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THE DYNKIN SYSTEM GENERATED by BALLS in Rd CONTAINS ALL BOREL SETS Let X Be a Nonempty Set and S ⊂ 2 X. Following [B, P. 8] We

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THE DYNKIN SYSTEM GENERATED by BALLS in Rd CONTAINS ALL BOREL SETS Let X Be a Nonempty Set and S ⊂ 2 X. Following [B, P. 8] We 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 generated2 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 433 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 434 MIROSLAV ZELENY´ 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE DYNKIN SYSTEM CONTAINS ALL BOREL SETS 435 (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: 0 p;j 1 ∩ 0 p;j1 ∅ 6 j=1 j=1 [ l\=1 p[=l [ l\=1 p[=l @ A @ A We may assume that u<v. Suppose on the contrary that ∞ ∞ ∞ ∞ ∞ ∞ x Du Dv : ∈ 0 p;j1 ∩ 0 p;j 1 j=1 j=1 [ l\=1 p[=l [ l\=1 p[=l @ A @ A 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 436 MIROSLAV ZELENY´ 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 ">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) eachA ball from has its center in L, (c) L = H, A (d) each∩ ballA from has its diameter less than ", (e) for everyS x HA there exists r>0 such that B(x; r) intersects only finitely many elements∈ from .
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