Singular measures and the key of G

Publicacions Matem`atiques, Vol. 44 (2000), 483–489 SINGULAR MEASURES AND THE KEY OF G Stephen M. Buckley and Paul MacManus Abstract We construct a sequence of doubling measures, whose doubling constants tend to 1, all for which kill a Gδset of full Lebesgue measure. 0. Introduction A non-zero Borel measure νis said to be doubling if there is a constant C≥1 such that C−1≤ν(I) ν(J)≤C,(1) whenever I,Jare adjacent intervals of the same length. We call the smallest C=Cνfor which this condition holds, the doubling constant of ν. A measure is a multiple of Lebesgue measure if and only if its doubling constant is 1. It was shown in [BHM] that if U⊂[0,1]nis open and |∂U|= 0, then νn(U)→|U|whenever νnis a sequence of probability measures on [0,1]nwhose doubling constants tend to 1. In particular, if Uis an open subset of [0,1] of full measure, then νn(U)→1. We will show, amongst other things, that there exists a Gδset Gin [0,1] of full measure, and a sequence νnof measures whose doubling constants tend to 1, yet νn(G)= 0 for all n. We can even choose the measures to be “renormalizations” of a single measure νwhich “fit the gaps in G”asakeyfitsalock. We wish to thank the referee for drawing our attention to the paper of Kakutani. 1. Definitions and basic results There is an easy way, essentially due to Kahane [Kh], to generate doubling measures. Let Qconsist of all intervals on [0,1) of the 2000 Mathematics Subject Classification. 60G30. The first author was partially supported by Enterprise Ireland. 484 S. M. Buckley, P. MacManus form [m4−k,(m+ 1)4−k), where m, k are non-negative integers, and set Q(j) to be the subset of Qconsisting of those intervals of length 4−j. For any I∈Qthe four children are labeled I0,I1,I2,I3, moving from left to right. Now consider HI(x)=     1,x∈I1 −1,x∈I2 0,otherwise. The product I∈Q(1+aIHI) converges weak-∗to a doubling, probability measure µ, provided that supI∈Q |aI|<1. We call any such measure µ aKahane measure and write µK= supI∈Q |aI|. Furthermore, the doubling constant Cµtends to 1 as µKtends to 0; in fact, if µK≤ 1−, then there is a constant c, dependent only on , such that Cµ≤ 1+cµKwhenever for some >0. For our purposes it will be sufficient to consider Kahane measures for which all of the coefficients aIat any given scale are equal and µK≤ 1−for some >0, which we assume to be fixed from now on. We denote this class of measures by M, or simply M. Then every measure in Mis of the form ∞ j=1(1 + ajRj) where Rj=I∈Q(j)HI;itis convenient to introduce the notation cj(µ)≡aj. We will focus on those measures µ∈Mfor which cj(µ)→0asj→∞, and we label these M0. For µ∈Mand n=0,1,2,... the measure µn∈Mhenceforth denotes the element of Mwith cj(µn)=cj+n(µ), j∈N. The measures µnare “renormalized” versions of µ; in fact, if S⊂[0,1) is a measurable set and fSis the periodic function with period 1 whose restriction to [0,1) is the characteristic function of S, then µn(S)=1 0fS(4nt)dµ(t). Given µ∈M 0, it follows from the estimate in the last paragraph that the sequence of doubling constants (Cµn) has limit 1. Thus every µ∈M 0 is optimally doubling at small scales in the sense that ν=µsatisfies (1) with C=Cµnwhenever I,Jare adjacent intervals with |I|=|J|≤4−n. The following result is a special case of a result of Kakutani [Kk, Corollary 1]. Theorem A. Let µ, ν ∈M, with aj=cj(µ),bj=cj(ν), for all j∈N. If (aj−bj)∞ j=1 lies in l2, the class of square summable sequences, then µνµ, otherwise µ⊥ν. Singular Measures and The Key of G485 In fact, when νis Lebesgue measure and (an)∈l2above, more is true: µlies in the Muckenhoupt class A∞, and in particular µhas density lying in Lp([0,1]) for some p>1; see [Bu] and [FKP].1 Kakutani proves this result by careful analysis, but let us pause to prove the singularity part of this result using the Lyapunov version of the Central Limit Theorem [Bi, Theorem 27.3] which we now state. Theorem B. Suppose that {Xn}∞ n=1 is a sequence of independent random variables, and that the moments E(Xn)=en,E(Xn−en)2=σ2 n= 0, and E|Xn−en|3=τ3 nare finite for each n.Let sn=n  i=1 σ2 i1/2 ,t n=n  i=1 τ3 