On Absolute Continuity∗

journal of mathematical analysis and applications 222, 64–78 (1998) article no. AY975804

Zolt´an Buczolich†

Department of Analysis, Eotv¨ os¨ Lorand´ University, Muzeum´ krt. 6–8, H–1088, Budapest, Hungary

View metadata, citation and similar papers at core.ac.ukand brought to you by CORE provided by Elsevier - Publisher Connector Washek F. Pfeffer ‡

Department of Mathematics, University of California, Davis, California 95616

Submitted by Brian S. Thomson

Received June 5, 1997

We prove that in any dimension a variational measure associated with an additive continuous function is σ-finite whenever it is absolutely continuous. The one-dimensional version of our result was obtained in [1] by a dif- ferent technique. As an application, we establish a simple and transparent relationship between the Lebesgue integral and the generalized Riemann integral defined in [7, Chap. 12]. In the process, we obtain a result (The- orem 4.1) involving Hausdorff measures and Baire category, which is of independent interest. As variations defined by BV sets coincide with those defined by figures [8], we restrict our attention to figures. The set of all real numbers is denoted by , and the ambient space of this paper is m where m 1 is a fixed integer. In m we use exclusively the metric induced by the maximum≥ norm . The usual inner product of · x; y m is denoted by x y, and 0 denotes the zero vector of m. For an ∈ · x m and ε>0, we let ∈ B x y m x y <ε : ε = ∈ x − *The results of this paper were presented to the Royal Belgian Academy on June 3, 1997. †This author was supported by the Hungarian National Foundation of Scientific Research, Grant T019476 and FKFP 0189/1997. E-mail: [email protected]. ‡E-mail: [email protected]. 64

The closure, interior, boundary, and diameter of a set E m are denoted by cl E; int E; ∂E, and d E , respectively. ⊂ The Lebesgue measure in m is denoted by λ; however, for E m,we ⊂ write E instead of λ E . A set E m with E 0 is called negligible. ⊂ = We say sets A; B m overlap if A B > 0. Unless specified otherwise, the words “measure”⊂ and “measurable” ∩ as well as the expressions “almost all,” “almost everywhere,” and “absolutely continuous” always refer to the Lebesgue measure λ. The s-dimensional Hausdorff measure in m is denoted by H s. A set m m 1 T of σ-finite measure H − is called thin. Thin sets will play an important⊂ role in our exposition. Unless specified otherwise, a number is an extended real number, and a function is an extended real-valued function.

1. CHARGES

A cell is a compact nondegenerate subinterval of m, and a figure is a finite (possibly empty) union of cells. If A and B are figures, then so are A B, ∪ A B cl int A int B and A B cl A B : = ∩ = − m 1 Each figure A has the perimeter A H − ∂A , and the unit exterior defined, in the obvious way,=m 1-almost everywhere on . normal νA H − ∂A The family F of all figures is topologized as follows. For n 1; 2;:::; topologize the families =

F A F A B 0 and A

Additivity: F A B F A F B for each pair of nonoverlapping ﬁgures A and B. ∪ = + Continuity: Given ε>0, there is an η>0 such that F C <εfor each ﬁgure C B1 0 with C < 1/ε and C <η. ⊂ /ε 66 buczolich and pfeffer

To illustrate the concept of charge, we give two important examples.

(1) If is a locally integrable function on m, let f F A Afdλ for each figure A. By the absolute continuity of the Lebesgue integral,= F is a charge, called the indefinite Lebesgue integral of f . R (2) If is a continuous vector field on m, let v F A ∂A v m 1 for each figure . Then is a charge, called the =of [7,· νA dH − A F flux v Proposition 11.2.8]. R

2. DERIVATES

Given an η>0, we say a ﬁgure A is η-regular if

A >η: d A A The closer η is to 1/ 2m , the more cube-like are η-regular figures. In particular, no η-regular figure is empty. Let x m, and let F be a real-valued function defined on F . For a positive η<∈ 1/2m, let

inf sup F B DηF x =δ>0 B B where B is an η-regular figure with x B and d B <δ. The number ∈ sup DF x DηF x =0<η<1/2m is called the upper derivate of F at x. Using an argument similar to [10, Chap. IV, Theorem 4.2], it is easy to show that the functions and , DηF DF defined on m in the obvious way, are measurable. If DF x D F x , we denote this common value by DF x , and say F =−is derivable− at 6=±∞x; the number DF x is called the derivate of F at . When for all positive 1 2 , we say that x Dη F x < η< / m F is almost derivable at x (cf.+∞ [7, Section 11.7]); in particular, F is almost derivable at x whenever D F x < . The term “almost derivable” is justified by the following result proved +∞ in [2, Theorem 3.3].

