Ordinal Rankings on Measures Annihilating Thin Sets - California

13 Pages·2010·0.86 MB·English

Ordinal Rankings on Measures Annihilating Thin Sets
Author(s): Alexander S. Kechris and Russell Lyons
Source: Transactions of the American Mathematical Society, Vol. 310, No. 2 (Dec., 1988), pp.
747-758
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2000989 .
Accessed: 20/05/2013 12:59
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
.
American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to
Transactions of the American Mathematical Society.
http://www.jstor.org
This content downloaded from 131.215.71.79 on Mon, 20 May 2013 12:59:54 PM
All use subject to JSTOR Terms and Conditions
TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 310, Number 2, December 1988
ORDINAL RANKINGS ON MEASURES
ANNIHILATING THIN SETS
ALEXANDER S. KECHRIS AND RUSSELL LYONS
ABSTRACT. We assigna countableo rdinaln umbert o each probabilitym ea-
sure whicha nnihilatesa ll H-sets. The descriptive-setth eoreticst ructuroef
this assignmenatl lowsu s to show that this class of measuresi s coanalytic
non-BorelI.n additioni,t allowsu s to quantifyth ef ailureo fR ajchman'sc on-
jecture.S imilarr esultsa re obtainedf orm easuresa nnihilatinDgi richletse ts.
A closet subset E of the unit circleT R/Z is called an H-set ift heree xistsa
sequence {nk} of positivei ntegerst endingt o oo and an interval( i.e., a nonempty
open arc) I C T such that fora ll k and all x E E, nkx 0 I. These sets play a
fundamentarl ole as examples of sets of uniquenessf ort rigonometrisce ries [KL;
Z, ChaptersI X, XII]. A (Borel) probabilitym easure, uo n T is called a Rajchman
measurei f ,(n) -+ 0 as Il -+ oo, where ,u(n)= fTTe(-nx)dp(x), e(x) = e2rix.
We denote by R the class of such measures. These measuresh ave also been very
importantt o the studyo fs ets ofu niqueness.I n particulare, veryR ajchmanm easure
annihilatese verys et of uniqueness,h ence every H-set. Aftere stablishingt hese
relationships[ Rl, R2], Rajchman conjecturedt hat, in fact, the only measures
which annihilatea ll H-sets are those in R. This, however,i s false [Li, L2, L3,
L5]. Here, we shall quantifyh ow distantR ajchman's conjecturei s fromt he truth.
Given a class ' of closed subsets of T, denote by WJ" the class of probability
measureso n T whicha nnihilatea ll sets in i?: , E F" X* VE E ' (,(E) = 0).
Thus R C H', where H denotes the class of H-sets. Denote by PROB(T) the
compact, metrizables pace of (Borel) probabilitym easureso n T with the weak*
topology. It is easy to check that R is a Borel, in fact II3 (i.e., F6), subset of
this space. We establish in ?3 that H' is a rI1 (i.e., coanalytic) but not Borel
subset ofP ROB(T). This is the firste xampleo f a naturalc lass ofm easureso fs uch
complexityk nownt o the authorsa nd it highlightts he distinctionb etweenR and
H'.
Our method of proof actually providesq uite a bit of furtherin formationon
the relationshipb etweenR and H'. In ?1, we assign to each ,uE H' a countable
ordinaln umberh (s) whichm easuresi n somes enset hec omplexityof t hev erification
that ,u annihilatesa ll H-sets. We show that h has certaind efinabilitypr operties,
namely,i t is a Il1-rank (see [KL]). In ?3, usingt he techniquesd evelopedi n [L3 and
L4], we prove that the rank h is unboundedi n w1, the firstu ncountableo rdinal;
Receivedb y the editorsS eptember1 0, 1987.
1980 MathematicsS ubject Classification( 1985 Revision). Primary 43A46, 03E15; Secondary
43A05,4 2A63,5 4H05.
