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LIMITATIONS on the EXTENDIBILITY of the RADON-NIKODYM THEOREM 0. Notation and Basic Definitions Throughout the Paper X and Y PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 8, Pages 2491{2500 S 0002-9939(03)07046-1 Article electronically published on March 11, 2003 LIMITATIONS ON THE EXTENDIBILITY OF THE RADON-NIKODYM THEOREM GERD ZEIBIG (Communicated by N. Tomczak-Jaegermann) Abstract. Given two locally compact spaces X; Y and a continuous map r : Y X the Banach lattice 0(Y ) is naturally a 0(X)-module. Following ! the Bourbaki approach to integrationC we define generalizedC measures as 0(X)- 1 linear functionals µ : 0(Y ) 0(X). The construction of an L (µ)-spaceC and ! the concepts of absoluteC continuityC and density still make sense. However we exhibit a counter-example to the natural generalization of the Radon-Nikodym Theorem in this context. 0. Notation and basic definitions Throughout the paper X and Y will denote locally compact spaces and r : Y X will be a fixed continuous function. ! We use standard notation as in [3, 5]. In particular (X) stands for the space of C0 all complex-valued continuous functions on X which vanish at infinity and b(X) is the space of all complex-valued bounded continuous functions on X. The injectiveC and the projective tensor products are respectively denoted ˇ and ^ .Themost relevant feature of the injective tensor product for the present⊗ work is⊗ that we can identify (X Y )with (X) ˇ (Y )[2]. C0 × C0 ⊗ C0 Recall that a Banach module M over the Banach algebra 0(X) (under the pointwise operations) is a Banach space together with a contractiveC bilinear action- : map 0(X) M M.IfX is compact this action is subject to the usual axioms: α.(β.mC)=(×αβ):m−! and 1:m = m for all `scalars' α, β in (X)andallelements C0 m of M.IfX is not compact, in which case the algebra 0(X)doesnothavea unit element, the second axiom is replaced by the non-degeneracyC requirement that ('j:m)j converges to m,where('j)j is any contractive approximate identity for 0(X). Clearly 0(X)isa 0(X)-module under the pointwise operations. C Given two BanachC modulesC M and N over (X), a bounded linear map T : C0 M N is said to be 0(X)-linear if T (α.m)=α.T (m) for every scalar α 0(X) and! every m M. TheC Banach space of all (X)-linear maps from M2toCN is 2 C0 denoted L 0(X)(M;N). C Received by the editors March 20, 2002. 2000 Mathematics Subject Classification. Primary 46B22; Secondary 46J10, 46E30. Key words and phrases. Banach module, Radon-Nikodym Theorem, Riesz Theorem. c 2003 American Mathematical Society 2491 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2492 GERD ZEIBIG 1. Generalized measures µ and their L1(µ)-spaces The starting point for our work is the fact that the continuous function r : Y X ! yields a natural (X)-module structure on (Y ), the action being α.f := (α r) f C0 C0 ◦ · for α 0(X)andf 0(Y ). Consider2 C the special2 caseC where X is the one-point space . Now the function {∗} r : Y X is necessarily the constant function r , the Banach algebra 0(X) ! ≡∗ : C identifies with the field C and the action 0(X) 0(Y ) 0(Y ) `is' the vector C × C −! C space action of C on 0(Y ). Hence any 0(X)-linear map µ : 0(Y ) 0(X) iden- C C C ! C tifies with a usual bounded linear functional on 0(Y ). The Riesz Representation Theorem [4] in turn identifies µ with a finite RadonC measure on Y. This special case motivates Definition 1.1. A generalized measure µ on Y is a bounded 0(X)-linear map µ : (Y ) (X). We say that µ is positive if µ(f) 0 for everyC f 0. C0 ! C0 ≥ ≥ Example 1.1. If X is the one-point space, the generalized measures on Y simply identify with the finite Radon measures on Y . For more examples assume that r : Y X is the canonical projection pr : X Z ! Z × X for some locally compact space Z. Every Radon measure µsc : 0(Z) C on Z induces a generalized measure µ : (X Z) (X)asthecompositeC ! C0 × ! C0 id (X) ˇ µsc C0 ⊗ µ : 0(X Z) 0(X) ˇ 0(Z) 0(X) ˇ C 0(X): C × ≡ C ⊗ C −−−−−−−! C ⊗ ≡ C We have µ(f)(x)= f(x; :) dµsc ZZ for f 0(X Z)andx X.Inparticularµ is positive if µsc is positive. To2 getC more× generalized2 measures on X Z we can start with any bounded × linear map Tµ : 0(Z) 0(X): we obtain a generalized measure µ by taking the composite C ! C ˇ id (X) Tµ multiplication µ : (X Z) (X) ˇ (Z) C0 ⊗ (X) ˇ (X) (X): C0 × ≡ C0 ⊗ C0 −−−−−−−! C0 ⊗ C0 −−−−−−−−−! C0 We then have µ(f)(x)=Tµ(f(x; :))(x)forf 0(X Z)andx X. Again µ is 2 C × 2 positive if Tµ is positive. It can be shown that every generalized measure µ : (X Z) (X)comes C0 × ! C0 from a unique map Tµ : 0(Z) b(X): if X is compact we simply have Tµ(g)= µ( g pr )forg (ZC). If X! isC merely locally compact, we use a contractive ◦ Z 2 C0 approximate identity of 0(X)toobtainTµ. We have in fact an isometric identi- fication of Banach latticesC L 0(X)( 0(X Z); 0(X)) L( 0(Z); b(X)): C C × C ≡ C C We now come to the definition of the space L1(µ) for some fixed positive gener- alized measure µ. To begin we observe that we can define a semi-norm on (Y ) C0 by setting f 1 := µ( f ) 0(X),where : 0(X) denotes the usual sup-norm. The subadditivityk k followsk fromj j k theC classicalk trianglekC inequality. Indeed, let f;g be in 0(Y ).Sinceµ( f + g ) is a positive element of 0(X) there is some x0 in X such C j j C that µ( f + g ) 0(X) = µ( f + g )(x0)=δx0 µ ( f + g ), where δx0 denotes the k j j kC j j ◦ j j Dirac measure at x0.Asδx0 µ : 0(Y ) C `is' a positive measure in the usual ◦ C ! License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXTENDIBILITY OF THE RADON-NIKODYM THEOREM 2493 sense, the triangle inequality assures us that 1 f + g 1 = δx0 µ ( f + g )= f + g L (δx µ) k k ◦ j j k k 0◦ 1 1 f L (δx µ) + g L (δx µ) = δx0 µ ( f )+δx0 µ ( g ) ≤kk 0◦ k k 0◦ ◦ j j ◦ j j µ( f ) 0(X) + µ( g ) 0(X) ≤k j j kC k j j kC = f 1 + g 1 : k k k k We denote by L1(µ) the completion of the normed space obtained from ( (Y ); C0 : 1) by quotienting out the null-space of the semi-norm : 1. k k 1 k k By the construction of our space L (µ) we have a canonical map i : 0(Y ) 1 C ! L (µ). This map is injective precisely when the semi-norm : 1 on 0(Y ) is actually a norm, and this occurs when µ has \full support" in a sensek k whichC will be made precise in a subsequent paper [7]. Example 1.2. a) If X is the one-point space our positive generalized measure µ identifies with a positive (finite) Radon measure on Y .ThespaceL1(µ)inthe sense we just defined now identifies with the usual space L1(µ). b) It can be shown that in the case where Y r X is the identity X id X, 1 −! −! the space L (µ) identifies with a certain 0(U). Specifically U is the complement 1 C in X of the subset m− ( 0 ), where m b(X) denotes the function such that µ(f)=f m for every f f g (X). 2 C · 2 C0 Proposition 1.1. The space L1(µ) is naturally a (non-degenerate) Banach (X)- C0 module and a Banach lattice, in such a way that the canonical map i : 0(Y ) 1 C ! L (µ) is a morphism of 0(X)-modules and of Banach lattices. Furthermore this mapC satisfies the universal property that for every Banach 0(X)-module M and every contractive morphism T :( 0(Y ); : 1) M of semi- k k ! normedC (X)-modules there exists a unique contractiveC morphism H : L1(µ) M 0 ! of (X)C-modules such that the following diagram commutes: C0 T 0(Y ) /M C < i H L1(µ) Proof. L1(µ)isa (X)-module. • C0 Recall that the 0(X)-module structure of the semi-normed space ( 0(Y ); : 1) is given by α.f :=C (α r) f for α (X)andf (Y ). As C k k ◦ · 2 C0 2 C0 α.f 1= µ( α r f ) 0(X)= µ( α r f ) 0(X)= µ( α : f ) 0(X) k k k j ◦ · j kC k j |◦ ·| j kC k j j j j kC = α µ( f ) 0(X) α 0(X) µ( f ) 0(X) = α 0(X) f 1 ; kj j j j kC ≤k kC k j j kC k kC k k 1 the nullspace K := : − ( 0 )ofoursemi-norm : 1 is algebraically an ideal k k1 f g k k in (Y ), so the module action of (X) on (Y ) quotients to a (X)-action on C0 C0 C0 C0 0(Y )=K.
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