Theorem 2.1. Let F be a charge, and let E be the set of all x m at ∈ which F is almost derivable. Then F is derivable at almost all x E. ∈ on absolute continuity 67

3. VARIATIONS

A partition is a collection (possibly empty) P A1;x1 ;:::; A ;x = p p where A1;:::;A are nonoverlapping figures and x A for i 1;:::;p. p i ∈ i = Given η>0 and a nonnegative function δ on E m, the partition P is called ⊂ - if each figure is -regular, η regular Ai η δ-fine if x E and d A <δx for i 1;:::;p, i ∈ i i = if p . in E i 1 Ai E = ⊂ If P is δ-fine, then each x belongs to the set x E δ x > 0 . A non- S i ∈ x negative real-valued function δ defined on a set E m is called a gage on if the 0 of ⊂is thin. The following E zero set Zδ x E δx δ proposition, established= in [6],∈ connectsx = partitions with charges.

Proposition 3.1. Let F be a charge, and let 0 <η<1/2m. For any gage δ on a figure A, there is an η-regular δ-fine partition A1;x1 ;:::; A ;x in int A such that p p p F A A <ε: i i 1 [= Let F be a function on F , and let E m. For 0 <η<1/2m, let ⊂ p inf sup VηF E F Ai ; δ = P i 1 X= where is a gage on and is an -regular δ E P A1;x1 ;:::; Ap;xp η δ-fine partition, and = sup V F E VηF E : ∗ =0<η<1/2m It is easy to verify that the functions and , defined in the obvious VηF V F way, are Borel regular measures in m [2, Lemma∗ 4.6] that vanish on thin sets. The measure V F is called the critical variation of F. For charges, the critical variation ∗V F extends the classical variation VF [10, Chap. III, Sect. 4] from figures to∗ arbitrary subsets of m. We state this explicitly in the following proposition proved in [2, Lemma 4.4].

Proposition 3.2. If F is a charge, then V F A VF A for each fig- ∗ = ure A. If F is a function on F and A F , we define a function F L A on F ∈ by the formula F L A B F A B for each B F . Similarly, if µ is = ∈ a measure in m and A m, we define a measure µ L A by the formula ⊂ 68 buczolich and pfeffer

µ L A B µA B for each B m. Since the boundary of any figure is thin, it is= easy to∩ verify that ⊂ L L and L L Vη F A VηF A V F A V F A = ∗ = ∗ for each figure A.Acharge in a figure A is a charge F with F F L A. = Similarly, a measure in a set E m is a measure µ with µ µ L E. Recall from [10, Chap. III, Sect.⊂ 12] a charge F in a figure= A is called absolutely continuous (abbreviated as AC) if given ε>0, there is a 1>0 such that

F B <ε B B X∈ for each finite collection B of nonoverlapping subfigures of A with . Also recall that a charge is AC if and only if is the in- B B B <1 F F definite∈ Lebesgue integral of a function f defined on A. In particular, the Pclassical variation VF of an AC charge F is finite. Proposition 3.3. A charge F in a figure A is AC if and only if V F is AC and finite. ∗ Proof. Let F be AC and ε>0. Choose a 1>0 as in the definition above. Given a negligible set E A, find an open set U so that E U and . There is a positive⊂ function on such that ⊂ U <1 δ E Bδ x x U for each .If is a -fine partition, ⊂ then x E P A1;x1 ;:::; Ap;xp δ P is a partition∈ in U =by the choice of δ. Thus p p