Keyw ordsa ndp hrasesR. ajchmanm easurest, hins ets,s ets ofu niquenessr, ank.
Researcho ft he firsta uthorp artiallys upportedb y NSF grantD MS8718847.
The seconda uthorw as partiallys upportedb y an NSF PostdoctoraFl ellowship.
?1988 American Mathematical Society
0002-9947/88 $1.00 + $.25 per page
747
This content downloaded from 131.215.71.79 on Mon, 20 May 2013 12:59:54 PM
All use subject to JSTOR Terms and Conditions
748 A. S. KECHRIS AND RUSSELL LYONS
that is, Va < w1 3, E H' (h(,u) > a). It followsf romt he BoundednessT heorem
forf II-ranks (see [KL, V.1]) that H' is not Borel.
Denote by Hl- the class of all ,u E H' with h(,u) < a. Thus Hj C HL C
*@C H.- C . C H' ... (a < d < w1), H' aW HO-L and HO-L C H' for
each a < w1, so that {HO-} is a stratificationof H' in a hierarchyo f increasing
complexityT. he "simplest"m easuresi n H' are those in HiL. In ?2, we provet hat
these include the Rajchman measures: R C HiL. This bound gives quantitative
meaningt o the statementt hat R is only a small part of H'. We also show that
anotherc anonicalc lass of measuresi n H', the so-calledq uasisymmetrimc easures,
belongt o Ha-. We do not knowi ft heyb elongt o Hi. An interestincgo nsequenceo f
theseu pperb ounds is that the techniquesi n [L3] forp rovingm easurest o be in H'
are demonstratedt o be morep owerfuiln a quantitatives ense than the traditional
techniquesu sed (and amplified)i n ?2.
We conclude this paper by establishing,i n ?4, analogous resultsf ort he class
D' ofp robabilitym easuresa nnihilatinga ll Dirichlets ets. (Recall that a closed set
E C T is a Dirichlets et [LP] ift herei s a sequenceo fp ositivei ntegers{n k} tending
to oo such that supXEEI Ifnkx-I+I 0, whereI I xjj = dist(x,Z ).) D' has been studied
beforeu ndert he name YI (see [HMP, pp. 212-215, 242-247]).
NOTE. Afterc ompletiono f this manuscriptw, e learnedt hat B. Host, A. Lou-
veau, and F. Parreau establisheds everal monthse arliert han us that DL is not
Borel; theyu sed the characterizationof DL giveno n p. 243 of [HMP]. Theirw ork
is unpublished.
1. A fl1-rank on H'. Let K(T) be the space of closed subsetso f T witht he
standard Hausdorffm etric. This is a compact space. As a subset of K(T), it is
verifiedin [KL, IV.2.7] that H is EO (i.e., Gb7).
We next computet he followingu pper bound fort he complexityo f H'.
PROPOSITION 1 . 1. The set H' is flI1( i.e., coanalytic)i n thes pace PROB(T).
PROOF. Since H is Borel and
, E H1 X VE E K(T) [E E H =t p(E) =O ]
for, uE PROB(T), it is enought o show that the relation
{(,u,E ): , E PROB(T)&E E K(T)&,(E) = 0}
is Borel (in fact,G o) in the space PROB(T) x K(T). Now, if {Vn} is an open basis
fort he topologyo f T whichi s closed under finiteu nions,t hen for, uE PROB(T)
and E E K(T),
Ip(E) = O X Vm3n(E C Vn&M&(T\Vn>) mi)
Since {E: E C V} is open in K(T) fore ach open V C T, while {,u:p ,(V) > a} is
open in PROB(T) fore ach a E R and V open in T, the resultf ollows. O
Our main resulti s that H' is not Borel,t husd etermininegx actlyt he complexity
of H'.
We willu se below notationc oncerningf inites equencesa nd treesa s in [KL, IV.1].
