' rf_ { w}. Let x E X n 5'1,. The n there exists a n open neighbourhood U of x such that w ~ U. By the lemma of Urysohn there exists a continuo us fun ction J E C0 (I() , 0::; f::; 1, such that f(x) = 1 and f(Uc) = {0}. Since U is not in the comple me nt of 5 1, we have p(f) > 0. 0 But this is a contradiction topE C0 (J\.) . Thus 5'1, C {w}. 0 Now C0 (I\)' is norm and lattice isomorphic to C(!\T /C0 (I\.) = M(I\.)/[liw]; let J1 E M(I\) and let [t be the corresponding element in M(F)/[bw ]- Also, let p 0 := Jll co( K)· Since 9J1(J\) is an AL-space the isometry of the map Jlo ~ fi foll ows from II /t [[=infaE III [[f.t + o:liw [[=infaEIII( [[ Po II +fo:i) =II Po[[. On the other hand we have 9J1(I\.) = 9J1b(.X) EB [liw] where the projection P: 9J1(K)--+ Mb(X) is given by P(p) = Xx ·1t, and where II Jl· 11=11 P1, II+ II(!- P)Jt [[ , since M(I\) is an AL-space and P is a ba nd projection (0::; P ::; I). This implies that 9J1(I\)j[8w] is norm and lattice isomorphic with Mb(X). Thus we have C'o(l\T =: 9:11(1\')/[bw] as well as 9J1 &(X) =: M(I\')/ [bw], which implies that C0 (K)' can be id entified with M&(X). :30 For the foll owing we recall the notions used in C hapter I, Section 3. L always denotes the compact space occurring in the Riesz isomorphism C (!\")' 1 =: C (!,). ThP prope r, continuous map : L----> 1\ wa.s defin ed in Proposition I. C.:3. 1 as wr ll as Z := <1>- 1( {w}) and }' := L \ Z . By Lemma I. C.3.3 each class Yw· Further let L 0 be the set L \ {w}. Thus we have t he ba nd decompos ition C (L) = C(L 0 ) EB [X;;;.J. From 9J1(K) = 9J1b(X) EB [bw] (proof of Proposition 1.2) we obtain a nothe r band decomposition of 0 0 1 0 0 C(L): C(L) = 9J1(K )' = [bw] EB 931 b(X) , wh ere [(1 - P)'t (0) = [( [ - P)(9J1(/\' ))] = [bw] = {hE 9J1 (K)' : h(w) = 0} = 931 b(X)'. 1.3 PROPOSIT IO N. 931 b(.X)' is Haua ch lalli ce iso 11101phtc to C( L 0 ). PROOF. By the preceding it is e nough to s how that 931 v( X )0 = [\;;;-],s in ce a band decomposition of a Riesz space is uniquely determined by either one of its s un1ma 11cl s. Since 931(1\) = 9J1 b(X) EB [bw] and hence 9J1b(X ) is closed in 931(1\ ), the quotie nt space 9J1(1\-)/9J1b(X) is Hausdorff. Thus 9J1 (K)/9J1b (X) is isomorphic to IR [17 , 1.3 .1], a nd therefore one-dime ns iona l. T his implies that 0 0 9Jl b(X) is one-dimensional sin ce (9J1(1\)/ 9J1 b( .X ))' =: 931 v(X) . Thus the po lar of 9J1b(X) is a. one dimensiona l band of continuous fun ctions in C (L). From the decomposition of the function XL into characteri stic fun ctions it. fo llows t.ha.t. t he s upport of its compone nt. in 931 b( X) 0 is a n isolated point z of L. Tn fact, let XL = h1 +h2 , h 1 E 931 b(X) 0 ,h." E [b u.,]0 Sinct' h 1 l. h2 we have XL= h i+h~; hence h1 = hi and h 2 = h3, which implies that. h 1 a nd h 2 a re co ntinuous characteristic fun ctions, hence have open and closed support. Assume now that. the s upport of h 1 contains two distinct points. 0 Then by the lemma. of U rysohn there exist two linearly indepe nde nt fun ctions in 9J1b( .X) . But this is impossible since 9J1 b(X) 0 is one-dimensional. Suppose:: :f w. The n :: is a n isolated point of} ', sin ce by l.C.:3.:3 Y contain s a ll isolated points of L other t ha n w. Now the point fun ('t iona. l 8, is order continuous on K(V): le t A J 0 , A C C (L), and a.ssurne that limA 8, (J) = limAf(::) =: n > 0. The n Cl,\, :::; A which is a contradictio n. Thus by I. C. l.3 a nd l.C.:3. J , 8. is a n eleme nt of931 v( X) . But by assumpt io n < b. , \ ,>= 0 which contradicts < b, ,x, > = l. The refore, finally, it follows that::= wand 9J1 b(X)0 = [xw-l C C(L). :3 1 2. Imbedding of the Borel and Ba ire Functions into !.h e Bidual of K( X) The notation 111 this section is the sanw as explain(' d in the previ ous section, s ubsequent. to Proposition 1. 2. We now recall from [19] t.lw ma in resu Its used througho ut this chapter. If .C denotes the set. of a ll bounded , lower semi conl.inu ous real fun ctions ;:: 0 o n A' a nd if M denotes the set of all bounded real fun ctions ;:: 0 on 1\· , the evaluation map has an extension to .C a nd ;\If defin ed as fo ll ows [19 , Defint.i o n A. :,! .l]: ¢}(g)= s up{ ¢*(h)= inf{J)( y) : 0 :S h :S y , !I E .C} (hEM). If R. denotes a Ri esz s ubspace of rn; K, conta ining J, of bounded real fun ctions on 1\ and if ¢* is additive on R.+ , then its linear extension J : R. - C (l\') is a n isometric lliesz isomorphis m for the supremum norm ofR. and the standard norm ofC(X), respectively [19, A. 3.5]. 13y [19 , B.2.6] the Riesz spaces R. with this property in clude the space B(A') of a ll bounded Borel fun ctions on J\. a nd J : B(J\) - C(I\) has the fo ll owing properties: 2.1 THEOREM. Th e map The purpose of this section is to show tha t J(Bb(X)) , wh ere Bb(X) de notes the doubly bounded Borel fun ctio ns (Definition A.l.4), is a. Dedekind cr-complete lli e :;~ s ubspace of K(Y) . 