i1/3 . If limn→∞ tn/sn=0, then Yn≡n i=1(Xi−ei)/snconverges in distribution to the standard normal distribution. In this paragraph we employ the notation of Theorem A. The functions Rnare independent as random variables on [0,1] with respect to ν, and so the functions fn= log[(1 + anRn)/(1 + bnRn)] are also independent. A little calculation with the power series expansion for log(1 + t) gives Eν(fn)≡en=−(an−bn)2 4(1 −b2 n)+O(|an−bn|3), Eν(fn−en)2≡σ2 n=(an−bn)2 2(1 −b2 n)+O(|an−bn|3), Eν|fn−en|3≡τ3 n=|an−bn|3 2 1+b2 n (1 −b2 n)2+O(|an−bn|4). Thus if sn,t nare as in Theorem B, limn→∞ |an−bn|= 0, and (an− bn)∞ n=1 /∈l2, then t3 n/n i=1 |ai−bi|3and s2 n/n i=1(ai−bi)2are bounded above and below by positive, finite constants that are independent of n. It is then routine to deduce that limn→∞ tn/sn= 0; one simply splits the sum at a point beyond which |an−bn|is very small and uses the estimate · l3≤ · 2/3 l2· 1/3 l∞. Thus Theorem B is applicable in the case Xn=fn. Since n i=1 eiis much larger than sn for large n, it follows that Yntends to −∞ in ν-measure and thus ∞ n=1(1+anRn)/(1+bnRn) converges in ν-measure to the zero function. Set {PN}to be the partial products of this infinite product. We have 1These references only say that µlies in dyadic A∞but, since µis a doubling measure, this implies that µ∈A∞. 486 S. M. Buckley, P. MacManus just seen that this sequence of functions converges to zero in ν-measure. However, PN(x)=µ(IN(x))/ν(IN(x)), where IN(x) is the unique element of Q(N) containing xand so, by the Radon-Nikodym theorem, {PN}converges ν-a.e. to the Radon-Nikodym derivative of µwith respect to ν. Consequently, the Radon-Nikodym derivative is zero ν-a.e., and so µ⊥νwhenever (aj−bj)/∈l2. We are mainly interested in Theorem A when νis Lebesgue measure. In this case if the sequence (cj(µ)) has limit zero but does not lie in l2, then µis a singular measure which is optimal doubling at small scales. The mere existence of such a measure may seem a little surprising and was only recently established (using different techniques) by Cant´on [C] and Smith [S]. There is an obvious bijection, A, between Qand the set of finite sequences whose terms lie in {0,1,2,3}. We will refer to A(I) as the address of I. The jth term in the address is Aj(I). For I∈Q, we let E(I) consist of the union of the intervals J∈Qfor which A2j(J)=Aj(I) for all j. So the odd terms in A(J) are arbitrary and the even terms are specified. If I∈Q(j), E(I) consists of 4jelements of Q(2j). For n=0,1,2,... and I∈Q,Tn(I) consists of those intervals J∈Qfor which An+j(J)=Aj(I) for all j. So the first nterms of Jare arbitrary and the remainder are specified. When I∈Q(j), Tn(I) consists of 4n elements of Q(j+n). Note that if Iand Jare disjoint, then E(I) and E(J) are disjoint, as are Tn(I) and Tn(J). For any set Bthat is a union of disjoint elements Iof Q, we define E(B) to be the union of the E(I), and we define Tn(B) similarly. It is easy to check that |E(B)|=|B|and that |Tn(B)|=|B|. Let Σjbe the collection of subsets of [0,1) that are unions of elements of Q(j). Any set B∈Σmis said to be j-indifferent if whenever B⊃I∈ Q(m) and Jis one of the three elements of Q(m) for which A(J) and A(I) differ only in the jth place, then J⊂B. Equivalently if S(B)isthe set of sequences of length mgiven by A(I) for each I∈Q(m), I⊂B, then Bis j-indifferent precisely if S(B) is measurable with respect to the σ-algebra generated by the sets Sk,l ={(ai)m i=1 :ak=l},1≤k≤m, k =j, l ∈{0,1,2,3}. The point of this definition is that if Bis j-indifferent, then µ(B)doesnot depend on the cj(µ). In particular, if B∈Σm, then E(B)isj-indifferent for all odd numbers jand all even j>2m, and Tn(B)isj-indifferent for all j≤nand all j>n+m. Singular Measures and The Key of G487 2. Construction of µand G Our main result is as follows. Theorem 1. There exists a measure µ∈M 0on the interval [0,1) and aGδset Gcontained in [0,1) which have the following properties: (a) µ([0,1)) = 1,|G|=1and µ(G)=0. (b) µn(G)=1for all odd n∈Nand µn(G)=0for all even n∈N. Taking νn=µ2n, we immediately get Corollary 2. There exists a Gδset Gin [0,1] of full measure and a