We denote by N* the set of positivei ntegers.T o defineo ur rank h: H' -+ w1,w e
This content downloaded from 131.215.71.79 on Mon, 20 May 2013 12:59:54 PM
All use subject to JSTOR Terms and Conditions
RANKINGS ON MEASURES ANNIHILATING THIN SETS 749
associate firstt o each s E PROB(T), each I whichi s a rationali ntervali n T, and
each E E Q+ = {x E Q: x > O}, a tree T" s on Seq N* as follows:
T (no, nj,. . nk): Vi < k(ni E N*)&Vi < k(ni+,/ni > 2)&
pI({x E T: Vi < k(nix 0 I)}) > E}.
Notice that ifI D I' and E > E', then TI' C T"'I'.
REMARK. The condition" ni+i/ni > 2", instead of the expected one, "6ni+1>
ni", has the effecto f thinningd own the trees for normalizationp urposes. For
example,i n Theorem2 .4 below,w e obtain that h(,i) = 1 forR ajchman measures,
rathert han h(,u) < w.
PROPOSITION 1.2. Let it E PROB(T). Then it E H' X VI VE(T is well
founded).
PROOF. If for some I, , TI E has an infiniteb ranch no,I n2i,2 . .., then ,u(E) >
E > 0 forE = {x E T:Vi(nix 0 I)}. As E E H, we have it 0 H'. Converselyi,f
As H, let E E Q+, E E H be such that ,u(E) > E and let O < no < n <" and
I be such that nkx I fora ll k E N, x E E. By goingt o a subsequence,w e can
O
assume that ni+i/ni > 2 fora ll i. Then (no,n l, . . , nk) E TI ' fora ll k, i.e., TI"6
is not well founded. E
For each well-foundedtr ee T, we definei ts height,h t(T), as follows.F irst, for
each s E T, definei ts heighti n T, ht(s,T ), by
ht(s,T ) = 0 if s is terminali n T, i.e., has no propere xtensioni n T,
ht(s,T ) = sup{ht(sA(y),T ) + 1: sA(y) E T}
= sup{ht(t,T ) + 1: t E T, t D s} otherwise.
Then let ht(T) = ht(0, T). (This definitionis slightlyd ifferenftr omt hat used in
[KL, V.1], whereo ne definesh t(s,T) = 1 if s is terminali n T.)
For each ,uE H', definet he rankf unctionh '(,u) = sup{ht(T,,') + 1: I a rational
intervala nd E E Q+}. Clearlyh ': H' -- w1. We will showf irstt hat h'(,u) is always
a limito rdinal,t hus of the formw . a fors ome a < wi. We will then defineh (,)
by h'(p) = w h(p).
LEMMA 1.3. For each ,u E H', h'(,u) is a limito rdinal.
PROOF. First notice that h'(,u) > w fora ll ,u E H'. Indeed, given any O <
no < ni < < nk with ni+ /ni > 2, one can easily findI and E such that
(no, .. , nk) E TI,'. To completet he proof,i t is enought o show that if , E H',
h'(,) > w * a (a > 1), and N E N*, then h'(,u) > *a + N. Since h'(p) > w *a ,
therea re I and E withh t(T,,') > w oa. Fix next a sequence0 < no < *. < nN-1
with ni+,/ni > 2 and a rational intervalI ' C I such that p({x:Vi < N - 1
(nix 0 I')} > 1 - /2.
We claimt hati fS = {(n)At E T,{e: n > 2nN-1} U{0}, thenh t(S) > w*a. This is
because fors ome s E T,,', we have ht TI 6) = w. a and thuss up{ht(ssA+ (n), TI'
1} = wea. Since the supremumis not attaineda nd sA(n)At E T,, => (n)At e TI
it followst hat ht(S) > w a.
Now let
T = {(no,..., nN-l)AS s E S}.