A Ai Ai U <1; i 1 ≤i 1 ≤ = = and we have X X p p

F Ai F A Ai <ε: i 1 = i 1 X= X= This implies V F E εand, as ε is arbitrary, V F E 0. Since V F is a measure in A,∗ it is≤ AC. In view of Proposition 3.2,∗ = ∗ V F m V F A VF A < : ∗ = ∗ = +∞ Conversely, assume V F is AC and finite. Given ε>0, a standard argument [9, Theorem 6.11]∗ shows there is a 1>0 such that V F E <εfor ∗ each Borel set E with E <1. Select a finite collection B of nonoverlapping subfigures of with . Proposition 3.2 implies A B B B B <1 = ∈ F B VFS B P V F B V F B <ε: ∗ ∗ B B ≤ B B =B B = X∈ X∈ X∈ [ on absolute continuity 69

Corollary 3.4. A charge F in a figure A is the indefinite Lebesgue integral of a function f on A if and only if V F is AC and finite. ∗

4. THE MAIN RESULT AND ITS APPLICATION

A support of a measure µ in m is the set supp µ m G m G is open and µ G 0 : = − ⊂ x = Clearly µ U > 0 for each open set U m with U supp µ . Since [ ⊂ ∩ 6= Z m is Lindelof,¨ µ m supp µ 0. We say E m is a supporting set if E > 0 and E supp −λ L E . = ⊂ In order to prove= our main theorem, we need a technical result concern- ing Hausdorff measures. It reconciles a particular problem arising from the testy relationship between measure and category. We postpone its proof to Section 5. Theorem 4.1. Let 0 0. There is a supporting compact set C E and a G set H C ⊂ δ ⊂ satisfying the following conditions: (1) H is dense in C and H 0; = (2) given Z m with H s Z 0, we can find a supporting compact ⊂ = set C C Z so that H C is dense in C . Z ⊂ − ∩ Z Z Lemma 4.2. Let F be a charge, and let 0 <η<1/2m.IfV Fis AC, η then E x m D F x is a negligible set. η = ∈ x η = +∞ Proceeding towards a contradiction, suppose 0. As is Proof. Eη > Eη measurable, there is a supporting compact set and a negligible C Eη Gδ set H C satisfying the conditions of Theorem⊂ 4.1 with m 1 0 and an open set U m so that D U and the set D x D U δx >t is dense⊂ in D U and hence∩ 6= in ZK U. t = ∈ ∩ x ∩ ∩ The family B of all η-regular figures B with F B B/K U and d B Bi ; i 1 K U i 1 ≥ = ∩ = and there is an integerX 1 with p X 1. As p i 1 F Bi > B1;x1 ;:::; is an -regular ≥-fine partition,= the arbitrariness of implies Bp;xp η δ δ V F H 1, a contradiction. P η ≥ 70 buczolich and pfeffer

Corollary 4.3. Let F be a charge. If V F is AC, then F is derivable almost everywhere. ∗ If is AC, then so is for each positive 1 2 . Ob- Proof. V F VηF η< / m serve ∗

∞ m E x D1/nF x = n 2m 1 ∈ x =+∞ =[+ is the set of all x m at which F is not almost derivable. Since E is negligible by Lemma∈ 4.2, the corollary follows from Theorem 2.1. Theorem 4.4. Let F be a charge. If V F is AC, then up to a thin set, m is ∗ 1 2 the union of compact sets K1;K2;::: with V F Kn < for n ; ;:::: ∗ +∞ = In particular, V F is σ-finite. ∗ Proof. By [2, Proposition 4.2 and Corollary 4.8], the equality V F E DF x dλ x ∗ = E Z holds for each measurable set E m. Corollary 4.3 implies D F x ⊂ = DF x < almost everywhere. Thus m is the union of a negligible set and the+∞ sets E x B 0 DF x k : n;k = ∈ n x ≤ As 2 m for 1 2 the measure is -finite. V F En;k k n k; n ; ;:::; V F σ We∗ complete ≤ the argument by= showing that, up to a thin∗ set, each set m E with V F E < can be covered by closed sets C1;C2;::: with ⊂ ∗. To this end,+∞ choose a positive 1 2 , and find a gage V F Cn < η< / m δ∗on E so that+∞ p 1 F Ai /n Cn En E Zδ , it remains= ∈ to showx . Define= a positive function= ∪ n∞ 1 En V F Cn < δn on = by letting ∗ +∞ Cn S δx if x E ; ∈ n δn x 1 = if x C E : n ∈ n − n Choose an -regular -fine partition .If η δn B1;y1 ;:::; Bp;yp x1;:::;xp are distinct points of , and are disjoint cubes with for En K1;:::;Kn xi Ki i 1;:::;p, let ∈ = p A K B K : i = i ∪ i j j 1 [= on absolute continuity 71

