Publicacions Ma em`a iques, Vol. 44 (2000), 483–489
SINGULAR MEASURES AND THE KEY OF G
S ephen M. Buckley and Paul MacManus
Abs ac
We cons uc a sequence o doubling measu es, whose doubling
cons an s end o 1, all o which kill a Gδse o ull Lebesgue
measu e.
0. In oduc ion
A non-ze o Bo el measu e νis said o be doubling i he e is a con-
s an C≥1 such ha
C−1≤ν(I)
ν(J)≤C,(1)
whene e I,Ja e adjacen in e als o he same leng h. We call he
smalles C=Cν o which his condi ion holds, he doubling cons an
o ν. A measu e is a mul iple o Lebesgue measu e i and only i i s
doubling cons an is 1.
I was shown in [BHM] ha i U⊂[0,1]nis open and |∂U|= 0,
hen νn(U)→|U|whene e νnis a sequence o p obabili y measu es on
[0,1]nwhose doubling cons an s end o 1. In pa icula , i Uis an open
subse o [0,1] o ull measu e, hen νn(U)→1. We will show, amongs
o he hings, ha he e exis s a Gδse Gin [0,1] o ull measu e, and a
sequence νno measu es whose doubling cons an s end o 1, ye νn(G)=
0 o all n. We can e en choose he measu es o be “ eno maliza ions”
o a single measu e νwhich “fi he gaps in G”asakeyfi salock.
We wish o hank he e e ee o d awing ou a en ion o he pape
o Kaku ani.
1. Defini ions and basic esul s
The e is an easy way, essen ially due o Kahane [Kh], o gene -
a e doubling measu es. Le Qconsis o all in e als on [0,1) o he
2000 Ma hema ics Subjec Classifica ion. 60G30.
The fi s au ho was pa ially suppo ed by En e p ise I eland.
484 S. M. Buckley, P. MacManus
o m [m4−k,(m+ 1)4−k), whe e m, k a e non-nega i e in ege s, and se
Q(j) o be he subse o Qconsis ing o hose in e als o leng h 4−j.
Fo any I∈Q he ou child en a e labeled I0,I1,I2,I3, mo ing om
le o igh . Now conside
HI(x)=




1,x∈I1
−1,x∈I2
0,o he wise.
The p oduc I∈Q(1+aIHI) con e ges weak-∗ o a doubling, p obabili y
measu e µ, p o ided ha supI∈Q |aI|<1. We call any such measu e µ
aKahane measu e and w i e µK= supI∈Q |aI|. Fu he mo e, he
doubling cons an Cµ ends o 1 as µK ends o 0; in ac , i µK≤
1−, hen he e is a cons an c, dependen only on , such ha Cµ≤
1+cµKwhene e o some >0.
Fo ou pu poses i will be sufficien o conside Kahane measu es o
which all o he coefficien s aIa any gi en scale a e equal and µK≤
1− o some >0, which we assume o be fixed om now on. We
deno e his class o measu es by M, o simply M. Then e e y measu e
in Mis o he o m ∞
j=1(1 + ajRj) whe e Rj=I∈Q(j)HI;i is
con enien o in oduce he no a ion cj(µ)≡aj. We will ocus on hose
measu es µ∈M o which cj(µ)→0asj→∞, and we label hese M0.
Fo µ∈Mand n=0,1,2,... he measu e µn∈Mhence o h deno es
he elemen o Mwi h cj(µn)=cj+n(µ), j∈N. The measu es µna e
“ eno malized” e sions o µ; in ac , i S⊂[0,1) is a measu able se
and Sis he pe iodic unc ion wi h pe iod 1 whose es ic ion o [0,1)
is he cha ac e is ic unc ion o S, hen µn(S)=1
0 S(4n )dµ( ). Gi en
µ∈M
0, i ollows om he es ima e in he las pa ag aph ha he
sequence o doubling cons an s (Cµn) has limi 1. Thus e e y µ∈M
0
is op imally doubling a small scales in he sense ha ν=µsa isfies (1)
wi h C=Cµnwhene e I,Ja e adjacen in e als wi h |I|=|J|≤4−n.