2.2 LE!viMA. }' is dense iu La. PROOF. We consider the AL-space 9J? b(X). By Propos itio n 1. 3 9J1b(X)' is isomorphic to C (L0 ) and by [18 , II.8.3] ~J1 6 (X) is (isomorphic to) t he ba nd of a ll order continuo us linear forms on C(La). Since each order in terval in 9J? b(X) is weakl y compact [1 7 , V.8.6 Coro ll ary] the topology o(C( La), 9J?b( .X)) is consist.ent with the duality < C(La), ~ bCX) > [17 , 1V .3.2]. Now we suppose that Y is not dense in La. Le t. J: E La\ Y . The n by the le mma. of U rysohn there exists a n h E C'(La) s uch t hat h( x ) = l a nd h(Y) = {0}. Hence hE K(Y).L . This implies that 3:,! B := K(Y).l.l :f. C(Lo). Since C(Lo) is order complete (as dual of ~1b()()) we have C(L0 ) = B + B.l, B :f. {0} and B.l :f. {0}. Since the band projec tion P: C(L0 )-+ B is continuous for the lattice topology o(C(Lo), ~1b(X)) (18, 11.5.2] we also have the topologica l direct sum C(Lo) = B ffi B.l. But 0 0 then ~b(X) = (B.l) ffi B , wh ere o denotes polars with res pect to < C (L 0 ),~ 6 (X) >,and we have 0 0 0 B ,(B.l)o :f. {0} (sin ce B, B.l :f. {0}). Let 0 :f. JL E /J = (.((Y).l.l) . Since K(Y) C K(Y).l.l we have (K(Y).l.l )° C (K(Y))D. Thus JL(K(Y)) = {0} whi ch contradicts the fact that K(Y) separates ~b(X). Therefore Y is dense in L 0 . 2.3 PROPOSITION. ¢; maps Xw onto Xw-· PROOF. We first prove the equ a lity ¢(Xx) =sup{¢(/): 0 ~ I ~ I, IE A:(X)} , wh ere¢ denotes the evaluation map ¢: C' (I\' ) ~ C (L). Since Xx is a bounded , lower semi-continuous functio n on X , it follows from the definition of ¢that ¢(xxl =sup{¢(!): 0 ~ f ~ X.x,f E C'(J\.)} =sup{¢ (!): 0 ~ I~ 1, / E C'o( K)}. Let A:={! E C'o(I\): 0 ~ f ~ 1} and B := {f E K(X): 0 ~ I~ 1}. Then B C A; hence supA¢(!) 2: sups¢(!). In order to prove the other direction of the in equality let Jl· E 9J1b(Xl+ · Since JL is order continuous on C(Lo) we have supA¢(f) ~ sups¢(/) if[ J.L(supA ¢ (f)) ~ J.l(sups ¢ (f)) or supA < ¢ (f),JL > ~ sups< ¢ (f),JL >for allj.t E ~1b(X)+- Let now f E A be fix ed and let c > 0. We choose c1 > 0 such that. c1 < ¢(e), J.1 > ~ c. Let W := {t EX : f(t) 2: <1}. W is compact in f(. IfU := {t EX: f(t) > c:l},c1 > c2 > 0, then U is an open neighbourhood of W, and Wand J\. \ U are disjoint. closed sets in /\.. He nce there exists an open neighbourhood V of W in I\ such that V and /\ \ U are disjoint closed sets. By the lemma of Urysohn there exists a function f' E C'(J\·), 0 ~ f' ~ 1, f'(t) = 1 for all l E Wand f'(s) = 0 for a ll sEA.\ V. Thus h := f'lx E K(X) sin ce the support of / 1 is a closed s ubset of V , he nce compact in I\ , and therefore in X (the support of I 1 is a subset of X). Let g := f · f'- Then glx E K(X) , g ~ f, g(t) = f(t) ift E Wand f ~ g+ c1e. Hence< ¢ (/), p. > ~ < ¢ (g) , p. > +ct < ¢(e),JL > ~ < ¢(g),JL >+c. This implies that< ¢(g) , p. > 2: < ¢ (f), JL >-<::and therefore sups< ¢(g) , Jl > 2: supA < ¢ (/), Jt >. Since this holds for a ll I'· E ~1 b( X)+ , the desired eq uality is established. 33 Now from J(xx) =sup{¢(f): 0::; f ::; l,f E IC(X)} = s up{h' : 0::; h' ::; 1, h' E K(Y ) } =:hit fo ll ows that hE C( L) and h(t) = 1 for a ll t E Y. But. Y is dense in Lo by Lemma 2. 1; thus h(t) = 1 for a ll tELa. Since X Lo E C(L) (i.e. , sin ce XLo is a cont inu ous ttpper bo und o f ¢(A) a nd ¢(B)) we have h(w) = 0. But Xw = e- Xx , and we obta in J( yw) = J(e- Xx) = e- h = Xw-· 2.4 PROPOSITION. J maps !h e spa ce B(S) of bou11d cd JJm -el fun ctions into C(Lo). PROOF. Since B(J\.) = /J(S) tfl [\J (Proposit ion A.3.1) a nd C(L) = C(Lo) tfl [xw-l a re band decompos itions the assertion foll ows from Proposition 2.3. ln fad, iff E [xJj_ = B(X) we have l¢ (fll 1\ lxw-1 = 1¢ (!)1 1\ IJ(xw ll = J( lfll 1\ J( lxwll = J( lfl 1\ lxw I)= 0; hence¢(!) E C(La). 2.5 PROPOSITION. J maps th e spa ce I]o( X) of doub ly bounded /Jorel fun ctions into K (Y). PROOF. Let k E Bb(.X). By t he lemma of Urysohn t hen' exists a n f E K(X)+ such t hat lk l::; f . Hence l¢(k) l = ¢(1 kl)::; JU) , a nd ¢ Ul by the definition o f J is a n element of K (Y). But K(Y ) is a n ideal in C(L); therefor e J (l.:) E A:(Y ). 2.6 NOTATION. By 1/• let us denote the restriction of J to Bb(S ). We shall hencefo rth not distingu ish between k E B b(,Y) a nd "!/; (!.:). r or p. E ~Jl(X) , we sha ll wri te < 11 , k > =: J kd1t. Thus via the mappin g ¢>, we have defin ed t he in tegra l of an arbitra ry, doubly bounded Bore l fun ctio n k on }{ wit h respect to a n arbi t rary Radon measure fl E 9J7 (S ); equivalently, every R adon measure on X is extended to a measure (see below) o n t hf' rin g of doubly bounded Borel sets in X . 