Clearly, it is possible to choose so close to and so small that xi yi Ki A1;x1 ;:::; A ;x is an η-regular δ-ﬁne partition, and p p p p 1 2 F Bi < F Ai

Definition 4.5. A function f on a figure A is R-integrable whenever there is a charge F in A satisfying the following condition: given ε>0, we can find a gage δ on A so that

f xi Ai F Ai <ε i 1 − X= for each -regular -ﬁne partition in . The ε δ A1;x1 ;:::; Ap;xp A charge F, which is uniquely determined by f , is called the indeﬁnite R-integral of f .

Note that neither the R-integrability of f nor the indeﬁnite R-integral of f depends on the values f takes on a negligible set. If f and g are R-integrable and have the same indeﬁnite R-integral, then f g almost everywhere. =

Theorem 4.6. A charge F in a ﬁgure A is an indeﬁnite R-integral of a function f on A if and only if V F is AC. In which case DF f almost ∗ = everywhere in A, the function f is measurable, and V F E E f dλ for ∗ = each measurable set E A. ⊂ R Proof. Suppose V F is AC, and denote by N the negligible set of all x A at which F is not∗ derivable (Corollary 4.3). Let ∈ 0ifxN; f x ∈ = DF x if x A N; ∈ − and choose a positive ε<1/2m. There is a gage δ on N such that N p

F Ai <ε i 1 X= for each -regular -fine partition . Making ε δN A1;x1 ;:::; Ap;xp N larger and smaller, we may assume and for δN ∂A Zδ Bδ x x A ⊂ N N ⊂ 72 buczolich and pfeffer each int . Given , find a 0 so that , x N A x A N 1x > B1 x A and ∈ ∩ ∈ − x ⊂ F B f x <ε B − for each -regular figure with and . Define a gage on ε B x B d B <1x δ A by the formula ∈ δ x if x N; δ x N ∈ = 1 if x A N: x ∈ − Let be an -regular -fine partition, and note P A1;x1 ;:::; Ap;xp ε δ P is a= partition inA by the choice of δ. From the inequality p

f xi Ai F Ai F Ai f xi Ai F Ai 1 − = + − i xi N xi A N X= X∈ ∈X− <ε εA ε 1 A + i≤ + xi A N ∈X− we deduce F is the indeﬁnite R-integral of f . Since DF f almost everywhere, f is measurable by [10, Chap. IV, Theorem 4.2].= Finally [2, Theo- rem 4.7] implies for each measurable set . V F E E f E A Conversely, suppose∗ =F is the indeﬁnite R-integral of a function⊂ f on A, and select a negligible setR N A. With no loss of generality, we may assume f x 0 for each x N.⊂ Given a positive ε<1/2m, there is a gage δ onA=such that ∈ p

f xi Ai F Ai <ε i 1 − = for each -regular -fineX partition in . Choose ε δ A1;x1 ;:::; Ap;xp A a gage on so that , and for each 1 N 1 δN; N ∂A Z1 B1 x x A int . Now if ≤ ∩ ⊂ is an ⊂-regular -fine x N A Q B1;y1 ;:::; Bq;yq ε 1 partition,∈ ∩ then is a -fine= partition in and for 1 . Q δ A yj N j ;:::;q Hence ∈ = q q

F Bj f yj Bj F Bj <ε; j 1 = j 1 − X= X= and we conclude V F N ε. The arbitrariness of ε implies V F is AC. ∗ ≤ ∗ Theorem 4.6 in conjunction with Corollary 3.4 provides an elegant com- parison of the generalized Riemann and Lebesgue integrals. Corollary 4.7. Let F be a charge in a figure A. If V F is AC, then F is the indefinite R-integral of DF . • ∗ If V F is AC and finite, then F is the indefinite Lebesgue integral of DF . • ∗ on absolute continuity 73