The ollowing esul is a special case o a esul o Kaku ani [Kk,
Co olla y 1].
Theo em A. Le µ, ν ∈M, wi h aj=cj(µ),bj=cj(ν), o all j∈N.
I (aj−bj)∞
j=1 lies in l2, he class o squa e summable sequences, hen
µνµ, o he wise µ⊥ν.
Singula Measu es and The Key o G485
In ac , when νis Lebesgue measu e and (an)∈l2abo e, mo e is ue:
µlies in he Muckenhoup class A∞, and in pa icula µhas densi y lying
in Lp([0,1]) o some p>1; see [Bu] and [FKP].1
Kaku ani p o es his esul by ca e ul analysis, bu le us pause o
p o e he singula i y pa o his esul using he Lyapuno e sion o
he Cen al Limi Theo em [Bi, Theo em 27.3] which we now s a e.
Theo em B. Suppose ha {Xn}∞
n=1 is a sequence o independen an-
dom a iables, and ha he momen s E(Xn)=en,E(Xn−en)2=σ2
n=
0, and E|Xn−en|3=τ3
na e fini e o each n.Le
sn=n

i=1
σ2
i1/2
,
n=n

i=1
τ3
i1/3
.
I limn→∞ n/sn=0, hen Yn≡n
i=1(Xi−ei)/sncon e ges in dis ib-
u ion o he s anda d no mal dis ibu ion.
In his pa ag aph we employ he no a ion o Theo em A. The unc-
ions Rna e independen as andom a iables on [0,1] wi h espec o ν,
and so he unc ions n= log[(1 + anRn)/(1 + bnRn)] a e also indepen-
den . A li le calcula ion wi h he powe se ies expansion o log(1 + )
gi es
Eν( n)≡en=−(an−bn)2
4(1 −b2
n)+O(|an−bn|3),
Eν( n−en)2≡σ2
n=(an−bn)2
2(1 −b2
n)+O(|an−bn|3),
Eν| n−en|3≡τ3
n=|an−bn|3
2
1+b2
n
(1 −b2
n)2+O(|an−bn|4).
Thus i sn,
na e as in Theo em B, limn→∞ |an−bn|= 0, and (an−
bn)∞
n=1 /∈l2, hen 3
n/n
i=1 |ai−bi|3and s2
n/n
i=1(ai−bi)2a e bounded
abo e and below by posi i e, fini e cons an s ha a e independen o
n. I is hen ou ine o deduce ha limn→∞ n/sn= 0; one simply
spli s he sum a a poin beyond which |an−bn|is e y small and
uses he es ima e ·
l3≤ ·
2/3
l2·
1/3
l∞. Thus Theo em B is ap-
plicable in he case Xn= n. Since n
i=1 eiis much la ge han sn
o la ge n, i ollows ha Yn ends o −∞ in ν-measu e and hus
∞
n=1(1+anRn)/(1+bnRn) con e ges in ν-measu e o he ze o unc ion.
Se {PN} o be he pa ial p oduc s o his infini e p oduc . We ha e
1These e e ences only say ha µlies in dyadic A∞bu , since µis a doubling measu e,
his implies ha µ∈A∞.
486 S. M. Buckley, P. MacManus
jus seen ha his sequence o unc ions con e ges o ze o in ν-measu e.
Howe e , PN(x)=µ(IN(x))/ν(IN(x)), whe e IN(x) is he unique el-
emen o Q(N) con aining xand so, by he Radon-Nikodym heo em,
{PN}con e ges ν-a.e. o he Radon-Nikodym de i a i e o µwi h e-
spec o ν. Consequen ly, he Radon-Nikodym de i a i e is ze o ν-a.e.,
and so µ⊥νwhene e (aj−bj)/∈l2.
We a e mainly in e es ed in Theo em A when νis Lebesgue measu e.
In his case i he sequence (cj(µ)) has limi ze o bu does no lie in l2,
hen µis a singula measu e which is op imal doubling a small scales.
The me e exis ence o such a measu e may seem a li le su p ising and
was only ecen ly es ablished (using diffe en echniques) by Can ´on [C]
and Smi h [S].