2.7 THEOREM. The mapping 'lj> defin es a sequentially order· con tinuous Riesz isomorphism of Bb(X) onto a Dede l.:ind a- C01 11 ]) lete Ries:: subspace of K(Y) C 9J1(X)' . PROOF. By Theorem 2.1 and Proposit ion 2.4 J(B( X )) is a De cl ckind a-complete Riesz s ubspace of C(L0 ). Sin ce Bb(X) is an ideal in B( S ) a nd sin ce J is a Riesz isomorphism, J (Bb(X)) is an id eal in J(B(.X )). Therefore 'lj>( Bb(S)) is an ord er a-complete Riesz subspace of C(L0 ). 2.8 COROLLARY. Let (f,) be a bounded sequence of Bb(X) which co nve rg es on X pointwise to f and such that all fn vanish outside a suitable compact su bset of X. Then we ha ve for all ll E 9J1(X): limn P.(/n) = p(J). 34 PROOF. [t is enough to consider the case wh ere 0 ::S fn I f , f n, f E Bb(X). Then by Theorem 2.7 lj; (f) =supn ·~P (J, ) . Since each J1 E 9Jl(X) is o rd er continuous on K(Y) we have tt(f, ) = [p( 1/J (f,))] ~ p.(f). 3. Regularity For the sake of completeness we prove this well kn own lemma.. 3.1 LEMMA. L e t~ be a O"-alg ebm of !!U bscts of a sci S and let {p., : ~ E !} be a directed (::S) family of countably additiue set fuu ctions ~- iF: +. Th cu Jt( ! l) := s up,E / J.t., (A) , A E :L , defin es a counta.bly additive sel fun ction on I:. PROOF. We show that p. is 0"-a.dditivc. To this <' ncl , let, {A n } be a. sequ ence of disjoint subsets of X. Set A = U ~=l A,.. Since J.L ,(A) = L ~= l p,(An) ::S 2::::= 1 tt(A,) for all t E I it fo ll ows that p(A) ::S 2::: ~= 1 p.(A,). On the other hand , sm ce {p , : t. E I} is a directed fa mily, 11 is finite ly additive (p.(A U B) =sup,Efp,(AUB) =sup,E J[p,(A)+ p , (fl)] =sup,E t/t, (A)+sup,E tfl, (B) = J.L( A)+p(B) , A, BE I:) and we have for each k E 1"1 k k k 2..::>(A,.) = p( U A,)= sup ft.,( U A,.) ::S sup J.t,(A) = p.(A) . 1 n=l n=l L n=l ' 3.2 DEFINITIONS. Let X be loca lly co mpact anrllet 23 b( X ) be lit e ring of relative ly co mpact Borel sets ·in X . (a) A (positzve) Borel measure Jt on X is a cou ntably additive se l fun ction p. : 23b(X)---+ ~ (respec- liv e ly ~ +). (b) A positive Bore/measure Jt 011 .\" is ca lled regular· if J1 sa tisfies th e following two conditions: (i) p.(U) = sup{p.(C'): C' C X co mpacl aud C C U} f or· each open se t U E 23 b(X). (ii) p(B) = inf{p(U): U C 23b(X) open and B C U } for each Borel se t BE 23b(X). Finally, an arbitrary Bor·e l measure p ts called r·egula1· if th e total variation IPI of p satisfies the preceding conditions. 35 3.3 NOTE. By t he preced in g defi ni t ion, a Borel measure on X is necessaril y fini te-valu ed . In these circumstances, for posit ive measures I'· conditio n (i) of 3.:! is equi valent wit h (i') f.l (B ) =sup{p(C') : CC X contpac t. a nd C C B } for each Borf' l set BE !B b(X). A proof can be fo un d, for exam ple, in [1] . See a lso Remark 3.8 below . 3.4 PROPOSIT ION. Let {t he an order defi nes a co unlahly additive set functio n fL on 'B b(X ) with values in IH', (i .e., a Borel m eas ure on X in the sense of Defi nition 3. 1). Conversely, if p is a Borelm.easure 011 X , th en f - J fdft defin es an order P ROOF. Let p be a n order u-cont inuous lin ea r form on B 11 ( X ). We show t hat f.l is u-addi tive. To t hi s end , let {An } be a sequence of d isj oin t s ubsets of 'B b( X ) such t ha t. A:= U;::'=1 An E !B b(X). We have 00 n n f.l (A) = f.l (XA) = f.l (L \A , )= ft( s up L \A,)= s up f.l (L XA,) n=l n i =l n i= l n 00 =sup L!'(A , ) = Lfl(A ; ). n i = I i= I Conversely, let p be a Borel measure a nd le t. J, - f bf' a n o rder co nve rgent sequence in Bb(X) . Since f a nd J, , 11 E J\1, a re doubly bo un ded Borel fu nctions , t he respective in teg rals exist. Moreover , by t he definition of order co nvergent sequences [18 , 11.1 .7] a nd by t he Lebesgue Dominated Convergen ce T heorem it foll ows th at J f , dft --+ J f df.t. NOTE. We s hall hencefort h not distin guis h between t hese two versions a nd , for an order u - continuous linea r form fl. on Bb(X) , we s ha ll write p.(A ) = fi.(,\ _.,) = JA dft. Mo reover , we note that the 13 orel m ea ~ ur e s on X cons titute a ba nd in the Riesz dua l of Bb(X). 3.5 T H EOR E M . Lel A be a non-void , direc ted (S) f amily of positive Borel m eas ures, and let 1/(B) := s up,,EA f.t( B ) be finite f or each B E 'B b( X ). Theu I/ is a positive Borel m easure which is regular iff ea ch p E A is regu la r. P RO O F. It is cl ear by Lemma 3. L t ha t. I/ is co unt.ab ly add it ive on 'B b(X ). Suppose I/ to be regula r a nd let BE 'B b(X) and c > 0. T hen t here exists a. com pact set C a nd a n ope n set U E 'B b(X) such t hat C C B C [I and p( U \ C ) < f. T hus 0 S ft( {I \ C) S //( U \ C ) < f. whi ch s hows t hat each p EA is regul ar . 