This content downloaded from 131.215.71.79 on Mon, 20 May 2013 12:59:54 PM
All use subject to JSTOR Terms and Conditions
750 A. S. KECHRIS AND RUSSELL LYONS
Clearlyh t(T) > w *a + N. We claimt hatT C TIe!2, thush t(TIe ) > w a + N
.
and we are done. Evidently,e verye lemento f a sequence (no,... , nN- 1)As E S is
at least twicea s big as its predecessor.A lso, we have
,u({x: nox I'& * * * &nN-lX ? I')} > 1 - E/2
and, if (no,..., nN-1,fnN,..., nN+m) E T, then
1({fx nNX ? IV&... &fnN+mX ? IV}
> ,u({x: nNX ? I& ... &nN+mX ? I}) > E.
Therefore
,u({x: nox I'& ... &nN+mX ? I'}) > E/2
and our proofi s complete. O
As mentionedb efore,w e may now defineh (,u)f or, uE H' by h'(,u)= wh().
The followingf acte stablishest he basic definabilitypr opertieso f this rank.
PROPOSITIN 1.4. The rankh :H ' -1 w1 is a f1l -ranko n thef 1l set H1.
PROOF. Recall that ifX is a Polishs pace, P C X is a f1 set in X and p: P -- Wi
is a ranko n P, thenw e say p is a fIl-rank if,l ettingp (x) = w1 fora ll x ? P, we
have that the relations
x <?, y x E P&p(x) < p(y),
* <*, y x E P&p(x) < p(y) (Wx ()
are fl' (as subsets of X2).
A typicale xample of a HI1-ranki s the following(s ee [KL, V.1]): Identifying
treeso n SeqN* witht heirc haracteristifc unctionsw, e can view thema s members
of the space 2Seq N* (= {0, 1Seq N'). Let WF C 2Seq N be the set of well-founded
treeso n SeqN*. Then WF is f1l. Moreover,t he rankT 4 ht(T) is a f11-ranko n
WF.
Going back to h now, notice that for, u,v E H', h(,u)< h(v) - h'(,u)< h'(v),
so it is enought o showt hat h' is a HIl-ranko n H'. Again let h'(,u)= wi if, H'.
Then
8 h' H Wh( p) < h'(v)
X, E H & VI E 3J ,6 [ht(T ,,) < ht(T J')]
, E H'& VI,E 3J,6 [TI E WF&ht(T,, ) < ht(Tj,f6)]
and
/I< h' v psE H'& h'(p) < h'(v)
':
X , E H & 3 J,6 VI I, [TI E WF & ht( T,,':) < ht( Tv")]
Since H1 sets are closed underc ountablei ntersectionasn d unionsa s well as Borel
pre-imagesa nd ht is a HI1-ranko n WF, it is enought o show that fore ach fixed
pair I, e, the map ,u e T is a Borel map from PROB(T) into 2Seq . Since
2Seq N* has the productt opology,t his reduces to showingt hat fore ach fixeda =
(no, .. . , nk) with 2ni < ni+ , the set
{u E PROB(T): s E TI'} - {,u E PROB(T): ,(F) > E},
This content downloaded from 131.215.71.79 on Mon, 20 May 2013 12:59:54 PM
All use subject to JSTOR Terms and Conditions
RANKINGS ON MEASURES ANNIHILATING THIN SETS 751
whereF -= nfl.0{x: nix 0 I}, is a Borels et. SinceF is closed,l et F = nfV, ,,V, n
decreasinagn d openi n T. Then
p(F) > c -? Ve' < c Vn (p(Vn) > e'),
so actuallyt hiss et is a G6 and we are done. O
2. Measures of low rank. Let A = {f E C(T):f E 1'(Z)} withI lfIlA=
ilf1. For f E A,p ER, ande>0, define
Z
N(f,) = min N > 0: If(n)l < }
lnl>N
and
N'(,u, ) = min{N > 0: inj > N =- Ift(n)I< e}.
We write( Tmf)(x) = f(mx). Thus,I ITmfIIA= IlfI A and N(Tmf,e)= mN(f,e)
form > 1.
