PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 2, Pages 433–437 S 0002-9939(99)05507-0 Article electronically published on September 23, 1999

THE DYNKIN SYSTEM GENERATED BY BALLS IN Rd CONTAINS ALL BOREL SETS

MIROSLAV ZELENY´

(Communicated by Frederick W. Gehring)

Abstract. We show that for every d N each Borel subset of the space Rd with the Euclidean metric can be generated∈ from closed balls by complements and countable disjoint unions.

Let X be a nonempty set and 2X. Following [B, p. 8] we say that is a Dynkin system if S⊂ S (D1) X , (D2) A ∈S X A , ∈S⇒ \ ∈S (D3) if A are pairwise disjoint, then ∞ A . n ∈S n=1 n ∈S Some authors use the name -class instead of Dynkin system. The smallest Dynkin σ S system containing a system 2Xis denoted by ( ). Let P be a metric space. The system of all closed ballsT⊂ in P (of all Borel subsetsD T of P , respectively) will be denoted by Balls(P ) (Borel(P ), respectively). We will deal with the problem of whether (?) (Balls(P )) = Borel(P ). D One motivation for such a problem comes from measure theory. Let µ and ν be finite Radon measures on a metric space P having the same values on each ball. Is it true that µ = ν?If (Balls(P )) = Borel(P ), then obviously µ = ν.IfPis a Banach space, then µ =Dν again (Preiss, Tiˇser [PT]). But Preiss and Keleti ([PK]) showed recently that (?) is false in infinite-dimensional Hilbert spaces. We prove the following result.

Theorem 1. Let d N, and let Rd be equipped with the Euclidean metric. Then ∈ (Balls(Rd)) = Borel(Rd). D This theorem was partially proved by Olejˇcek ([O]), who proved it for d =2, 3 (the case d = 1 is easy and well-known). Several of Olejˇcek’s ideas will also be used in our proof of the general statement. In fact, we prove a slightly more general result: Each Borel subset of the space Rd (with the Euclidean metric) can be generated from closed balls by countable monotone unions, countable monotone intersections and countable disjoint unions.

Received by the editors February 11, 1998. 1991 Mathematics Subject Classification. Primary 28A05, 04A15. This research was supported by Research Grant GAUK 190/1996 and GACRˇ 201/97/1161.

c 1999 American Mathematical Society

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Each open ball is a countable increasing union of closed balls; we could use the open balls as well. The paper [PK] contains a remark on the result, proved independently by Jackson and Mauldin, that (?) holds even for every Banach space of finite dimension. We start with definitions of auxiliary notions. Let X be a nonempty set. We say that 2X is a monotone system if the following hold: S⊂ (M1) if A A A ..., A ,then ∞ A , 1 ⊂ 2 ⊂ 3 ⊂ n ∈S n=1 n ∈S (M2) if A1 A2 A3 ..., An ,then ∞ An . ⊃ ⊃ ⊃ ∈S Sn=1 ∈S We say that 2X is a ?-system if satisfies (M1), (M2) and (D3). The smallest S⊂ D S T ?-system containing 2Xis denoted by ?( ). We say that A Rd is a sphere D T⊂ D T ⊂ of dimension t if there exists a closed ball B Rd with a center x Rd and an ⊂ ∈ affine subspace V Rd of dimension t + 1 containing x such that A = V ∂B.We define ⊂ ∩

d t = A R ; A is a Borel set, which can be covered by S { ⊂ countably many spheres of dimension t . } d Let A, B R .WesaythatAis a subset of B modulo t (the notation A ⊂ S ⊂ B mod t)ifA B t.WesaythatA=Bmod t if A B mod t and S \ ∈S S d ⊂ S B A mod t. The Euclidean open ball with center x R and radius r>0is denoted⊂ by BS(x, r). ∈ Lemma 2. Let X be a nonempty set, and let 2X beasystemclosedwith respect to finite unions and finite intersections. ThenV⊂ the smallest monotone system containing is closed with respect to countable unions and countable intersections. V Proof. We define ( ) = the smallest monotone system containing , M V V = A ( ); B : A B ( )andA B ( ) and M1 { ∈MV ∀ ∈V ∪ ∈MV ∩ ∈MV } = A ( ); B ( ): A B ( )andA B ( ) . M2 { ∈MV ∀ ∈MV ∪ ∈MV ∩ ∈MV } It is easy to see that and are monotone systems. We also have M1 M2 V⊂M1⊂ ( ). Hence 1 = ( )and 2 ( ). This gives ( )= 2 and MthereforeV ( )M is closedM withV respectV⊂M to finite⊂M unionsV and finite intersections.M V M Thus ( ) is closedM V with respect to countable unions and countable intersections. M V Lemma 3. Each Dynkin system is a ?-system. D Lemma 4. Each Borel subset of a metric space P can be generated from open subsets of P by countable monotone unions and countable monotone intersections. The proof of Lemma 3 is obvious and will be omitted. Lemma 4 is well-known (see [K, p. 344] and Lemma 2).

