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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p.
Download (direct link): anintroduction1982.djvu
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Proof, (due to A. Garsia) Clearly Fs e L]{(m). For 0 < n < N we have FK > f, so UFn > Ujn by positivity, and hence UFN -r f >f„+i. Therefore
UFx{x) + f{x) > max f„{x)
1 £*i£jY
— max fn(x) when F,v(x) > 0 0&n£N
I Measure-Preserving ТглПМОРЛШЮПл
Tin» / i s - I- I',\ 1)11 ■ • [л-: /-"л l-V) > 0}, i?o film > j ( I1 silm — J ^ Ul'silm
• r
— j ^ I- л i/m — J ( L l's dm since Fv = 0 on A'Vl.
j^ 1\ dm — j UTVdm since F-V > 0 and hence UFS > 0.
> 0 since |jL'|| < 1. □
Corollarj 1.16.1. Let T.X —> X he nwasurc-prcxijruiiuj. If ge Lj.(ni) and
[ 1 "'J
Ux = <л e ,Y: sup V <j[T\x)) > a ( n> i n i= о
then
I $ dill > Уш(В, г A)
JIlar\A
if T~1 1 = .1 din! m{A) < x.
Proof. VVe lirsl prove lhis result under tire assumptions m(X) < cc and -1 = .Y. Let f — >j — y.. then — |J v - о ( v' I'\(v) > 0J so that jBi /dm > 0 by Theorem 1.16 and therefore у dm > y.iinbJ. In the general case we apply the above to T\A to get \A„,iai)dm > %m(A n BJ. □
Proof of Tiii:oki;M 1.14. VVe first assume m(X)<oc. By considering real and imaginary parts it su(Vices to consider / For such an / let
jS*U) = Iimsiip„ ., U/'U V;,= o/(T4)and./#(x) = limmf„.,oc,(l/)i)X:'ri/(T‘x). We have /* T - f* and /„ T = /* because if а„(х) = (1/л) У^оДТх) then ((n + 1), ii)di: n(.v) — u„(Tx) — f(x)/n. We have to show that /* = /* a.e. and that they belong to L'(m).
For real numbers a, /> let Ey /, = [.v e X\ fjx) < /? and a < /*(x)]. Since (xUJx) < / *(.v)j = (J{/*&{!> < у and a, /) both rational} we shall show m(Ea ji) — 0 if l> < a because then we shall have/* = /* a.e. Clearly T"1 Eajt = EaJ and if we put Ba = {x e X |sup„-., (l/л) У>~^/(Т‘х) > a} then £a/J о Bx = £a p. From Corollary 1.16.1 we get
j/.„ = J,„ n/J„ felm - am(Eu: n = “'«(£«>•)•
Therefore _ / <//» > аш(
If we rcplace f, a, /> by -/, — fi, —a, re nectivcly. then since (—/)* = -/* and (-/\ = -/* wc get
| ^ f dm < РпфЕХш0).
Therefore am(/;ii(,i < lini(Exll), so if /f < a then m(Ey fl) - 0. This gives/* = /* a.e. as we explained above. Therefore (l/и) о /(Гх) —► /* a.e.
M.miu;
39
Го 41 low /'* с L'\iu) we use liie part of Faion's lemma lhal asserts lim c/„ e if [у,,] is a (poimwise) comergenl sequence of non-negative integrable functions with lim inf |(/„dm < x. Let gj.x) = |( 1 /л) У)'=(5/(7',(л'))|- Then | j. lldm so wc can apply the Fa ton lemma to assert lim„_ ^ г/„(л) = „(1 n) £"~l,flT'x)\ = |./ *| belongs lo L‘M It remains to show that | / dm = \ f*dm if m(X) < со. Let D'l = {x e X:(k n) < f :'(\i < Ik -t- 1)//;,’ where к e Z, n > 1. For cach omall с > 0 we have D'l n B[kiu)_,. = D’l and by Corollary 1.1C-1
Г j dm > ( — <: |шШ") so lhal Г f dm > - ni(Dl).
J,y£ \n J n
Then j ^ j
L I * dm < - - - /н(Щ) < - in(D'k) * L
л jj J"i<
by the above mequalii). Summing over к gives \ x f* dm < (m(X)/'n) + j, / dm. Since lln.s holds for all n > 1 we have fA J * dm < ] v f dm. Applying 111 is to — /'in.slcad of / gives | v (— /)* din < jA- —/ dm so that J.y/* dm > |A prim. Hut /, — /* a.e. so j A / * dm = |A f dm. This finishes the proof when m( \ i < cc.
When m(X) = со 'he above proof is valid once we have shown thai ///(£,.,,) < x. when (i < y. (Wc need to show this to apply Corollary 1.16.1.) Suppose firstly that a > 0. Let Ce J be any set with С с £a/J and m(C) < oc. (Such a sei exists bccause X i* c-finite.) Then h = / — e L‘(»3) so by the maximal ergodic theorem
f (I — я/с)^ 0 f°r all N > 1.
(The function Hv is ussociatcd to /i by Theorem 1.16.) But C^EuP<z Ijx‘ //Y(.v) > 0; and therefore \x\f\clm > can(C). Hence m(C) < (bx) JA \ f\dm for each С £/А with С с: Ea-lt and m(C) < oc. Since X is o-finite we have т(Е,л1.) < oc. If a < 0 then /? < 0 so we can apply the above with — / and —ji instead of/ and a to get m(EaiP) < cc. □
§1.7 Mixing
If Г is a measure-preserving transformation of a probability space we have deduced from the ergodic theorem that T is ergodic ifT V/l, В e
1 1
lim - Yj ‘A n B) = m(A)m(B).
II ~t Gf.j W |'=0
We can make changes in the method of this convergence to give the following notions.
40
1 Mca^uef-Prcscrving Tranbl'orm.iiions
DiTmiiiun 1.5. Let 7'be a measure-preserving transformation ofa probability space {X'.Sijn).
(i) T is wciik-n’ixiKt] if V.'l, В e Л
j и -1
lim - У \m(T~'A n B) — ш(/4)ш(В)| = 0.
n
(ii) T is struiuj-im King if V/l, В e 3#
lim m{T~"A n В) — т[А)тШ). h~* f
Remarks
(1) Every strong-mixing transformation is weak-mixing and every weak-mixing transformation is crgodic. This is because if {a,,} is a sequence of real numbers then lim„_ , «„ = 0 implies
1 1
lim - ^ lf/,-1 = 0
" ,*Го
and this second condition implies
1 1 i;m >_ iij = 0.
ii-.л n i - ()
(2) Am L.\ample of an crgodic transformation which is not weak-mixing is given by a rotation T(z) = a: on the unit circle K. This will be proved at the end of this section but one can sec it roughly as follows. If A and В are two small interval.s on К then T ‘A will be disjoint from В for at least half of the values of i so that (l/n) £"=(| \m(T~‘A n B) — m(A)m(B)\ > \m(A)m(B) for large ii. From this one sees that, intuitively, a weak-mixing transformation has to do .some “stretching.”
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