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Unbounded Weighted Radon Measures and Dual of Certain Function Spaces with Strict Topology

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Unbounded Weighted Radon Measures and Dual of Certain Function Spaces with Strict Topology 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.
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