Lemma 5. Let m N, let L Rd be a closed set, let t N,t d, and let G be a ∈ ⊂ ∈ ≤ subset of L. Suppose that there exists a sequence of systems of closed balls n n∞=1 such that {B } (i) is a union of m systems such that each of them is disjoint, Bn (ii) the center of each ball from ∞ is in L, n=1 Bn (iii) L n G and G = L n mod t 1 for every n N, ∩ B ⊂ ∩ S B S − ∈ S S

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(iv) n, n0 N,n0>n C n C0 n :C0 C or C0 C = , ∀ ∈ ∀ ∈B ∀ ∈B 0 ⊂ ∩ ∅ (v) limn + sup diam C; C n =0. → ∞ { ∈B } ˜ ? d ˜ ˜ Then there exists G (Balls(R )) with G G and G = G mod t 1. ∈D ⊂ S − Proof. Let = m k,whereeach k is a disjoint system of closed balls. Put Bn k=1 Bn Bn k 1 S − ∞ k = C k; C i ,Dk= k , Dn,j { ∈Bn 6⊂ Bs} n,j Dn,j i=1 s=j [ [ [ [ for every k 1,...,m ,j,n N,and ∈{ } ∈ m ∞ ∞ ∞ ∞ G˜= Dk ,Z=G . p,j \ Bn j=1 n=1 k[=1 [ l\=1 p[=l \[

Using (iii) we obtain Z t 1.Letx G Z. Then there exist a sequence (Cn)n∞=1 ∈S− ∈ \ of closed balls, an increasing sequence (pn)n∞=1 of natural numbers, k N and j N such that ∈ ∈ k for every , x Cn pn n N • ∈ k∈B1 ∈ x/ − ∞ i . • ∈ i=1 s=j Bs k ˜ It is easyS to verifyS thatS x ∞ ∞ Dp,j.ThuswehaveG Gmod t 1.On ∈ l=1 p=l ⊂ S − the other hand the conditions (ii) and (v) imply that ˜ ,since is closed. T S G L L Using this and the condition (iii) we obtain ⊂

∞ ∞ G˜ = L G˜ L = L G. ∩ ⊂ ∩ Bn ∩ Bn ⊂ n=1 ! n=1 [ [ [  [  Now we want to prove that G˜ ?(Balls(Rd)). At first we prove that ∈D

∞ ∞ ∞ ∞ ∞ ∞ Du Dv = whenever u = v.  p,j  ∩  p,j ∅ 6 j=1 j=1 [ l\=1 p[=l [ l\=1 p[=l     We may assume that u

∞ ∞ ∞ ∞ ∞ ∞ x Du Dv . ∈  p,j ∩  p,j  j=1 j=1 [ l\=1 p[=l [ l\=1 p[=l     u This means that there exist j N and j0 N such that x l∞=1 p∞=l Dp,j and v ∈ ∈ ∈u x ∞ ∞ D . There exist p>j0 and a ball C with x C. There l0=1 p0=l0 p0,j0 p,j ∈ v ∈D T S ∈ exist p0 >pand a ball C0 with x C0. But this and (iv) imply that T S ∈Dp0,j0 ∈ C C u. This is a contradiction with C v . 0 p 0 p0,j0 Now⊂ ⊂ it is sufficientB to show that ∈D S ∞ ∞ ∞ k ? d Dp,j (Balls(R )) for every k =1,...,m. ∈D j=1 [ l\=1 p[=l k We fix k 1,...,m and observe that the system = ; ∞ is ∈{ } V { X X⊂ n=1 Bn} contained in the system of all disjoint unions of closed balls in Rd and is closed with respect to finite intersections and finite unions. Moreover,S Dk S for every p,j ∈V

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k p, j N. Using Lemma 2 we obtain that the set j∞=1 l∞=1 p∞=l Dp,j is contained in the∈ smallest monotone system containing and hence also V S T S ∞ ∞ ∞ k ? d D (Balls(R )). p,j ∈D j=1 [ l\=1 p[=l

Lemma 6. Let L Rd be an affine subspace of dimension t or a sphere of dimen- ⊂ sion t.LetG Lbe relatively open in L. Then there exist m N andasequence ⊂ ∈ of systems of closed balls ∞ satisfying the conditions (i) – (v) in Lemma 5. {Bn}n=1 Proof. At first we prove the following Claim.

