Unbounded weighted Radon measures and dual of certain function spaces with strict topology S. Maghsoudi and A. Rejali Abstract Let X be a C-distinguished topological space, and let ! be a weight func- tion on X. Denote by Cb(X; !) the space of all real-valued functions f with e f=! 2 Cb(X), and by Cb(X; !) the space of all real-valued continuous func- tions f such that f=! is bounded. We introduce certain locally convex topolo- e gies on Cb(X; !) and Cb(X; !), and as our main results we determine their duals.1 Introduction Let X be a C-distinguished topological space; that is, a Hausdorff topological space X such that the space of real-valued bounded continuous functions on X, Cb(X), separates points of X. By a weight function on X we mean a Borel measurable function ! : X ! (0; 1) such that !¡1 is bounded on the compact subsets of X. Let Cb(X; !) be the space of all real-valued functions f with f=! 2 Cb(X), and e let Cb(X; !) be the space of all real-valued continuous functions f such that f=! is bounded. It is easy to see that, with the usual pointwise operations and the norm e k:k! defined by kfk! := kf=!k the spaces Cb(X; !) and Cb(X; !) are Banach spaces. In the special case ! = 1 the space Cb(X; !) is the usual space Cb(X). In his seminal work [5], Buck defined the strict topology ¯ on Cb(X) for a locally compact space X. Later, Giles [6] extended the result of Buck to the completely regular spaces. The strict topology ¯ is the locally convex topology defined by the seminorms P', where P'(g) = supf '(x) jg(x)j : x 2 X g (g 2 Cb(X)); 12000 Mathematics Subject Classification: Primary 46E10, 46E27 Secondary: 46A03, 46A40. Key words: C-distinguished space, strict topology, strong dual, weighted function space, weighted Radon measure. 1 2 S. Maghsoudi and A. Rejali and ' varies through the set of all positive bounded Borel measurable functions vanishing at infinity. An interesting and important property of the strict topology is that the dual space of (Cb(X); ¯) can be identified with the space of all finite regular Borel measures on X; see [6] and [7] for more details. Many authors have studied the so-called strict topologies; see for example [1], [3], [8], [9], [10], [14], [15], and [16]. Our aim in this paper is to introduce and study locally convex topologies ¯(X; !) e e and ¯(X; !) on Cb(X; !) and Cb(X; !), respectively. We will show that the dual of these spaces can be identified in a natural way with a Banach space of weighted Radon measures on X. 1 Space of Weighted Radon measures Let us recall some basic concepts and results from measure theory as given in [4]. The σ-algebra generated by the open subsets of X is called the σ-algebra of Borel sets and denoted B = B(X). A (positive) Radon measure is a Borel measure ¹ : B(X) ¡! [0; 1] such that ¹(C) is finite for C 2 K(X), and ¹ is inner-regular; that is, ¹(B) = supf ¹(C): C ⊆ B; C 2 K(X) g (B 2 B(X)); where K = K(X) denotes the family of all compact subsets of X. The set of all + + positive Radon measures on X is denoted by M (X). By Mb (X) we denote the set of all positive bounded Radon measures on X. We need the following simple lemma. Lemma 1.1 Let ' : X ! [0; 1] be a Borel measurable function on the Hausdorff space X such that is bounded on the compact subsets of X. If ¹ 2 M +(X) then + ¹' 2 M (X), where Z ¹'(B) = ' d¹ (B 2 B): B Proof. It is clear that ¹' is a positive measure which is bounded on compact subsets. For each compact subset K of X, let ¹K = ¹ÂK . It is clear that ¹K is a Radon measure. Directing the family K by inclusion, we get ¹K % ¹. Hence, by Exercise 1.29 in [4], Z Z supf ' d¹ : K ⊆ B; K 2 Kg = supf ' d¹K : K ⊆ B; K 2 Kg K Z K = ' d(¹ÂB) ZX = ' d¹; B Unbounded weighted Radon measures and dual of 3 for each Borel subset B. This proves the lemma. 