36 Suppose now each tt E A to be regular. First we show that for each open set U E !Bb(X) v(U) = sup{v(C): C compact and C C U} . Let f > 0. By the definition of v there exists a J.l such that ( 1/(U) - p( U ) < ~ and by the regularity of J.l there exists a compact set C C U such tha t ( p.( U)- p(C) < 2. Thus v(U)- v(C) :S 1/(U) -tt(C) :S p(U)- p(C) + c/2 < t. It remains to show that for each BE !Bb(X) 1/(B) = inf{1/(U): U open and B C U} . Let f > 0 and U E IBb(X) such that B CU . Then by the definition of I/ there exists an pEA such that ( 1/(U)- p(U) < . 2 Since pis regular there exists an open set U1 E !Bb(X) such that B C U1 C U and Since //- pis a positive measure on !B b(_\.), we obtain J.l(B)] < f/2 + t/2 =f. This implies that 1/ is regular. Let J.l be a positive Borel measure on X and let C be a compact subset of X. If Pc is defined by Pc (B) := p(B n C), B E !B b( X), then ttc is a positive Borel measure on X . 3.6 COROLLARY. Let ft be a posilzve Bor'elrneasure on X . fJ and only if each of the measures Pc is T'egular, then p is regular. PROOF. Since for every relatively compact set B C X there exists a compact set C such that B C C, we have tt(B) = supc JJ. c (B),C C X compact. The assertion follows now from the previous theorem. 37 3.7 THEOREM. The set of all regular Borel m easures on X constitutes a band zn the order dual (Riesz dual) of Bb(X). PROOF. By Proposition 3.4 the se t of all regular Borel measures on X is contained in the order dual Bb(X)* of Bb(X). Le t p and IJ be regular 13ore l measures on X and let o: ,{3 E JP?. . Further let f > 0 and B E !Bb(X) . Then there exists a compact set C and a n open set U E !Bb(X) such that C C B C U and lo:llfti(U \C) < f/2 a.s we ll as 1;3 11/JI(/J \ C ) < c/2. He nce lo:p. + ,81JI(U \C)< c. There fore the regular Borel measures form a vector subspace in Bb(X )*. Let now p b e a regular Bore l measure anciiJ E Bb(X)* such that IIJI::; 114 The n IPI = lftl V IIJI whi ch by Theore m 3.5 implies that IIJI is regular. This shows that the regular Bore l m easures form an ideal in Bb( X)*. Finally, let 0::; ft a I ft , p E B 0(X)* ,Jt 0 regular Borel measures. Since p(B) is finite for each BE !Bb(X) , the asse rtion follows from Theore m :3.5. 3.8 REMARK. The definition of a Bore l measure give n in 3.2 is not traditional a nd requtres justification. First, a countably additive set fun ction IB(X) ~ ~ +(= [0 , + oo]) is traditiona lly called a regular Borel measure if it satisfies condition (i') of 3.3 as well as (ii) of 3.2 for a ll Bore l sets B C X . The fact is now that in gene ral , such an exte nsion does not exist for a give n Radon m easure p 2 0 on X; thus, a Riesz re presentation theore m (in the sense o f C.2.:2 below) docs not hold in all locally compact spaces. (It is well known that it does hold if X is countable at infinity- in particular, when X is compact). For details, see Bauer [3], Ange r- Bauer [2], C ou rreges [7]. Halmos [10] copes with this situation by considering the We observe in conclusion that the Riesz Represe ntation Theore m proved in Aliprantis and Burkinshaw [1 , Theorem 28.3] holds in all lo cally compact spaces X , but at the expe nse of weake ning the regularity concept through requiring inner regularity for open se ts only. 38 C. The Riesz Representation Theorem 1. Regular Bo rel Measures a nd the Ri esz Dual of Bb(X) We recall that by Proposition B. 2.5 Bb (X) can be id entifi ed with a. su bspa.ce of K(Y). Let J.l E 9J1(X)+ and let lie be defin ed by lieU) := p.( \ c f).f E K(X) , C C X compact. Then we have the following result. 1.1 LEMMA. For all co mpact sels CC X, 011e has ft c E 9J1t,(X). PROOF. Let P: K(Y)- K(V) be the projection P(h.) := \ c h , h E K(Y) ,O ~ P ~I. Then the adjoint P*: K(Y)*- K(Y)* is a. band projection. The order continuous dual K(Y)Q 0 of K(Y) is a. band in K(Y)* and by Theo rem l. C. l.3 A.:(}') ~ 0 can he id entifi ed with 9J1(X ). Since bands remain inva riant under band projections we ha.v(' P*( 9J1(X)) C 9J1(.X). This implies J.t e E 9J1(X) sin ce < P*~i , h > = < li , Ph > = < J.t , >.. c h > = < J-l e, h >, h. E K(r"). Now for f E K(X), I< J.t c, f > I =I< li ,Xc f >I= lfc fd~il ~ p.(C')suptE C' II(t)l ~ jt( C') II I ll oo· Thus fl·e E 9J1b(X) . 1.2 LEMMA. Every positive m cas'!l1·e Jt E 9J1(X) is /h e supremum. of all fl·e , CC X co mpact. PROOF. We have to show that for IE K(X)+ Jt(f) =supcfie (f) . LetS denote the support ) of f . Since Sis compact it follows that. ft.(!)= p(fy5 = p 5 (f) =S UJ),; P·c (f). By Proposition n.:J.4 we have 9J1(X) c B b (X) ~. The next. theorCIII s hows that. 9J1(.X) is identical with the set of a ll regular 13 orel measures. 