Claim. Let L Rd be as in Lemma 6, H L be relatively open in L and ε>0. ⊂ ⊂ Then there exist m N (depending only on the dimension of Rd) and a system of closed balls such that∈ A (a) is a union of m systems such that each of them is disjoint, (b)A each ball from has its center in L, (c) L = H, A (d) each∩ ballA from has its diameter less than ε, (e) for everyS x HA there exists r>0 such that B(x, r) intersects only finitely many elements∈ from . A Proof of the Claim. Let be a system of all closed balls B Rd such that the U 1 ⊂1 center of B is in H, B L H and min 2 dist(center(B),L H), 4ε diam B<ε. Since H is relatively open∩ ⊂ in L,wehave{ L =H. \ }≤ ∩ U We use Besicovitch’s theorem ([Z, p. 9]) to obtain m N (depending only on d S ∈ m the dimension of R ) and disjoint systems 1,..., m such that k=1 k and m A A m A ⊂U L ( k=1 k)=H. It is easy to check that = k=1 k satisfies (a) – (e) and the∩ proof of theA Claim is over. A A S S S S

We will construct the desired systems n,n N.Putε=1andH=G.Using the Claim we obtain a system of closed ballsB ,∈ which satisfies (a) – (e) and hence B1 also the conditions (il0)–(vil0)forl=1below.

(il0) For every n =1,...,l, n is a union of m systems such that each of them is disjoint, B l (ii0) the center of each ball from is in L, l n=1 Bn (iii0) L n G and G = L n mod t 1 for every n =1,...,l, l ∩ B ⊂ ∩ SB S − (iv0) n, n0 1,...,l ,n0>n C n C0 n : C0 Cor C0 C = , l ∀ S ∈{ } ∀S ∈B ∀ ∈B 0 ⊂ ∩ ∅ (v0)supdiam C; C 1/n for every n =1,...,l, l { ∈Bn}≤ (vi0) for every n =1,...,l and for every x there exists r>0 such that l ∈ Bn B(x, r) intersects only finitely many elements from n. S B Suppose that we have defined systems n,n=1,...,l, satisfying (il0)–(vil0). Put B 1 H = ∂B; B L and ε = . Bl \ { ∈Bl} ∩ l +1 [ [  The condition (vil0)givesthatHis a relatively open subset of L. Applying the Claim to H and ε we obtain a system of closed balls . Bl+1

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Let B be a closed ball with the center in L.Then∂B L is a sphere of dimension t 1 or a singleton or an empty set since we use Euclidean∩ balls. This implies that − ∂B; B l L t 1.Thuswehave { ∈B}∩ ∈S− S L l=L l+1 mod t 1. ∩ B ∩ B S − Since G = L l mod t 1[we conclude[G = L l+1 mod t 1. The remaining ∩ B S − ∩ B S − part of condition (iiil0+1) and conditions (il0+1)–(iil0+1), (ivl0+1)–(vil0+1) are easy to check. S S

Proposition 7. Let d N, and let Rd be equipped with the Euclidean metric. Then ∈ ?(Balls(Rd)) = Borel(Rd). D Proof. We will prove by induction over t the following Claim, which implies Propo- sition 7.

Claim. Let L Rd be an affine subspace of Rd of dimension t or a sphere of ⊂ dimension t and let B L be a Borel set. Then B ?(Balls(Rd)). ⊂ ∈D Let t = 0. Then the Claim obviously holds since B is empty or a singleton or a two point set. Suppose that we have proved the statement for t = p 1. We will deal with the case t = p.IfG Lis a relatively open set in L, then− Lemmas ⊂ ? d 5 and 6 show that G = G1 G2,whereG1 G2 = ,G1 (Balls(R )) and ∪ ∩ ∅ ∈D ? d G2 p 1. The induction hypothesis easily implies that G2 (Balls(R )). ∈S− ∈D Thus each relatively open subset of L is contained in ?(Balls(Rd)). This fact and D Lemma 4 show that each Borel subset of L is contained in ?(Balls(Rd)). Thus the Claim is proved. D Proof of Theorem 1. Theorem 1 follows immediately from Proposition 7 and Lemma 3. I thank O. Kalenda, D. Preiss, L. Zaj´ıˇcek and an anonymous referee for helpful discussions and comments. References

[B] H. Bauer, Probability theory and elements of measure theory, Academic Press, 1981. MR 82k:60001 [K] K. Kuratowski, Topology I, Academic Press, 1966. MR 36:840 [O] V. Olejˇcek, The σ-class generated by balls contains all Borel sets, Proc. Amer. Math. Soc. 123 (12) (1995), 3665–3675. MR 96c:28001 [PK] D. Preiss, T. Keleti, The balls do not generate all Borel sets using complements and count- able disjoint unions (to appear). [PT] D. Preiss, J. Tiˇser, Measures in Banach spaces are determined by their values on balls, Mathematika 38 (1991), 391–397. MR 93a:46080 [Z] W. Ziemer, Weakly differentiable functions, Springer-Verlag, 1989. MR 91e:46046

Faculty of Mathematics and Physics, Charles University, Sokolovska´ 83, Prague 186 00, Czech Republic E-mail address: [email protected]

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