5. PROOF OF THEOREM 4.1

Throughout this section a number s with 0 2 remain fixed. For n 0; 1;:::; the product − = m k k 1 i ; i ; n +n i 1 N N = where are integers,Y is called an - . Denote by the k1;:::;km n cube Q n family of all -cubes, and let .For 12 select an n Q n∞ 0 Q n i ; ;:::; integer = = = S 2 m pi i m s is and let N + pi − + ηi : ≥ m s = 2Nm pi i 1 − + − A set C m is called amiable if it is compact and supporting, and if there are positive⊂ integers such that for 1 2 we have ni i ; ;:::; ni 1 >ni pi i and whenever = meets . We prove+ Theorem+ + 4.1 C Q Q/ηi Q Q ni C by showing ∩ ≥ that each measurable∈ set of positive measure has an amiable subset, and that amiable sets have the desired properties. Lemma 5.1. Let K Rm be a compact set with K > 0 and let k 0 be ⊂ ≥ an integer. Suppose Q Q and ν > 1 are chosen so that K Q > Q /ν j ∈ j ∩ j j j for j 1;:::;k.Ifν>1, then for each sufficiently large integer n 1 there = ≥ is a nonempty compact set K K such that K Q > Q /ν for j n ⊂ n ∩ j j j = 1;:::;k, and K Q > Q /ν for each Q Q n which meets K . n ∩ ∈ n If is a density point of , there is an integer 1 such Proof. x K K nx that ∈ for each containing . Letting≥ K Q > Q /ν Q n n Q n x ∩ ∈ ≥ x E x K n n ; n = ∈S x x ≤ the Lebesgue density theorem implies lim , and hence lim En K En for 1 . Thus for each= sufficiently large integer ∩ Qj K Qj j ;:::;k =1 we∩ have =0 and for 1 . Fix such n En > En Qj > Qj /νj j :::;k an≥n, and let ∩ = K Q Q n E Q and K K K: = ∈ x n∩ 6= Z n = ∩ As the family is finite, is compact, and since , the set K Kn En Kn[ Kn is not empty and for ⊂1 .If Kn Qj En Qj > Qj /νj j ;:::;k Q meets , then ∩ ≥. Hence∩ =and ∈. Qn Kn Q K Kn Q K Q En Q In particular, contains∈ an with ∩ =, which∩ implies ∩ 6= Z Q x K nx n Kn Q K Q > Q/ν. ∈ ≤ ∩ = ∩ Corollary 5.2. Each measurable set E m with E > 0 has an ami- ⊂ able subset. Proof. Select a compact set K E with K > 0. Using Lemma 5.1, construct inductively positive integers⊂n and nonempty compact sets K K i i ⊂ 74 buczolich and pfeffer so that for i 1; 2;:::; we have = ni 1 >ni pi i; Ki 1 Ki; + + + + ⊂ and whenever and meets . Having Ki Q > Q /ηk k i Q Q nk Ki done this,∩ let , and observe≤ ∈ K0 i∞1 Ki = = T lim Q K0 Q Ki Q ∩ =i ∩ ≥ η →∞ k for each Q Q n which meets K0. Now K0 > 0 since K0 , and it ∈ k 6= Z suffices to let C supp λ L K0 . To justify the inductive= construction, first apply Lemma 5.1 to k 0 and = ν η1, and obtain an integer n1 1 and a nonempty compact set K1 K = ≥ ⊂ such that K1 Q > Q /η1 for each Q Q n1 which meets K1. Next assume positive ∩ integers and nonempty∈ compact sets n2;:::;ni K2;:::;Ki have been constructed so that for j 2;:::;i, we have = 1 nj >nj 1 pj 1 j ;KjKj1; − + − + − ⊂− and whenever meets and 1 . Since Kj Q > Q /ηk Q Q nk Kj k j only finitely∩ many i meet∈ , Lemma 5.1 implies≤ there≤ is an Q k 1 Q nk Ki integer ∈ and= a nonempty compact set such that ni 1 >ni pi i Ki 1 Ki + + whenever+S meets and+ 1 ⊂ 1. Ki 1 Q > Q /ηk Q Q nk Ki 1 k i + ∩ ∈ + ≤ ≤ + Let C m be an amiable set, and let n be the associated sequence ⊂ i of positive integers. For each integer i 1 and each Q Q n p select ≥ ∈ i + i a Q Q n p i with Q Q, and let ∗ ∈ i + i + ∗ ⊂ int and ∞ ∞ Hi Q∗ HC Hi: = = 1 Q Q ni pi j i j ∈ [ + \= [= Each is a dense open subset of m, and so is a dense subset i∞j Hi HC Gδ m = m n p of by the Baire category theorem. Every Q Q 0 contains N i+ i connectedS components of , and each connected∈ component of has Hi Hi measure m ni pi i . Thus mi 2 i by our choice of , N− + + Q Hi N− < − N and ∩ =

