SINGULAR MEASURES AND THE KEY OF G

We construct a sequence of doubling measures, whose doubling constants tend to 1, all for which kill a $G_{\delta}$ set of full Lebesgue measure.


Introduction
A non-zero Borel measure ν is said to be doubling if there is a constant C ≥ 1 such that whenever I, J are adjacent intervals of the same length.We call the smallest C = C ν for 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] n is open and |∂U | = 0, then ν n (U ) → |U | whenever ν n is a sequence of probability measures on [0, 1] n whose doubling constants tend to 1.In particular, if U is an open subset of [0, 1] of full measure, then ν n (U ) → 1.We will show, amongst other things, that there exists a G δ set G in [0, 1] of full measure, and a sequence ν n of measures whose doubling constants tend to 1, yet ν n (G) = 0 for all n.We can even choose the measures to be "renormalizations" of a single measure ν which "fit the gaps in G" as a key fits a lock.
We wish to thank the referee for drawing our attention to the paper of Kakutani.form [m4 −k , (m + 1)4 −k ), where m, k are non-negative integers, and set Q(j) to be the subset of Q consisting of those intervals of length 4 −j .
For any I ∈ Q the four children are labeled I 0 , I 1 , I 2 , I 3 , moving from left to right.Now consider otherwise.
The product I∈Q (1+a I H I ) converges weak- * to a doubling, probability measure µ, provided that sup I∈Q |a I | < 1.We call any such measure µ a Kahane measure and write µ K = sup I∈Q |a I |.Furthermore, the doubling constant C µ tends to 1 as µ K tends to 0; in fact, if µ K ≤ 1 − , then there is a constant c , dependent only on , such that C µ ≤ 1 + c µ K whenever for some > 0.
For our purposes it will be sufficient to consider Kahane measures for which all of the coefficients a I at 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 M is of the form ∞ j=1 (1 + a j R j ) where R j = I∈Q(j) H I ; it is convenient to introduce the notation c j (µ) ≡ a j .We will focus on those measures µ ∈ M for which c j (µ) → 0 as j → ∞, and we label these M 0 .For µ ∈ M and n = 0, 1, 2, . . . the measure µ n ∈ M henceforth denotes the element of M with c j (µ n ) = c j+n (µ), j ∈ N. The measures µ n are "renormalized" versions of µ; in fact, if S ⊂ [0, 1) is a measurable set and f S is the periodic function with period 1 whose restriction to [0, 1) is the characteristic function of S, then µ n (S) = 1 0 f S (4 n t) dµ(t).Given µ ∈ M 0 , it follows from the estimate in the last paragraph that the sequence of doubling constants (C µn ) has limit 1.Thus every µ ∈ M 0 is optimally doubling at small scales in the sense that ν = µ satisfies (1) with C = C µn whenever I, J are adjacent intervals with |I| = |J| ≤ 4 −n .
The following result is a special case of a result of Kakutani [Kk,Corollary 1].
In fact, when ν is Lebesgue measure and (a n ) ∈ l 2 above, more is true: µ lies in the Muckenhoupt class A ∞ , and in particular µ has density lying in L p ([0, 1]) for some p > 1; see [Bu] and [FKP]. 1akutani 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.
n=1 is a sequence of independent random variables, and that the moments and E|X n − e n | 3 = τ 3 n are finite for each n.Let In this paragraph we employ the notation of Theorem A. The functions R n are independent as random variables on [0, 1] with respect to ν, and so the functions are also independent.A little calculation with the power series expansion for log(1 + t) gives are bounded above and below by positive, finite constants that are independent of n.It is then routine to deduce that lim n→∞ t n /s n = 0; one simply splits the sum at a point beyond which |a n − b n | is very small and uses the estimate Set {P N } to be the partial products of this infinite product.We have just seen that this sequence of functions converges to zero in ν-measure.However, P N (x) = µ(I N (x))/ν(I N (x)), where I N (x) is the unique element of Q(N ) containing x and so, by the Radon-Nikodym theorem, {P N } converges ν-a.e. to the Radon-Nikodym derivative of µ with respect to ν.Consequently, the Radon-Nikodym derivative is zero ν-a.e., and so µ ⊥ ν whenever (a j − b j ) / ∈ l 2 .We are mainly interested in Theorem A when ν is Lebesgue measure.In this case if the sequence (c j (µ)) has limit zero but does not lie in l 2 , then µ is 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 Cantón [C] and Smith [S].
There is an obvious bijection, A, between Q and 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 A j (I).For I ∈ Q, we let E(I) consist of the union of the intervals J ∈ Q for which A 2j (J) = A j (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 4 j elements of Q(2j).For n = 0, 1, 2, . . .and I ∈ Q, T n (I) consists of those intervals J ∈ Q for which A n+j (J) = A j (I) for all j.So the first n terms of J are arbitrary and the remainder are specified.When I ∈ Q(j), T n (I) consists of 4 n elements of Q(j + n).Note that if I and J are disjoint, then E(I) and E(J) are disjoint, as are T n (I) and T n (J).For any set B that is a union of disjoint elements I of Q, we define E(B) to be the union of the E(I), and we define Let Σ j be the collection of subsets of [0, 1) that are unions of elements of Q(j).Any set B ∈ Σ m is said to be j-indifferent if whenever B ⊃ I ∈ Q(m) and J is 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) is the set of sequences of length m given by A(I) for each I ∈ Q(m), I ⊂ B, then B is j-indifferent precisely if S(B) is measurable with respect to the σ-algebra generated by the sets The point of this definition is that if B is j-indifferent, then µ(B) does not depend on the c j (µ).In particular, if B ∈ Σ m , then E(B) is j-indifferent for all odd numbers j and all even j > 2m, and T n (B) is j-indifferent for all j ≤ n and all j > n + m.