LEMMA 2.1. For f,g E A and e > 0,
N(fgXE< ) (fX21 1911A) 91 2| If |IA)
PROOF. Denotet het ermso n the rightb y N1 and N2. If ik+ 11> N1 + N2,
thenI kI> N1 or 11> N2. Therefore
Z
tfg(n)l= (k)(1)
lnI>Nl+N2 lnl>N,+N2 k+l=n
< >j If(k)g(l)I + > If(k)'(1) < e. O
IkI>Nj 11I>N2
IEZ kEZ
LEMMA 2.2. For f, g E A, , E R, and
n > 2 max
(N (f, IlN 9 ') 82 f A111|
we have
/I
f Tngd -I f d g()()0 ) <
PROOF.I ndeed,w e have
J|
f *T ngd 1a- Jf di = + )
-f(0)
I
<I1I9 (-r)I Z If(-l)fz(nr+1)1+ 1f(-l)t(nr+1+) 11
r$O j1j1n/2 1jL>n/2
< E
sincei fI ll < n/2a nd r 7&O , inr+ 11> n/2. 0:
This content downloaded from 131.215.71.79 on Mon, 20 May 2013 12:59:54 PM
All use subject to JSTOR Terms and Conditions
752 A. S. KECHRIS AND RUSSELL LYONS
LEMMA 2.3. If,uER,fEA, f(O) 1,I lflA >1, K >2 ml,m2, ... ,mK E
NK*
1~~~~~~~~~~~~ A
for < j < K, then
f
flnT m3 f d1f'(O) K <
1j=
PROOF. Let Fk = HXj1T=m 3f . We shalls howt hat
]
(2.1) Fk+idp -JFkcd4 f(O) <H
for0 < k < K, whereF o 1. Multiplyinbgo ths idesb y f(O) Kk1 and adding
overk yieldst hed esiredin equalityN. ow( 2.1) willf ollowif
m ,?1 > 2max (N(Fk, 2K IfIIA)N (Iu 2KIIfIIk+'))
That mk+1 is at least twicet he second termf ollowsi mmediatelyf romt he hypoth-
esis. Thus,i t sufficetso showt hat
)
mkKN(fX2K-K1I If IIK-1 )- ( 2K |I||f|IA
By repeateda pplicationof L emma2 .1,w e obtain
N(Fk, 2KIIL4) ? N(Fk_l 4KIifI1)+2 (Tmkf'4KIIfIIA IIFklIIA)
< ...< N (Tmlf kK ) +N (Tm2f l2kKIIk)
+N(Tm3f, 2k-1KIIfIA) + +N Tmk fl 4KI ifIlI)
< kmkN(f27f KH fEI)
This impliest he desiredi nequality. O
THEOREM 2.4. Ift'E R, thenh (1u)=1; i.e.,RCHiL.
PROOF.W es halls howt hath '(1u)< w. Let I be a rationailn tervaaln de E Q+
Choosef E A so that0 < f < 1, f = 1 on II, and f(O) < 1. Then lIlflA> 1 and
lI < f. Choose K so that f()K < e/2 and choose
L > 2K1092 max (2KN f, 2KI6 A~)2N ('4KIIA)
We claimt hath t(TI,,) < L. Indeed,i f (mb,..., m'1) E TI, thent herei s a
subsequence{ mj}JK=1O f{ Mr}L=tl such that
j+j > 2 ma KN(f ) N' 4 f )
This content downloaded from 131.215.71.79 on Mon, 20 May 2013 12:59:54 PM
All use subject to JSTOR Terms and Conditions
RANKINGS ON MEASURES ANNIHILATING THIN SETS 753
Thus
K
,u{x: Vi < L(m'x ? I)} < | l1 C (mjx) d,u(x)
j=l
< ? j f( mjx) dp(x) <() K + < E
1
a contradiction. a
Let us say that a probabilitym easure, uo n T is quasisymmetrico,r ,uE QS, if
fors ome C, wheneverI , and I2 are adjacent intervalso n T of the same length,
PIl < C - A2
(These measures are related to quasiconformaml appings; see [BA].) Note that
every, uE QS is continuous.E xamples includeR iesz products( forb ackgrounds, ee
[K, p. 107])
fl
H= (1 + Re{ake(nkX)})
k>l
withl imlakl < 1, nk+l/nk > 3, and eithernk lnk+l or nk+l/nk -* oo; we omitt he
proof.W e do not knoww hetherQ S C H#, but we now show that QS C Hj,.