lim ∞ 0 Q HC Hi : ∩ =j = →∞ i j X= We conclude is a negligible set. HC Q Q 0 Q HC = ∈ ∩ Observation 5.3. If Q Q n meets C, then C Q Q /2for each S ∈ i ∩ ∗≥ ∗ connected component Q of H Q. ∗ i ∩ Proof. As C is an amiable set, 1 Q Q Q C Q Q C 1 Q ∗; − = − ∩ ≤ −η =2Nmpi i = 2 i + on absolute continuity 75 and hence

Q∗ Q∗ C Q∗ Q∗ C Q∗ Q C : ∩ = − − ≥ − − ≥ 2

Corollary 5.4. The set H C H is dense in C. = ∩ C Let 1 be an integer, and select a with and Proof. j Q Q ni i j . Then≥ by Observation∈ 5.3. We infer≥ each Q C Q C Hi ∩ 6= Z that meets∩ ∩meets6=Z also . Since each open subset Q i∞j Q ni C C i∞j Hi of ∈ m =is the union of a subfamily of ∩ = , the set is a i∞j Q ni C i∞j Hi relativelyS open dense subset of C. The= corollaryS follows from∩ the= Baire category theorem. S S If is a family of sets and is a set, we let .For A B AB A A A B the purposes of this section only, we introduce the= following∈ x terminology.⊂ Let Z m and ε>0. An ε-cover of Z is a nonoverlapping family ⊂ Z Q satisfying the following conditions: ⊂ (1) Z int Z; ⊂ (2) s ; B Z d B <ε ∈ S (3) s s for each . B Z d B 0. Proof. The existence of a nonoverlapping family C 0 that satisﬁes conditions 1 and 2 of an ε-cover of Z follows from [5, Theorem 5.1]. Proceed- ing inductively, for n 0; 1;:::; construct families C n and D n so that consists of all = for which s s, D n D Q n C n d D B C n d B and ∈ − ≤ ∈ D P 1 C n C n C n D: + = −D D n ∈[ Let and , and observe that is C n∞ 1C n D n∞ 1D n Z C D = = = = = ∪ a nonoverlapping family with C 0 Z; in particular, Z int Z. Furthermore,T S ⊂ ⊂ S S S ∞ d B s d B s d D s B Z = B C + n 1 D D n X∈ X∈ X= ∈X ∞ d B s d B s ≤ + 1 B C n D D n B C n D X∈ X= ∈X ∈X d B s d B s d B s <ε: = B C + B C 0 C = B C 0 X∈ ∈ X − ∈X 76 buczolich and pfeffer

Select a Q Q n Z, and observe that Q C n . Indeed, if k n is ∈ − 6∈ ≥ an integer, then C k and C n contain the same n-cubes; thus Q C n ∈ implies Q C, a contradiction. Since Q D n , we have ∈ 6∈ ∞ d Q s > d B s d B s d B s = + B C n Q B CQ k n D D k Q B C k D ∈X X∈ X= ∈X ∈X ∞ d B s d D s ≥ + B CQ k n D D k Q X∈ X= ∈X d B s d D s d B s: = + = B CQ D DQ B ZQ X∈ X∈ X∈

Observation 5.6. Let Q Q and let B Q be a nonoverlapping family 2∈ s ⊂ s such that Q B Q/ . Then d Q B B d B , and the equality ∩ ≥ ≤ ∈ occurs if and only if B Q . S = P Proof. As the other cases are clear, assume B consists of proper subsets m s of Q, and observe that d Q /d B N for each B B.AsN − >2, we obtain ≥ ∈ Q 2 B d Q s 2 m s m s B m s = d Q − ≤ d Q − = B B d Q − ∈ m s [ X d B − 2 2 d B s d B s < d B s: ≤ d Q ≤ Nm s B B − B B B B X∈ X∈ X∈