The e is an ob ious bijec ion, A, be ween Qand he se o fini e
sequences whose e ms lie in {0,1,2,3}. We will e e o A(I) as he
add ess o I. The j h e m in he add ess is Aj(I). Fo I∈Q, we le
E(I) consis o he union o he in e als J∈Q o which A2j(J)=Aj(I)
o all j. So he odd e ms in A(J) a e a bi a y and he e en e ms
a e specified. I I∈Q(j), E(I) consis s o 4jelemen s o Q(2j). Fo
n=0,1,2,... and I∈Q,Tn(I) consis s o hose in e als J∈Q o
which An+j(J)=Aj(I) o all j. So he fi s n e ms o Ja e a bi a y
and he emainde a e specified. When I∈Q(j), Tn(I) consis s o 4n
elemen s o Q(j+n). No e ha i Iand Ja e disjoin , hen E(I) and
E(J) a e disjoin , as a e Tn(I) and Tn(J). Fo any se B ha is a union
o disjoin elemen s Io Q, we define E(B) o be he union o he E(I),
and we define Tn(B) simila ly. I is easy o check ha |E(B)|=|B|and
ha |Tn(B)|=|B|.
Le Σjbe he collec ion o subse s o [0,1) ha a e unions o elemen s
o Q(j). Any se B∈Σmis said o be j-indiffe en i whene e B⊃I∈
Q(m) and Jis one o he h ee elemen s o Q(m) o which A(J) and
A(I) diffe only in he j h place, hen J⊂B. Equi alen ly i S(B)is he
se o sequences o leng h mgi en by A(I) o each I∈Q(m), I⊂B,
hen Bis j-indiffe en p ecisely i S(B) is measu able wi h espec o
he σ-algeb a gene a ed by he se s
Sk,l ={(ai)m
i=1 :ak=l},1≤k≤m, k =j, l ∈{0,1,2,3}.
The poin o his defini ion is ha i Bis j-indiffe en , hen µ(B)doesno
depend on he cj(µ). In pa icula , i B∈Σm, hen E(B)isj-indiffe en
o all odd numbe s jand all e en j>2m, and Tn(B)isj-indiffe en
o all j≤nand all j>n+m.
Singula Measu es and The Key o G487
2. Cons uc ion o µand G
Ou main esul is as ollows.
Theo em 1. The e exis s a measu e µ∈M
0on he in e al [0,1) and
aGδse Gcon ained in [0,1) which ha e he ollowing p ope ies:
(a) µ([0,1)) = 1,|G|=1and µ(G)=0.
(b) µn(G)=1 o all odd n∈Nand µn(G)=0 o all e en n∈N.
Taking νn=µ2n, we immedia ely ge
Co olla y 2. The e exis s a Gδse Gin [0,1] o ull measu e and a
sequence νno p obabili y measu es on [0,1] whose doubling cons an s
end o 1and o which νn(G)=0 o all n.
The oscilla o y beha iou o µn(G) desc ibed in Theo em 1(b) is all
he mo e ema kable since he measu es µna e eno malized e sions o
a single measu e µwhose doubling cons an s a e ending o one. The
idea is o cons uc G om se s ha a e indiffe en a odd le els n(and
hus ea such µnlike Lebesgue measu e), bu which a e concen a ed
in a eas whe e µnis small whene e nis e en.
P oo o Theo em 1: Le bbe any numbe s ic ly be ween 0 and 1. De-
fine νk o be he elemen o Mwhose coefficien s a e all 2−k. This
measu e is singula wi h espec o Lebesgue measu e. I ollows ha
o sufficien ly la ge nk, he e exis s Ak∈Σnk o which |Ak|≥1−bk
and νk(Ak)≤bk. We can assume ha he nka e inc easing o ∞.
Di ide he na u al numbe s in o consecu i e blocks B1,B
2,... o
leng h 2n1,2n2,... . Se aj=2
−kwhene e jis an e en numbe in
block Bk, and 0 o he wise. Define µ∈M
0by he equa ions cj(µ)=aj.
Now le mk=2n1+···+2nk−1 o k>1 and m1= 0. Thus mkis he
o al leng h o he blocks B1,...,B
k−1. Define Hk o be Tmk(E(Ak)).