THEOREM 2.5. QS C Hj-.
PROOF. Let , E QS, I C T be a rational interval,a nd E E Q+. Denote
T; 'I = {x: nx E I}. Then Tn 'I consistso fn equallys paced intervalss eparatedb y
the n intervalso fT n lIc; the ratioo ft he lengthso f adjacent intervalsis III/IICIa nd
thereforteh e ratio of theirp ,-measureiss bounded by a constant,C ', independent
of n. It followst hat if A is a finiteu niono f intervalst, hen
p(AnT,,lIC) < 2C'1(AnT,-1I)
fora ll largen . Let K be such that (2C1/(1+2C1))K < E. We claim that ht(T,j ) <
w * K, whichw ill completet he proof.
Indeed, if ht(TI 6) > w K, then there exist arbitrarilyla rge m such that
ht((m), TI'X) > w* (K-1). (Note that sAtAuAv E TI= tAvE TI,'.) Choose mo
such that
ht((mo)T, , ') > w*( K - 1)
and
,u(Tm1I c) < 2C',u(Tm I)
In a similarf ashion,w e may findi nductivelym k (1 < k < K - 1) such that
ht( (mO, ml... *,MI k), T,,':) > W * (K - 1-k)
and
(2.2) ,u(Ek_1 n T; lic ) < 2C',p(Ek_, n TT 1 I),
where
k
Ek= fnT; I
i=O
This content downloaded from 131.215.71.79 on Mon, 20 May 2013 12:59:54 PM
All use subject to JSTOR Terms and Conditions
754 A. S. KECHRIS AND RUSSELL LYONS
Now (2.2) impliest hat
,aEk < 2C'(pEkl - Ek)X
or
pEk < 2C0'pEk-1
Therefore
K
(I+2CI
whichc ontradictst he factt hat (MO, Mi1,..., MK_1) E TI,{. Z
3. Measures of large rank. For each a < w1,w e shall constructa canonical
treeT c,o fh eighta . Let Y be the set of strictlyin creasingm aps f: N* - N*. Any
f E Y induces a map f * on treesi n the obvious way. Given a < wl, p E]O,1 [ and
f E F, we shall constructa measure vc,,p,fE H' with h'-ranka t least a. This
showst hat H' is not Borel.
Given a tree T, let
(T) = {n: is (n E s E T)},
where,i f s = (nO,ln,...,fnk), we write n E s X 3i < k (n = ni), and let (T)C =
N*\(T). Definef n(x) = x2n+1 . 22n; notet hati fx and y are integrapl owerso f2
and n 0 m, then fn(x) #f m(y). For each limito rdinala < wi, choose a bijection
g&:a - N.
We constructo ur canonicalt reesT Q by inductionb, eginnningw ithT o = {0}. It
will turno ut that (TQ){ co nsistse ntirelyo fp owerso f 2. If T, have been constructed
for- y < a, let
{(22)As: s E fo*T3U} {0} ifa = ,3+ 1,
=U = <Qfac,Ts if a is a limit.
It is clear that ht(TQ) = a.