2 The following lemma is an easy application of Lemma 1.1. Lemma 1.2 Let X be a Hausdorff space, ' : X ! (0; 1) be a Borel measurable function, and ¹ be a positive Borel measure. Then (i) (¹')'¡1 = (¹'¡1)' = ¹. (ii) If '¡1 is bounded on the compact subsets and ¹' 2 M +(X), then ¹ 2 M +(X). Let X be a Hausdorff space, and let ! : X ¡! (0; 1) be a weight function + on X. Let Mb (X; !) be the set of all positive Radon measures ¹ on X such that + + + ¹! 2 Mb (X). Define the equivalence relation ‘»’ on Mb (X; !) £ Mb (X; !) by (¹; º) » (¹0; º0) if and only if ¹ + º0 = ¹0 + º: + + Let [¹; º] be the equivalence class of (¹; º) 2 Mb (X; !) £ Mb (X; !). Now we define + Mb(X; !) = f[¹; º]: ¹; º 2 Mb (X; !)g: Then Mb(X; !) with the operations defined by [¹; º] + [¹0; º0] = [¹ + ¹0; º + º0] and ( [¸¹; ¸º] if ¸ ¸ 0; ¸[¹; º] = [¸º; ¸¹] otherwise; 0 0 for [¹; º]; [¹ ; º ] 2 M(X) and ¸ 2 R, the norm k:k! defined by k[¹; º]k! = k¹! ¡ º!k; for [¹; º] 2 Mb(X; !), is a real normed space. Recall that k:k denotes the usual norm of a bounded Radon measure. + The following proposition states the elementary properties of Mb (X; !). Proposition 1.3 Let X be a Hausdorff space, and let ! : X ¡! (0; 1) be a weight function on X. Then + ¡ + (i) For each ´ 2 Mb(X; !) there exist unique measures ´ ; ´ 2 Mb (X; !) such that ´ = [´+; ´¡] and ´+ ? ´¡. (ii)(Mb(X; !); k:k!) is a Banach space. 4 S. Maghsoudi and A. Rejali + Proof. To prove (i), let ´ = [¹; º] for some ¹; º 2 Mb (X; !). Then θ := ¹! ¡ º! 2 Mb(X). By the Hahn decomposition theorem, there exist unique measures + ¡ + + ¡ + ¡ + + ¡1 θ ; θ 2 Mb (X) such that θ = θ ¡ θ and θ ? θ . Now set ´ = θ ! and ´¡ = θ¡!¡1. Thus ¹ + ´¡ = º + ´+, and so ´ = [´+; ´¡]. (ii) That (Mb(X; !); k:k!) is a normed space is straightforward. Now, we prove that (Mb(X; !); k:k!) is complete. To prove this, let (´n)n be a Cauchy sequence in + ¡ Mb(X; !) and note that k´nk! = k(¹n! ¡ ºn!) k + k(¹n! ¡ ºn!) k. Then, it is clear that (¹n! ¡ ºn!)n is a Cauchy sequence in Mb(X), and so converges to, say θ, + ¡1 ¡ ¡1 in Mb(X). Put ´1 = θ ! , ´2 = θ ! and ´ = [´1; ´2]. Then ´n ! ´; this follows from the fact that + ¡ k´n ¡ ´k! = k(´n ! ¡ ´2!) ¡ (¹n! ¡ ´1!) k + + ¡ ¡ · k´n ! ¡ θ k + k´n ! ¡ θ k: 2 For ´ 2 Mb(X; !), by the standard representation of ´ we mean the expression [´+; ´¡] stated in Proposition 1.3. 2 The dual Space of Cb(X; !) with ¯(X; !) topology We call a function ' : X ¡! [0; 1) an !-hood, if !' is a bounded Borel measurable function such that for each " > 0 there exists a compact subset K ⊆ X such that '(x) < "!¡1(x) for all x 2 X n K. The set of all !-hoods on X is denoted by Hd(X; !). Then, !-strict topology ¯(X; !) on Cb(X; !) is the locally convex topology generated by the seminorms fP' : ' 2 Hd(X; !)g, where P'(g) = supf'(x)jg(x)j : x 2 Xg for g 2 Cb(X; !). In the case ! = 1, ¯(X; !) coincides with the strict topology on Cb(X) defined in [5] and [6]. Note that ¯(X; !) topology is weaker than k:k! topology on Cb(X; !). So, ¤ (Cb(X; !); ¯(X; !)) by the norm kF k = supfjF (g)j : g 2 Cb(X; !); jjgjj! · 1g; is a normed space. The following result is a generalization of Theorem 4.6 in [7] to the more general setting of the weighted space of functions. Unbounded weighted Radon measures and dual of 5 Proposition 2.1 Let X be a C-distinguished space, and let ! : X ¡! (0; 1) be a weight function on X. Then the map ´ 7! I´+ ¡ I´¡ from Mb(X; !) onto ¤ + ¡ (Cb(X; !); ¯(X; !)) is an isometric isomorphism, where ´ = [´ ; ´ ] is the stan- dard representation of ´ and Z I¹(f) = f d¹ X + for f 2 Cb(X; !) and ¹ 2 Mb (X; !). ¤ ¤ Proof. Define T! :(Cb(X); ¯) ! (Cb(X; !); ¯(X; !)) by T!(g) = g!, where (g!)(') = g(!') for all ' 2 Cb(X). It is easy to see that T! is a well-defined linear isometric isomorphism. Also, the map ´ 7! ´+! ¡ ´¡! is a linear isometric + ¡ isomorphism from Mb(X; !) onto Mb(X), where ´ = [´ ; ´ ] 2 Mb(X; !). Now ¤ we only need to recall from [7] Theorem 4.6 that (Cb(X); ¯(X; 1)) is isometrically isomorphism to Mb(X). 2 ¤ The following example shows that the elements of (Cb(X; !); ¯(X; !)) cannot, in general, be represented by a signed measure on X. Example 2.2 Let m be the Lebesgue measure on R, and let ¹ = mÂ[0;1); º = mÂ(¡1;0) and ´ = [¹; º]. Also, let ! : R ! (0; 1) be defined by ( 1 if x 2 [¡1; 1] !(x) = 1=x2 otherwise: R R Define I(f) = R fd¹ ¡ R fdº for all f 2 Cb(R;!). Then ´ 2 Mb(R;!) and I 2 ¤ R (Cb(R;!); ¯(R;!)) , but there is no signed measure ³ on R such that I(f) = R f d³ for all f 2 Cb(R;!). f f 3 The dual Space of Cb(X; !) with ¯(X; !) topology e e In this section we define locally convex topology ¯(X; !) on the space Cb(X; !) and determine its dual space.