Then Hk∈Σmk+2nkand is j-indiffe en o all jexcep e en numbe s
la ge han mkand no la ge han mk+2nk, i.e., all e en numbe s in
Bk. Remo e he endpoin s o he in e als ha make up Hk o ge an
open se Uk. The se s Ukand Hkdiffe only by a coun able numbe
o poin s. Thus any doubling measu e gi es hem he same measu e
(doubling measu es on he line a e non-a omic). Se Gm=∞
k=mUk
and G=∞
m=1 Gm. This se Gis a Gδse .
We ha e |Hk|=|Ak|≥1−bk o all k, hence |Gm|= 1 o all m,
and |G|= 1. I nis odd and jis e en, hen cj(µn) = 0. Bu Hkis
j-indiffe en o all odd j, so i ollows ha µn(Hk)=|Hk|. As a esul ,
µn(G) = 1 whene e nis odd.

488 S. M. Buckley, P. MacManus
The se Hkis j-indiffe en o all jexcep e en jin Bkand cj(µ)=
2−k o hese excep ional in ege s. Thus µ(Hk)=νk(Ak)≤bk. Conse-
quen ly, µ(Gm)≤bm(1 −b)−1 o all m, and so µ(G)=0.
Suppose n−mis e en. Then cj(µn)=cj(µm) o “mos ” alues o j
in he sense ha o each k he numbe o places whe e he coefficien s
o size 2−kdo no ma ch up is bounded independen ly o k, indeed by
n−m. I ollows eadily om Theo em A ha µnµmµn.In
pa icula , µn(G) = 0 o all e en n.
Finally, we no e wo ac s abou he ela ionship be ween µnand µm.
Fi s , i n−mis odd, hen one o n,mis odd and he o he is e en.
Thus one o he measu es gi es ull measu e o G, while he o he gi es
Gze o measu e. In pa icula , µn⊥µm. Secondly, when n−mis e en,
he absolu e con inui y men ioned in he las pa ag aph o he p oo can
be s eng hened: he e exis s a cons an C, dependen only on n−m,
such ha C−1µm(E)≤µn(E)≤Cµm(E). I suffices o p o e his las
es ima e o E∈Q, in which case he es ima e ollows om he ac ,
ha cj(µn)=cj(µm) o “mos ” alues o j. We lea e he de ails o he
eade .
Re e ences
[Bi] P. Billingsley,“P obabili y and measu e”, hi d ed., Wiley
Se ies in P obabili y and Ma hema ical S a is ics. A Wiley-
In e science Publica ion, John Wiley & Sons Inc., New Yo k,
1995.
[Bu] S. M. Buckley, Es ima es o ope a o no ms on weigh ed
spaces and e e se Jensen inequali ies, T ans. Ame . Ma h. Soc.
340(1) (1993), 253–272.
[BHM] S. M. Buckley, B. Hanson and P. MacManus, Doubling
o gene al se s, Ma h. Scand. ( o appea ).
[C] A. Can ´
on, Singula measu es and he li le Bloch space, Publ.
Ma . 42(1) (1998), 211–222.
[FKP] R. A. Fe e man, C. E. Kenig and J. Piphe , The heo y
o weigh s and he Di ichle p oblem o ellip ic equa ions, Ann.
Ma h. (2) 134(1) (1991), 65–124.
[Kh] J.-P. Kahane, T ois no es su le ensembles pa ai s lin´eai es,
Enseignemen Ma h. (2) 15 (1969), 185–192.
[Kk] S. Kaku ani, On equi alence o infini e p oduc measu es, Ann.
o Ma h. (2) 49 (1948), 214–224.
Singula Measu es and The Key o G489
[S] W. Smi h, Inne unc ions in he hype bolic li le Bloch class,
Michigan Ma h. J. 45(1) (1998), 103–114.
Depa men o Ma hema ics
Na ional Uni e si y o I eland
Maynoo h, Co. Kilda e
I eland
E-mail add ess:[email p o ec ed]
E-mail add ess:[email p o ec ed]
P ime a e si´o ebuda el 26 d’oc ub e de 1999,
da e a e si´o ebuda el 19 de juny de 2000.