Given a set A C N*, definet he Bernoullic onvolution
+
)AAA A=n*E A [[62(0() o + -6(2 -n)]
where6 (x) representst he Dirac measurec oncentrateda t x. For a < wl, p E]O, [
f E Y, we constructp reliminarym easuresp a,p,f by induction.L et po,p,f = b(0)
fora ll p, f. If pj,pj have been constructedf ora ll -y < a, p E]0, 1[, and f E Y,
define
= () + (1 -) A( f - T.)
where
(0) = I' i(fl,v'd,fofo) if a=f+1,
1a,p,f * (,3, p f o fg. if a is a limit;
here,a nd below,w e are usingt he notation ( ) fors ubscripts.
Finally,d efine
va,tp,f = A(f*T.),C * pa,p,f
The reader should have in mind the followingp robabilistici nterpretationof
v,z,p,f. First definei .i.d. random variables Yn forn > 1 which take the values
This content downloaded from 131.215.71.79 on Mon, 20 May 2013 12:59:54 PM
All use subject to JSTOR Terms and Conditions
RANKINGS ON MEASURES ANNIHILATING THIN SETS 755
0, 1 each with probability1 /2. For A C N*, let YA = ZnEA Yn2-n. Let Zp be
independentr andomv ariablesf or0 < p < 1, independenta lso of {Yn}, whicht ake
the values 0, 1 with probabilities/ p and 1 - /p, respectivelyW. e definer andom
variablesX &,p,fb y induction. Let Xo,p,f 0. To defineX .p,fg ivenX .y,q,gf or
-7 < a, firstl et {XI,q,g:- 7 < a, q E]O,1 [,g E Y} be independentc opies of {X,y,q,g}
whicha re also independento f {Zp} and {Yn}. Now define
X,p,f (1-ZP)X(),f + Zp Y(f-T.)
where f
-() of if a = 3 + 1,
Z13<aX (M1,, f o fga(3)) if a is a limit.
Finally,d efine
Wa,p,f= Xa,p,f + Y(f
T.)c.
Thent hed istributionofs YA,X ,p,, X(pf I and W, p,fa re AA,1,P,fp,1l(f and
v,,p,f, respectivelyO. ne may show by inductiont hat Xc,,p,fc an be writteni n the
form
Xa,p,f = E Un2-n
n>1
fors ome {0, 1}-valued randomv ariablesU n with Un _ 0 if n 0 (f*T,).
The followingp ropositions hows that f*To C T)1' 'JPJ Combinedw ith Theo-
rem 3.2, this showst hat h'(va,pj ) > a.
PROPOSITION 3. 1. Va < w1 Vp Elo,1 [ Vf E Y Vs E f*Ta
(3.1) Va,p,f(E s) = /a,p,f(E s) > p
and
(3.2) /1a,p,f(E(f T,)c) = 1,
wheref or A C N*,
EA = x ET:Vn E A(2n-1x [i, 1i[ ) }
PROOF. We proceed by induction. Condition( 3.2) is easily established,s o we
concentrateo n (3.1). It is clear that va,p,fE, = /c,p,,fEs. Now (3.1) is trivial
fora = 0, so suppose it holds fora ll -7 < a. If a = 3 + 1 and s E f*T., then
s = (f(22))A30 fors ome so E (f o fo)*T3. Therefore
/',p,f(E8) > v4p'(/+/3i,,f o fo)(Es)
> \/,kL(f vIXf o fo)(ES0n E(((f o fo)*T)C))
=p /f( Xp, ffo fo)(E,0) (by (3.2))
by the inductiveh ypothesis.O n the otherh and, ifa is a limita nd s E f*Ta, then
s E (f o fg9c(f3))*Tff3o rs ome 3 < a. Therefore
p/,pj Es > \/pp(?) Es = Vpfi'(,3v,1 p,f o fg.0,))(Es) > p. L?
It remainst o provet hat v,p,f E H', forw hichw e use the methodso f [L3].
This content downloaded from 131.215.71.79 on Mon, 20 May 2013 12:59:54 PM
All use subject to JSTOR Terms and Conditions