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Fern⁄Ndez 289-295 Collect. Math. 48, 3 (1997), 289–295 c 1997 Universitat de Barcelona Moderation of sigma-finite Borel measures J. Fernandez´ Novoa Departamento de Matematicas´ Fundamentales. Facultad de Ciencias. U.N.E.D. Ciudad Universitaria. Senda del Rey s/n. 28040-Madrid. Spain Received November 10, 1995. Revised March 26, 1996 Abstract We establish that a σ-finite Borel measure µ in a Hausdorff topological space X such that each open subset of X is µ-Radon, is moderated when X is weakly metacompact or paralindel¨of and also when X is metalindel¨of and has a µ- concassage of separable subsets. Moreover, we give a new proof of a theorem of Pfeffer and Thomson [5] about gage measurability and we deduce other new results. 1. Preliminaries Let X be a Hausdorff topological space. We shall denote by G, F, K and B, respec- tively, the families of all open, closed, compact and Borel subsets of X. Let µ be a Borel measure in X, i.e. a locally finite measure on B. A set B ∈B is called (a) µ-outer regular if µ(B)= inf {µ(G):B ⊂ G ∈G}; (b) µ-Radon if µ(B)= sup {µ(K):B ⊃ K ∈K}. The measure µ is called (a) outer regular if each B ∈Bis µ-outer regular; (b) Radon if each B ∈Bis µ-Radon. A µ-concassage of X is a disjoint family (Kj)j∈J of nonempty compact subsets of X which satisfies 289 290 Fernandez´ ∩ ∈G ∈ ∩ ∅ (i) µ(G Kj) > 0 for each G and each j J such that G Kj = ; ∩ ∈B (ii) µ(B)= j∈J µ (B Kj)for each B which is µ-Radon. If each open set of finite µ-measure is µ-Radon, then X has a µ-concassage [2, Proposition 12.10]. If moreover µ is σ-finite, then every µ-concassage is countable. 2. Moderation and regularity In [3, Theorem 12.3, (ii)and Theorem 12.11] Gardner and Pfeffer have shown that if each open subset of X is µ-Radon, a σ-finite Borel measure µ in X is moderated when X is metacompact and when it is metalindel¨of and Martin’s Axiom and the negation of the Continuum Hypothesis are assumed. We prove the result mentioned in the abstract. Theorem 2.1 Let µ be a σ-finite Borel measure in X. If each open subset of X is µ-Radon, then µ is moderated whenever either of the following conditions is satisfied: (a) X is weakly metacompact; (b) X is paralindel¨of; (c) X is metalindel¨of and has a µ-concassage of separable subsets. Proof. Since µ is locally finite, the family U of all open subsets of X of finite µ- measure is an open cover of X and there is an open refinement V of U such that V ∪∞ V V ∈ N V = i=1 i with i point-finite for each i in case (a), is locally countable in case (b)and V is point-countable in case (c). (a)Let ( Kj)be a countable µ-concassage of X and assume that the family Vi,j = {V ∈Vi: µ (V ∩ Kj) > 0} ∈ N2 V ∪∞ W is uncountable for some (i, j) . Since i,j = k=1 k with Wk = {V ∈Vi: µ (V ∩ Kj) ≥ 1/k}, there are k ∈ N and distinct Vn ∈Vi,n∈ N, such that µ (Vn ∩ Kj) ≥ 1/k for each n ∈ N. It follows that µ (lim sup Vn) ≥ µ lim sup (Vn ∩ Kj) ≥ lim sup µ (Vn ∩ Kj) ≥ 1/k n n n Moderation of sigma-finite Borel measures 291 ∅ V V hence lim supn Vn = which contradicts the point-finiteness of i.Thus i,j is countable for each (i, j) ∈ N2. V { ∈V } V ∪∞ V Then the family 0 = V : µ(V ) > 0 is also countable since 0 = i=1 i,0 where, for each i ∈ N, Vi,0 = {V ∈Vi: µ(V ) > 0} which is countable because ∞ µ(V )= µ (V ∩ Kj) j=1 ∈G V ⊂∪∞ V V ∪ V−V for every V and so i,0 j=1 i,j. Set 0 =(Wn)and W0 = ( 0).If ⊃ ∈K V−V ⊂∪r W0 K , there are A1,..., Ar in 0 such that K i=1 Ai, hence r µ (K) ≤ µ (Ai)=0. i=1 Consequently, µ (K)= 0 for every K ∈Kcontained in W0 and µ (W0)= sup {µ (K): W0 ⊃ K ∈K}=0. ∪∞ ∈G Thus X = n=0 Wn with Wn for n =0, 1, 2,..., µ(W0)= 0 and moreover, µ (Wn) < + ∞ for each n ∈ N, hence µ is moderated. (b)Let ( Kn)be a countable µ-concassage of X. Each point of X has an open neighborhood which meets only countably many sets from V and, for each n ∈ N,a finite family of this neighborhoods is a cover of Kn, hence the family Vn = {V ∈V: V ∩ Kn = ∅} is countable for every n ∈ N. Then the family W { ∈V ∩ ∪∞ ∅} = V : V ( n=1 Kn) = W ∪ V−W ∪∞ is also countable. Set =(Vn)and V0 = ( ). Then X = n=0 Vn with Vn ∈Gfor n =0, 1, 2,..., ∞ µ (V0)= µ (V0 ∩ Kn)=0 n=1 and µ (Vn) < + ∞ for each n ∈ N, hence µ is moderated. (c)Let ( Kn)be a countable µ-concassage of X of separable subsets. For each n ∈ N there is a countable set An with A¯n = Kn and if V ∈Vthen V ∩ Kn = ∅ if and only if V ∩ An = ∅. Thus, for each n ∈ N the family Vn = {V ∈V: V ∩ Kn = ∅} 292 Fernandez´ coincides with the family {V ∈V: V ∩ An = ∅}. This proves that Vn is countable for every n ∈ N because V is point-countable and An is countable for each n ∈ N and, proceeding as in (b), the proof finishes. Corollary 2.2 Let µ be a σ-finite Borel measure in X. If each open subset of X is µ-Radon, then µ is outer regular and Radon when one of the following conditions is satisfied: (a) X is weakly metacompact; (b) X is paralindel¨of; (c) X is metalindel¨of and has a µ-concassage of separable subsets. Proof. It follows from above theorem and from [3, Theorem 6.7]. Remark 2.3. If Martin’s Axiom and the negation of the Continuum Hypothesis are assumed, all σ-finite Radon measures in metalindel¨of spaces are outer regular [2, Theorem 3.6 and 3, Theorem 12.11]. However, assuming the Continuum Hypothesis, it is possible to construct a σ-finite Radon measure in a metalindel¨of space which is not outer regular [3, Example 12.12]. Without Martin’s Axiom, we have shown that a σ-finite Radon measure µ in a metalindel¨of space X is outer regular whenever X has a µ-concassage of separable subsets. The following example shows the interest of this result. Example 2.4: For nonnegative integers k and n, we consider the points qk,n = −n −n { n} ∪∞ (k 2 , 2 )and we denote Qn = qk,n: k =0, 1, 2,..., 2 and Q = n=1 Qn.In X =[0, 1] ∪ Q we define a topology as follows: the points of Q are isolated, and a neighborhood base at t ∈ [0, 1] is given by the sets −n −n U (t, ε)={t}∪{qk,n ∈ Q:2|k 2 − t| < 2 <ε} where ε>0. With this topology the space X is not metalindel¨of (neither weakly metacompact nor paralindel¨of, hence), which may be seen by purely topological arguments, but is a direct consequence of the preceding corollary. Indeed, give −n x ∈ X, let f(x)=2 if x ∈ Qn and f(x)= 0 otherwise. Then the weighted counting measure µ in X with the weight f is a σ-finite Radon measure in X which is not outer regular (see [3, Example 12.7]). Since Q with the family of all U (t, ε),t∈ [0, 1],ε>0, is an open cover of X and Q is countable, each compact subset of X is countable, hence X has µ-concassages of separable subsets. Moderation of sigma-finite Borel measures 293 3. About gage measurability In [5] Pfeffer and Thomson define an outer measure v∗ by means of gages and introduce the concept of gage measurability which is different from the usual Carath´eodory definition. We shall expose, as briefly as possible, this process. Let S be a semiring of subsets of X and let v be a volume in S, i.e., a nonnegative real-valued function such that n v (A)= v (Ai) i=1 ∈S { }⊂S ∪n for each A and each disjoint collection A1,..., An for which i=1 Ai = A. A partition is a collection P = {(A1,x1),..., (Ap,xp)} where A1,..., Ap are disjoint sets from S and x1,..., xp are points of X. We say that P is anchored in a set E ⊂ X if {x1,..., xp}⊂E. A gage in a set E ⊂ X is a map γ that to each x ∈ E assigns an open neighborhood γ (x)of x in X.Ifγ is a gage in E ⊂ X, then a partition {(A1,x1),..., (Ap,xp)} anchored in E is called γ-fine whenever Ai ⊂ γ (xi)for i =1,..., p. If γ is a gage in E ⊂ X,welet p vγ (E)= sup v (Ai) i=1 where the supremum is taken over all partitions {(A1,x1),..., (Ap,xp)} anchored in E that are γ-fine. The map v∗ defined for each E ⊂ X by ∗ v (E)= inf vγ (E) γ where the infimum is taken over all gages γ in E, is an outer measure in X [5, Proposition 2.1].
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