1.3 THEOREM. An order u-continuous lin ear fonn I'· on B b( .\') zs an elem.en t of9J1(X) if and only if it is a regular Bore/measure on .\ . PROOF. Let J.t be an element of9J1(X). By Theo rem B.3.5 it is eno ugh to show that l~il E 9J1(X) is regular. Thus we can assume J.t 2: 0. Let C' be a. compact subset of .\. Then fl·c is regular. In fact: By Lemma. 1.1 J.l·e E 9J1 &( X) . Purt.h er there exists a. unique exte nsion lie of J.l·e to the one point compa.ct ifi ca.tion /\. of X such that. li,. ( {w}) = 0. By [20 , Theorem C.2.3] lie is regular if and only if lie is o(C'(A')'',9J1(I\.))-continuous. But. o(C'(A')",9J1(1\.)-continuity is equivalent to lie E 9J1(I\) sin ce o(C'(J\')'',9J1(I\.)) is consistent with < C'( X)" , 9J1(/\.) > . Since by Lemma. 1.2 we have J.l = SUPc J.l c, the a<;se rtio n follows now fr om Theorem B.3.5 . 39 Let now J-l E B b (X) ~ be regular. We have to s how t hat p E 9J1(X ). Since l;.tl is regul ar (Definition B.3.3) and hence p+ , J-l - are regul ar (B .3.5 ) we can ass ume p ;::::: 0. Let f-l o := PiA:( X )· T hen Po E 9J1(X) . Since 1/; (B b( X)) C A.:(Y) C 9J1(X)' (P ro position B .2.4), Jio := Po o 1j; is an ex tension of {t o to B b(X) , which is identified with fJ o by Note B 2.5. Thus we have lio E 9J1(X ). Now we cl aim t hat p. = p0 . Since ;.t and p0 a re countabl y addit ive it is enough to show that ;.t = p0 on t he open sets of Q3 b(X) (Definition A.l.l of Borel sets) . Let U E Q3 b(X) be open. T hen by the regul a ri ty of p. we have p.(U) = s up{Jt (C): C C .\compact a nd(.' C U} . ( 1) From the defini t io n of 1,6 (B.2.6) it fo ll ows that 1/; (yu ) =sup{ ¢ (!): 0 :S J :S Xu, f E K( X )} which implies (by t he definition of the elements of K(X)) that p0 (U) = sup{p.o(J) : 0 :S f :S \ u ,f E K(X )}. (2) Let f E K.( X ), 0 :S f :S Xu , and let S denote the compact s upport of f. T hen p.(S ) ;::::: p 0 (!) . On the other hand if C is a. compact s ubset of U t hen by the lemma of Urysohn there exi sts a continuous function 0 :S f :S Xu s uch that f(t) = 1 if I E C. T hu s p( C') :S Jt.n(f ). Therefo re (1) and (2) coin cid e and the theorem is proved . 1.4 PROPOSITION. Let p be a reg·ular Bo rel meas·ure on X. For every non- vo id dzrected (C) fam ily { Ua : a E A} of open se ts with. relat ively co mpact unz on U we ha ve J.l(U) = limJ.l( Ua ). oEA PROOF. By Theorem B .3. 7 p+ ancl1t - a. re regul a r sin ce p +, p - :S IPI· T herefo re we can assume p. :2: 0. Let ~n be t he fa mily of a ll compac t s ubsets of Ua, a E A , a nd let ~ 0 be the fa mily of all compact s ubsets of U. We cl ai Ill that ~ o = U a EA ~a . C learl y, U a EA ~ n C ~ o . In order to prove the reverse inclusion let Co be a. compact subset of U . Then {U a : a E A} is a. n open cover of Co which can be reduced to a. finite s ubcover Ua,, ... , Ucr" . By assumption the re ex ists a. n o pen set Uf3 , f3 E A , such that U 7= 1 Ucr , c u,J. Hence Co E ~/J whi ch proves t he assertion . Since f-l is regul a r we have p( U ) = s up C' E ~ o Jt( C ) and p.( Ua ) = supC' H a Jt(C ). Thus it follows from the preceding t ha t Jt(U ) = s up C' E ~ o ft.( C ) = SUJl oEA s upC' H n f.t(C) = supoEA p(Ua ) which completes t he proof. 40 In particular, if {G a : o· E A} is a non-void family of pairwise disjoint open sets with relatively compact union G then for every regul ar Borel measu re I'· we have fl ((,') = L ~t(G o ) · erE A (If A is infini te the sum is to be taken a long the filt er of cofi nite s ubsets of A). In C.2 it will be shown t hat t he property stated in Propsition 1.4 cha racteri zes regular Borel measures. 2. Topologica l Characterization of llcgu la. r Borel Measures and t he Riesz Representation Theorem 2.1 PROPOSITION. A linear fonll.fl. on Bb(S) zs a 1·egular /Jorrlmeasure if and only if~ is con- tinuous for o(Bb(X) , 9J1 (X)) (or any oth er topo logy consistent with th e duality < Bb(X) , 9J1 (X) > ). PROOF. By Theorem l.C.1.3 9J1 (X) is the order continuous dual of K(Y). Thus 9J1 (X) is a band in the order dual K (Y )* of A:( Y) [1 8, 11 .4 .3 Coro ll a ry J. Therefore and si nce the intervals in 9J1 ( X) are u( K(Y)* , K( Y) )-compact it. follows that the inte rvals in 9J1 ( X) are u(9J1( X) , K(Y) ) compact. K:(X) C Bb(X)). Thus u(9J1(X) , Bb(.\")) is a coarser ll a usclo rff topology a nd therefore identical with u(9J1(X) , K(Y)) on compact sets. li enee by tlw Mackey-Arens t heo rem [17, !V.3 .2] o(Bb(X) , 9J1( X)) is consistent with the duality < Bb (X ), 9J( ( .\" ) >. T he asse rtion follows now fr om Theorem 1.3. 2.2 T H EOREM (Riesz Representation T heorem). Every Radon measure ~ on X possesses a unique regular Borel extension "ji (Definition B.3. 