sequence νnof probability measures on [0,1] whose doubling constants tend to 1and for which νn(G)=0for all n. The oscillatory behaviour of µn(G) described in Theorem 1(b) is all the more remarkable since the measures µnare renormalized versions of a single measure µwhose doubling constants are tending to one. The idea is to construct Gfrom sets that are indifferent at odd levels n(and thus treat such µnlike Lebesgue measure), but which are concentrated in areas where µnis small whenever nis even. Proof of Theorem 1: Let bbe any number strictly between 0 and 1. Define νkto be the element of Mwhose coefficients are all 2−k. This measure is singular with respect to Lebesgue measure. It follows that for sufficiently large nk, there exists Ak∈Σnkfor which |Ak|≥1−bk and νk(Ak)≤bk. We can assume that the nkare increasing to ∞. Divide the natural numbers into consecutive blocks B1,B 2,... of length 2n1,2n2,... . Set aj=2 −kwhenever jis an even number in block Bk, and 0 otherwise. Define µ∈M 0by the equations cj(µ)=aj. Now let mk=2n1+···+2nk−1for k>1 and m1= 0. Thus mkis the total length of the blocks B1,...,B k−1. Define Hkto be Tmk(E(Ak)). Then Hk∈Σmk+2nkand is j-indifferent for all jexcept even numbers larger than mkand no larger than mk+2nk, i.e., all even numbers in Bk. Remove the endpoints of the intervals that make up Hkto get an open set Uk. The sets Ukand Hkdiffer only by a countable number of points. Thus any doubling measure gives them the same measure (doubling measures on the line are non-atomic). Set Gm=∞ k=mUk and G=∞ m=1 Gm. This set Gis a Gδset. We have |Hk|=|Ak|≥1−bkfor all k, hence |Gm|= 1 for all m, and |G|= 1. If nis odd and jis even, then cj(µn) = 0. But Hkis j-indifferent for all odd j, so it follows that µn(Hk)=|Hk|. As a result, µn(G) = 1 whenever nis odd. 488 S. M. Buckley, P. MacManus The set Hkis j-indifferent for all jexcept even jin Bkand cj(µ)= 2−kfor these exceptional integers. Thus µ(Hk)=νk(Ak)≤bk. Consequently, µ(Gm)≤bm(1 −b)−1for all m, and so µ(G)=0. Suppose n−mis even. Then cj(µn)=cj(µm) for “most” values of j in the sense that for each kthe number of places where the coefficients of size 2−kdo not match up is bounded independently of k, indeed by n−m. It follows readily from Theorem A that µnµmµn.In particular, µn(G) = 0 for all even n. Finally, we note two facts about the relationship between µnand µm. First, if n−mis odd, then one of n,mis odd and the other is even. Thus one of the measures gives full measure to G, while the other gives Gzero measure. In particular, µn⊥µm. Secondly, when n−mis even, the absolute continuity mentioned in the last paragraph of the proof can be strengthened: there exists a constant C, dependent only on n−m, such that C−1µm(E)≤µn(E)≤Cµm(E). It suffices to prove this last estimate for E∈Q, in which case the estimate follows from the fact, that cj(µn)=cj(µm) for “most” values of j. We leave the details to the reader. References [Bi] P. Billingsley,“Probability and measure”, third ed., Wiley Series in Probability and Mathematical Statistics. A WileyInterscience Publication, John Wiley & Sons Inc., New York, 1995. [Bu] S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340(1) (1993), 253–272. [BHM] S. M. Buckley, B. Hanson and P. MacManus, Doubling for general sets, Math. Scand. (to appear). [C] A. Cant´ on, Singular measures and the little Bloch space, Publ. Mat. 42(1) (1998), 211–222. [FKP] R. A. Fefferman, C. E. Kenig and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. Math. (2) 134(1) (1991), 65–124. [Kh] J.-P. Kahane, Trois notes sur le ensembles parfaits lin´eaires, Enseignement Math. (2) 15 (1969), 185–192. [Kk] S. Kakutani, On equivalence of infinite product measures, Ann. of Math. (2) 49 (1948), 214–224. Singular Measures and The Key of G489 [S] W. Smith, Inner functions in the hyperbolic little Bloch class, Michigan Math. J. 45(1) (1998), 103–114. Department of Mathematics National University of Ireland Maynooth, Co. Kildare Ireland E-mail address:[email protected] E-mail address:[email protected] Primera versi´o rebuda el 26 d’octubre de 1999, darrera versi´o rebuda el 19 de juny de 2000.