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 µ in a Hausdorff 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 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 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 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 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]. A set E ⊂ X is called gage measurable [5, Definition 3.1] if for each ε>0, there is a gage γ in X such that

p q v (Ai ∩ Bj) <ε i=1 j=1 for each γ-fine partitions {(A1,x1),..., (Ap,xp)} and {(B1,y1),..., (Bq,yq)} anchored in E and X − E, respectively. 294 Fernandez´

Throughout this section we shall use the notations of [5]. Thus, S∗ is the family of all gage measurable subsets of X and µ is the restriction of v∗ to the σ-algebra M of all subsets of X which are v∗-measurable in Carath´eodory sense. Then µ is a complete saturated and regular measure. The following result is [5, Theorem 4.7] with a more natural proof.

Theorem 3.1 If µ is σ-finite, then S∗ = M.

∗ Proof. By [5, Theorem 4.4], S ⊂M. Let E ∈M. There are sets En ∈Msuch ∪∞ ∞ ∈ N ∈ N that X = n=1 En and µ (En) < + for each n . Let ε>0. For each n there is G ∈Gwith E ∩ (X − E) ⊂ G and n n n −n µ Gn − En ∩ (X − E) <ε2 . ∩∞ − ∈F ⊂ Set F = n=1 (X Gn). Then F ,F E and ∞ ∞ E − F = E ∩ Gn = Gn − (X − E) n=1 n=1 ∞ ∞ = Gn − En ∩ (X − E) n=1 n=1 ∞ ⊂ Gn − En ∩ (X − E) n=1 and so ∞ µ (E − F ) ≤ µ Gn − En ∩ (X − E) <ε n=1 hence E ∈S∗ by [5, Lemma 4.10].  If µ is not σ-finite and Σ is the family of all µ-σ-finite elements of M, then the inclusion Σ ⊂S∗ holds when X is metacompact and when X is metalindel¨of and Martin’s Axiom and the negation of the Continuum Hypothesis are assumed [5, Theorem 4.11]. We shall establish that Σ ⊂S∗ holds also when one of the conditions (a),(b)or (c)from Theorem 2.1 or Corollary 2.2 is satisfied.

Theorem 3.2 The inclusion Σ ⊂S∗ is implied by either of the following conditions: (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 Corollary 2.2 of this paper proceeding as in [5, Theo- rem 4.11].  Moderation of sigma-finite Borel measures 295

References

1. R. Engelking, General Topology, PWN Warsaw, 1977. 2. R. J. Gardner and W. F. Pfeffer, Some undecidability results concerning Radon measures, Trans. Amer. Math. Soc. 259, 1 (1980), 67–74. 3. R. J. Gardner and W. F. Pfeffer, Borel measures, in Handbook of Set-theoretic Topology, Ed. K. Kunen and J. E. Vaughan, Nort-Holland, New York, 1984, 961–1043. 4. W. F. Pfeffer, Integrals and Measures, Marcel Dekker, New York, 1977. 5. W. F. Pfeffer and B. S. Thomson, Measures defined by gages, Canad. J. Math. 44, 6 (1992), 1303–1316.