2) such that l' (f) = JfdJi , f E K(X) . Precisely: Ji is th e uniqu e o(Bb(X) , 9J? (X))-conlinuous cxlenswu of I'· E 9J?(X ) to Bb(X) , and th e Riesz isomorp h ism ~ ___... Ji znduces an isometric isomor7Jhzsm of th e Banach lattice 9J1b(X) onto the norrned ideal of bounded regular Borel measures on .X. PROOF. The topology o(Bb(X) , 9J?(X )) is consistent with t lw duality < Bb(X), 9J1( X) > (proof of Proposition 2.1) . Hence by [17 , IV .! .3 and IV .3.3] K(.\" ) is o(Bb(X) , 9J1 (X))-dense in Bb (X). T herefore the first a:ssert ion follows from Proposit ion 2.1. 41 Since K(.X)~0 = Bb(X)+ it follows by the bipola r theore m [17 , IV.l.5) a nd by [17 , IV.3.3) that ,qX)+ is o(Bb(X),9Jl(X))-clense in Bb(.X)+· Thus p ~fl. is an isomorphism of Riesz spaces, because f.1 ?: 0 iff Ji?: 0. Let f.l E 9J1b(X ). Then II p II= sup{p.(f) : f E A.:( .\") , 0 :S III :S 1} and 1/II(.X) = sup{lf.li(C) : C C Xcompact}. As in the proof of Theorem 1.3 we conclude that II P· ll= 1/II(X) a nd hence, since 9J1b(X) is a normed lattice, that p ~fl. is an isometry. In the present context le t us note the following res ult. which s hows that supre ma of directed (:S) subsets of 9J1(X) can be inte rpreted in various ways. By ct we de note the family o f a ll compact s ubsets of X. The topology a(9J1(X) , '23 b( X)) (a(9J1(X),ct)) is the coarsest topology on 9J1(X) for which each of the mappings p- P.(\ 8 ) , 8 E '23 b(.\) (p ~ p(\c ),C E ct) o n 9J1(X) into IR is continuo us . 2.3 PROPOSITION. Let 0 :S p. E 9n(X). On th e interval [0 , p) th. e topologies a(9J1(X), 'Bb(X)) and a(9J1( X) , ct) coincide. In particular, if (Pa laEA is a dir·ecled ( :S) family in 911( X)+ , the following assertions are equivalent: (a) f.l = SUPa EA lla. (b) limoEA P·a(B) = p(B) fo r all B E '23b(X) . (c) lima EA f.l a (C) =It( C) fo r all C E ct . PROOF. In the proof of Propos itio n :.U it has been s hown that the intervals in 9J1(X) are a(9J1(X) , Bb(X))-compact. a(9J1(X) ,'B b( X)) is coarser than a(9J1(X) ,Bb(.X)) a nd a(9J1(X) , ct) is coarser than a(9J1( X) , 'B b(X)). Since both a re H a usdorrf topologies, the three topologies agree o n [O , p). Fo r the sam e reason they agree with a(9J1( ...\) , A.:(X)+), o n each inte rval [O ,p). By definition p = sup0 fi·a is equiva lent. to p.(f) = SUP a l' a (f) , f E K(X)+; lwnce to the a(9J1(X) ,K(X)+) converge nce o f l'a to ;t. The equivale nce of (a) ,(b) a nd (c ) foll ows now fr o m the de finition of the weak topologies. We finally give yet another cha racterizatio n of regular (positive ) Bo rel measures. 42 2.4 THEOREM. Let 11 be a positive Borel measure on X (D efinition B.2.3). 11 is reg·ular if and only if for every non-void directed (C) family { U cr : a E A} of open sets with r·elatively compact union U we have j.t( U ) = lim J.t( Un ). (* ) er E A PROOF. In Proposition 1.4 we showed that every regular Borel measure has property(*) · Let 0 :::; 11 E Bb(X) ~ with property (*). Let Jt 1 be the restriction of J.t to K(X) . Then p 1 E 9J1(X)+ · By Theorem 2.2 J.t 1 ha.s a. unique regula r Borel extens io n 7I1 to Bb(X). We cl a im that 11 = 'ji1 . Since 11 and 7I1 are 0'-a.dditive it is enough to show tha t they agree on ope n sets U E !B b(X) . From the hypothesis it follows that ft.(U ) = sup{ft.([f > 0]): 0 :::; J :::; \ ,, / E ..\:(.\)} because U = U{[f > OJ : 0 :::; f :::; ). u, f E A.'( X)}. On the other ha nd , sin ce 7I1 is regular, we have JI1 (U ) = sup{JI1(C ) : C C U, C compact} = s up{JtJ(f): 0 :::; J:::; xu ,J E K(X)}. The latte r equality has bee n proved in Theorem 1. 3. Let Sr denote the support off E K(X) , 0 :::; J:::; Xu . By the lemma. of Urysohn there exists a function g E K(X) , 0 ::=:; g ::=:; 1, such that xs1 ::=:; y :::; Xu. Thus J.t([f > OJ)::=:; f.ll(g) . Since p 1(f) ::=:; Jt([f > OJ) this proves the assertion. 3. Baire measures 3.1 DEFINITIONS. Let X be lo cally compact and let 'BbCX) be the nng of r·elatzv ely compact Baire sets in X . (a) A (positzv e) Bam: m easure ft. 011 X is a countably additive 8et fuu ctwn fl· : 'Bb( .X) ~ JR: (resp ec - (b) A po sitive Baire m easure fl· 011 X is called regular if ft satisfi es th e following two conditions: (i) J.t( U ) = sup{J.t(C ) : C C X com.pa cl flaire se t and C C U } for each open Baire set U E 'Bb(X) . (ii) p(B) = inf{p(U) : U C 'Bb(X)open and B C U} for each Baire set BE fh(X). Finally, an arbitrary Bair·e m easure J.l is called regular if th e total variation IPI of p satisfi es the prece ding conditzons. 43 3.2 REMARK . We identify the Baire measure J.l on X with the order u-continuous linear form f ---> J fdp it defin es on the Riesz space of do ubly bounded Baire fun ctio ns i]&(X) (Definition A.1.4). 