Lemma 5.7. Let Q Q n and let B Q be a nonoverlapping family ∈ i ⊂ of proper subcubes of Q such that Q∗ B Q∗ /2for each connected s ∩ ≥ s component Q∗ of Hi Q. Then d Q < B B d B . ∩ S ∈ Enumerate as where mp i , and let Proof. Q ni pi Q Q1P;:::;Qr r N for 1 + . We may assume that for an= integer with Qj∗ Qj Hi j ;:::;r k 0 = ∩, each = is contained in a , and no k r Q1;:::;Qk B B Qk 1;:::;Qr ≤ ≤ ∈ + is contained in any B B. Denote by C the family of all B B which contain one of the ∈ , and observe that each ∈is properly Q1;:::;Qk B B C contained in one of the .If 1 ,∈ then− Qk 1;:::;Qr k j r + + ≤ ≤

Q∗ 2Q∗ B Q∗ B ; j ≤ j ∩ = j ∩ Qj and Observation 5.6 implies [s [s. Now is the interior d Qj∗ B B d B Qj∗ ≤ ∈ Qj p i of an n p i -cube, and hence d Q∗ N i d Q for j 1;:::;r. i + i + Pj = − + = on absolute continuity 77

Since m s p si m s by our choice of p , we obtain − i − ≥ − i r r s s s s s pi i d B d B d Qj∗ r kd Q N− + = 1 ≥ 1 = − B B C j k B BQ j k ∈X− =X+ ∈X j =X+ s mp m s p si r k s m s r kd Q N− i N − i − − d Q N − : = − ≥ r On the other hand, k for 1 , and j 1 Qj C; Qj Q/r j ;:::;k = ⊂ = = d B d Q /N for each B B. Thus ≤ S ∈ S k k s m s Q Qj B d B d B − r =j 1 ≤B C =B C X= X∈ X∈ d Q m s Q − d B s d B s; m s s m s ≤ N − B C =d Q N − B C X∈ X∈ which yields s s m s. We conclude B C d B k/r d Q N − ∈ ≥ s s s s m s s d PB d B d B d Q N − >d Q : B B = B C + B B C ≥ X∈ X∈ ∈X−

Proof of Theorem 4.1. Without loss of generality, suppose m 1 Since each open subset of m is the union of a subfamily of , i∞1 Q ni the set is supporting whenever 0 for each = CZ Q CZ > Q i∞1 Q ni that meets C . Proceeding towards a contradiction,∩ suppose there∈S is= a Q Z ∈ Q n with C Q and C Q 0; in particular Q C SQ C i Z ∩ 6= Z Z ∩ = ∩ = ∩ ∩ G . Observe Q Z, because ∂Q Z G implies C int Q C Q 6∈ ⊂ ⊂ Z ∩ = Z ∩ 6= . As the set C is amiable, Q C Q/η > Q /2, and hence Z ∩ ≥ i Q Q Z Q Z Q G Q C > : ∩ Q = ∩ ≥ ∩ ≥ ∩ 2 A contradiction [ s [ s follows from Observation 5.6. d Q < B Z d B ∈ Q P 78 buczolich and pfeffer

It remains to show that H C is a dense subset of C . To this end, let ∩ Z Z j 1 be an integer, and select a Q Q n with i j and Q C .As ≥ ∈ i ≥ ∩ Z 6= Z observed previously, Q Z. Proceeding towards a contradiction, assume . Thus6∈ if is a connected component of , then Q CZ Hi Q∗ Q Hi Q ∩C ∩Q =ZG, and Observation 5.3 yields ∩ ∗ ∩ ⊂ ∗ ∩ Q Q∗ Z Q∗ Z Q∗ G Q∗ C : ∩ Q = ∩ ≥ ∩ ≥ ∩ ≥ 2

A contradiction[ s [ s follows from Lemma 5.7. We con- d Q < B Z d B clude each that∈ Q meets meets also . Since Q i∞j Q ni CZ CZ i∞j Hi every open subset∈ of= m isP the union of a subfamily of ∩ = , the set i∞j Q ni is aS relatively open dense subset of . The= propositionS fol- CZ i∞j Hi CZ lows∩ from= the Baire category theorem. S S

ACKNOWLEDGMENT

We are obliged to the referee for some important corrections.

REFERENCES

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