Construction of µ and G
Our main result is as follows.
Theorem 1.There exists a measure µ ∈ M 0 on the interval [0, 1) and a G δ set G contained in [0, 1) which have the following properties: = 1 for all odd n ∈ N and µ n (G) = 0 for all even n ∈ N.
Taking ν n = µ 2n , we immediately get Corollary 2. There exists a G δ set G in [0, 1] of full measure and a sequence ν n of probability measures on [0, 1] whose doubling constants tend to 1 and for which ν n (G) = 0 for all n.
The oscillatory behaviour of µ n (G) described in Theorem 1(b) is all the more remarkable since the measures µ n are renormalized versions of a single measure µ whose doubling constants are tending to one.The idea is to construct G from sets that are indifferent at odd levels n (and thus treat such µ n like Lebesgue measure), but which are concentrated in areas where µ n is small whenever n is even.
Proof of Theorem 1: Let b be any number strictly between 0 and 1. Define ν k to be the element of M whose coefficients are all 2 −k .This measure is singular with respect to Lebesgue measure.It follows that for sufficiently large n k , there exists We can assume that the n k are increasing to ∞.
Divide the natural numbers into consecutive blocks B 1 , B 2 , . . . of length 2n 1 , 2n 2 , . . . .Set a j = 2 −k whenever j is an even number in block B k , and 0 otherwise.Define The set H k is j-indifferent for all j except even j in B k and c j (µ) = 2 −k for these exceptional integers.Thus µ(H for all m, and so µ(G) = 0.
Suppose n − m is even.Then c j (µ n ) = c j (µ m ) for "most" values of j in the sense that for each k the number of places where the coefficients of size 2 −k do not match up is bounded independently of k, indeed by n − m.It follows readily from Theorem A that µ n µ m µ n .In particular, µ n (G) = 0 for all even n.
Finally, we note two facts about the relationship between µ n and µ m .First, if n − m is odd, then one of n, m is odd and the other is even.Thus one of the measures gives full measure to G, while the other gives G zero measure.In particular, µ n ⊥ µ m .Secondly, when n − m is 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 −1 µ m (E) ≤ µ n (E) ≤ Cµ m (E).It suffices to prove this last estimate for E ∈ Q, in which case the estimate follows from the fact, that c j (µ n ) = c j (µ m ) for "most" values of j.We leave the details to the reader.
µ ∈ M 0 by the equations c j (µ) = a j .Now letm k = 2n 1 +• • •+2n k−1 for k > 1 and m 1 = 0. Thus m k is the total length of the blocks B 1 , . . ., B k−1 .Define H k to be T m k (E(A k )).Then H k ∈ Σ m k +2n k and is j-indifferent for all j except even numbers larger than m k and no larger than m k + 2n k , i.e., all even numbers in B k .Remove the endpoints of the intervals that make up H k to get an open set U k .The sets U k and H k differ only by a countable number of points.Thus any doubling measure gives them the same measure (doubling measures on the line are non-atomic).