3.3 LEMMA. Let X be lo ca lly co mpact. The relatively compact open Baire sets of X fo rm a base of the topology of X. Similarly, each neighb o111·h. ood of any point ;t EX con tazn s a nezghbourhood of x which is a co mpact Baire set. PROOF. Let x be a n arbitrary point of X and let W be a relatively compact open neighbourhood of x. By the lemma of Urysohn there exists a continuo us fun ction 0::::; f ::::; 1 such that f(x) = 1 and f(y) = O,y Ewe. Now let U := [! 2: t] a nd V := [! > t],O < c < 1. U and V are Baire sets and neighbourhoods of x such that V C U C W. Since U is compact and V is open this proves the assertion. 3.4 LEMMA. Let,\ be loca lly compact. (a) A compact se-t C zs a. 8 aire set if and on ly if C zs a Cb -set. An open relatively compact set U is a Baire set if and only if U is an F17 -se l. (b) Let A C X be a relatively compact Ba ire set, G an open, F a compact subset of X such that F C A C G. There exists a compact Baire set F and an open relative ly compa ct Baire set G satisfying F C F C A C G C G. PROOF. (a) The first assertion of (a) fo ll ows fr om [10 , 51.0]. It re mains to show that an open relatively compact Baire set U is an F0 -set. Since by Lemma 3.3 every point in X has a compact Baire neig hbou rh ood t here exist. finitely ma ny compact Ba ire sets C,, l ::::; i ::::; n, which cover U. Let C := U7= 1 C; . C is a compact Baire set containing U. Hence C \ U is Ba.ire and compact. From the first assertion it foll ows that C \ U is a Gb-set which implies that U is an F 17 -set. (b) This st atement is proved in [20 , 0.2.2]. 3.5 THEOREM. A lzn ear fonn p on ih(X) zs a Baire m easure if and only if p is contznuous for o(Bb(X), 9J1 (.X)) (or any oth er topology co nsistent wilh th e duality < Bb(X),9Jl(X) >). PROOF. Let p E Bb(X)~ be a Baire measure, p.0 its restri ction to K(X ), and fio the unique o(Bb(X),9J1(X))-continuous extension of p. o E ~J1 (. \·) to Bb(X) (Theorem 2.2). By Proposition 2.1 fio is a regul ar Borel measure, hence countably addi t ive. Let I' J denote the restri ction of fio to Bb(X). Then 1'·1 is o(Bb(X), 9J1(X))-continuous and we claim that ft = f.t 1 . 44 Since 1-L and JLt are countably additive it is enough to s how that 1-L a nd 1-lt agree on e ve ry open relatively compact Baire set. L!'t. {! E lB I> (X) he open. By Le mma 3.4 U is an Fu-se t , say F, F, F, 11 U - Uoon=l "' Fn closec· l ' n C n+ 1· By thf'. lemma of U r y~o'" hn for each E l\l t here exists a functio n 0 < f n :::; 1 s uch that J(Fn) C {1} a. nd the s uppo rt. o f fn is conta in ed in U . Since Xu =limn- f n and s in ce f.l·kcx 1 = fLJ k(x 1 we have p.( U ) = lim ,,(f,) = lim P.l(fn ) = !'J (U). n-oo n.-N Thus p. is o(Bb(X) , ~1(X ))-cont inuous. Conve rsely, let 11 be o(Bb( X ), ~1 ( X ))-continuous. The n p. is bounded . Since orde r convergent sequences in fh(X) converge for o(ih(X),~1 (X)) (13.2.8) p. is co untably additive . 3.6 COROLLARY. Every Radon m easu1·e p. on X possesses a unique Ban·c eJ:tension Ji. Every Baire m easure on X is ( IJa ire) regular·. PROOF. By Theorem 2.2 every Radon measure I'· 0 11 X possesses a un1qu e regular Borel ex- tension Ji 1 . T he n p. := ji.1 la.(XI is the unique 13 a irf' ext<.' nsion as required. To prove that each Baire measure is ( Baire) regul a r, let p. be a Baire measure on X. By Definition 3. 1 we can assume that 1-L 2: 0. By Theorem 3.5 11 is o(l.'h( X) , 9J1 (.X))-continuous. S in ce Bb(X) is o(Bb(X) , 9J1( .X))-dense in Bb(X), p has a. unique o(Bb( .X),~1( .X ))- continuous exte nsion p. to Bb(X) ; by Proposition 2.1 p. is reg ul a r. T lw assertion fo ll ows now fr o m the definitio ns of regula rity a nd Baire regul a ri ty and fr om Lemma. 3.4(b). 3.7 C OROLLARY. Every Baire m easure possesses a unique regular Bor·el ext ension. In partie- ~dar , two regula r B orel m easur·es which co m cide on the 1·e lative/y co1n.pa ct Baire sets are identical. PROOF. The assert io n follows inunediately from the proof of Coro ll a ry 3.6. 3.8 REMARK. Let us o bserve that if X is (locall y compact a nd) countable at infinity then e very positive Baire measure possesses a. unique counta bly additive extens ion to lB (X) which is regul a r. The same is true for regul a r Borel measurf's. 45 4. Supplements It. is easy to see t ha t fo r each Borel measure I'· on X and each compact subset C of X, the mapping B ~ Pc (B) := p.(B n C), BE 'B b( X ), is a Bore l measure on X (Definition B.3.2). T he following is an extension of B.3.6. 4. 1 LEMMA. A Borel m easur·e I'· on X is re_qu lar· if and ouly if fl·c is re_qular f or eac h. compact s·ubs et C of X . PROOF. Let It be regular and let C0 be a fix ed , compact. t; Ub set of X . Given B E 'B&(X) a nd c > 0, there exists a compact set C and a n open set U in IB&(X) such t hat C C B C U and IPI(U\ C) llt coi( U \C)= llti((U \ C:) nCo)::; IP·I( U \C)< f holds. Thus fJ·Co is regu la r. Conve rsely, let f.l·c be regul ar for each compact. subset C of X . G iven B E 'B&(X) a nd c > 0, there exists a compact. set Co such that (.'0 is a neig hbourhood of B. Now sin ce f-1 Co is regul a r, we have C C B C U and II'·Co i(U \C) < r for suita bl e o pen a nd compact subsets U C Co ,C C Co respectively. But llti( U \C)= llti((U \C) nCo)= IPco i( U \ C)< c. Thus f.L is regular, as cl a imed. 4.2 THEOREM. Let X be any locally co mpac t spa cf and let (p 11 ) be a sequ ence of re_qular Borel measures such that for eac h BE 'B b( X) , lim 11 P·n(B) exists (in IR: ) T hen B-+ limn f.ln(B) defin es a re_qular Borel measure f.1 011 X . PROOF. By Lemma. 4. 1 (Jtn )c, n E f',f, is regul a.r for eac h compact s ubset C of X and by hypothesis fl·c (B) := lim,(p ,)c(B) = lim, P. n(C n B) , B E 'B &( X ), exists. It fo ll ows from [20, Theorem D.1.4] t hat llc is a regu Jar Borel measure fo r each co mpact C C X. Thus by Lemma. 4.1 , JL is regu lar. 4.3 CO ROLLARY. The baud of ail re_qu lar /Jorel measures in the Riesz d·ual of B&(X) is a(9Jl(X), Bb(X))-sequ entiai/y complete. 46 PROOF. By B.3.7 and C.1.3 the space 9J1(X) of a ll regular Borel measures is a band in Bb(X)*. Now, if (/tn) is a Cauchy sequence for 17(911( -Y) , IJb(X)) then for each BE 23b(X) li mn fln(B) exists (in JRl. ). Thus the assertion follows fr om Theorc111 4.2. Let ct be the family of a ll compact subsets of.\" a nd for any Fix ed C E ct let B(X, C) := {h E Bb(X): h(x) = 0 if x t:J. C}. Bb(X) is the union of the linear subspaces B(X,C) , where C runs through the compact subsets of X. Under the topology 'Ic generated by the uniform norm, B(X, C) is an AM-space with unit. lf 'I1 denotes the inductive topology on Bb(X) with respect to the family {(B(X, C), 'Ic) : C E ct} then (Bb(X) , 'I1 )' = Bb(X)* (since 'I1 coin cides with the order topology). Now we equip B(X,C) with the topology 'IM := T(8(X ,C),9J1(-Y)). Then we have the follow ing result of the inductive topology on Bb(S ). 4.4 PROPOSITION. Let 'I2 be th e znductzue topology on Bb(X) with respect to the family {(B(X,C), 'IM): C E ct} , where 'IM = T(B(X,C),9J1(X)). Then '.I2 is the Mackey topology T(Bb(X) , 9J1(X)). PROOF. By the de finition of the inductive topology T(B&(X) ,IJJI(X)) is coarser than 'I2 We now show that 'I2 is consistent. with the duality < Bb(X) ,9J1(X) >.By [17, II.6.1] a linear form on Bb(X) is 'I2-continuous if and only if its restriction to each of the id eals B(X, C) is 'IM-continuous, hence a regular Borel measure by C.2.1 (applied t.o the compact. space C). Thus by Lemma 4.1 the 'I2-continuous linear forms on Bb()\.. ) are precisely Lil t> regular Borel measures on X. Therefore the assertion follows from C.l.3. 4.5 PROPOSlTION . Suppose,\" is a locally compact space havwg property (B) such that 9J1(X) is barreled. Then the space Bb (X) is 17( Bb( X), 9J1( X) )-sequ entially co mplete. PROOF. We note First that each of the spaces IJ(X, C) (C: compact subset of X) is 17(B(X, C), 9J1(X))-sequentially complete [19 , B.2.7]. Thus it is enough to show that each 17(B&(X) , 9J1(X)) Cauchy sequence is contained and bounded in B(X , (.') for sollle compact C. But since 9J1(X) is barreled by hypothesis a 17( Bb( X), 9J1( X) )-bounded sequence is strongly bounded in 9J1( X)' [17, IV.5.2], hence order bounded in K(Y) by I.C.4.4, say by some hE K(Y)+. Now if S denotes the support of h and if C := 47 We close with the following remark. 1 4.6 PROPOSITIO . T he spa ce IJ 6(X) of doubly bouuded /Jon/ fuu ctions co iu cides with .\:(X) if and only if X is disc rete. PROOF. If ,\" is discrete the n by I. C.3.4 ,qx) is re fl exive. Conve rsely, if Bb(X) coincides with K(X) , we s uppose that X is not di sc rete. T hen t here ex ists an infinitP compact s ubset C of X. 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(22] TUCKER, C.T., Represc nt.at.ion of Baire FuncLions as Continuous Functions. Fund. Math. 101 , 181-188 ( 1978). (23] TUCI'"ER, C.T., Pointwise a nd Order Conve rge nce for Spaces of Continuo us Functions and Spaces of Baire Functions. Czechos lovak Math . .1 . :34 (109), 562-269 (1984). 50 VITA Hermine Engels was born on December 4, 1960 in Hatzfeld , Rumania. In 1981 she graduated from the List-Gymnasium in Reutlingen, Germany, and was admitted to the University of Tiibingen wh ere she received her Diploma. in Mathematics in August 1988. In J anuary 1989 she was accepted into the Ph. D. program of the Department of Mathematics of Florida Atlantic University. There she served as a teaching assistant until 1991. 51