AN INFORMATION COCYCLE FOR GROUPS OF NON-SINGULAR TRANSFORMATIONS.
3.1. NOTATION AND BACKGROUND.
For convenience o f presentation, throughout th is section we shall use (X,S,m) to denote a Lebesgue p ro b a b ility space.
Unless we have reason to be p a r tic u la r ly careful we sh all Id e n tify functions which coincide m -a.e.. I f A and 8 are sub-o-algebras o f S we say that A « 8 (mod m) 1f sup 1nf m(AaB) - sup In f m(AaB) ■ 0 .
Ac A Be 6 Be8 AeA
In p a rtic u la r we choose and f i x a countably generated o-algebra SQ c s w ith Sq ■ S (mod m) and such th at (X,Sq) 1s a standard Borel space. For a given sub-a-algebra A o f S , A w ill stand fo r the measurable p a r tit io n o f X given by A , and {m*;xeX} the associated decomposition o f m In to conditional measures. Since we shall have to manipulate these o b je cts very c a re fu lly we sh all describe them 1n a l i t t l e more d e ta il.
We choose a countably generated o-algebra AQ c Sq w ith Aq ■ A (mod m) and define an equivalence re la tio n on X by c a llin g two points x^ .X j c X e q u iva le n t, 1f and o nly 1 f, fo r every A c Aq , x^ e A <■> Xg « A . The equivalence class o f a point x « X w ith respect to th is equivalence
A
re la tio n 1s denoted by [ x ] A , and A 1s defined as the measurable p a r tit io n { [ x ] A ;x«X> o f X . The sets [ x ] A , x < X , are referred to as the atom o f A . I f p 1s a p ro b a b ility measure on (X.S ) equivalent to m (p * m) , the p a rtitio n A Induces a decomposition o f p Into con d ition a l measures: there e xis ts a fa m ily {p * ;x«X } o f p ro b a b ility measures on (X ,S Q) w ith the fo llo w in g pro pe rtie s. 1
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(11) For every B « SQ , the map x -► p*(B) 1s Ag-measurable, and
P(B) - fp J(B )d p (x) . (3 .2 )
A A
The objects A and {p " ;x « X } are e ss e n tia lly unique and Independent o f the choice o f Sq and Aq . Furthermore, I f f t L ^ X .S .p ) , and 1f Ep(f| A ) denotes the conditional expectation o f f , given A , w ith respect to the measure p , then
Ep( f | A ) ( x ) ■ j f dpj p -a .e . . (3.3)
The uniqueness properties o f conditional measures Im ply th a t, fo r m -a.e. x « X , the measures p* and m* are e q u iva le n t, and
dPî . di
Ei r f |A) V a .e . . (3.4)
I f we denote by pA and mA the re s tric tio n s o f p and m (re s p e c tiv e ly ) to the o-algebra A , then E J^JjjlA) 1s equal to the Radon-N1kodym
dp,
d e riva tive ^ on (X ,A ) . In p a rtic u la r we g e t, f o r equivalent " a
p ro b a b ility measures rn.p.q on (X ,S ) ,
Ep $ l A>-E„ < & l A> ■ E. ( i l * ) •
(3.5)I f 8 e S 1s another sub-o~a1gebra, we define A v 8 to be the refinement (A n B;A < A , B t 8} .
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3.2. DEFINITION.
Let A,B be sub-a-algebras o f S , and le t p be a p ro b a b ility measure equivalent to m on (X ,S ) , then the o o n d itio n a l information
o f 8 , given A , w ith respect to the measure p , 1s defined by
Ip (8 | A )(x ): - - log p J ( [ x ] Av8) , xeX . (3.6)
3.3. Remark.
An a lte rn a tiv e , but equivalent, d e fin itio n o f conditional Information Is contained 1n [P33.
C le a rly , Ip(B|A) < - p -a .e . 1f and o n ly 1 f, fo r p -a .e . x c X , P^(CxDa v b) > 0 o r » e q u iv a le n tly , I f 8 p a rtitio n s p -a .e . element o f A (e s s e n tia lly ) Into a countable number of subsets. Another way o f expressing
AvB
th is 1s to say that p -a .e . measure p£ 1s absolutely continuous with respect to p j . In t h is case we have, fo r p -a .e . x c X ,
dpAv8
log - j - ( y ) " V 8 |A)(y) * (3,7)
dpx
fo r pA-a .e . y * Cx]Av8 • as a consoRu«1^ 6 o f th* uniqueness of conditional measures.
For notatlonal convenience and b re v ity , we shall make the follow ing d e fin itio n .
3.4. DEFINITION.
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A a sub-a-algebra o f S , we define
I(p .m lA ): - - lo g E j j j j l A ) .
The next re s u lt re la te s the conditional Inform ation functions a ris in g from two equ ivalen t measures.
3.5. LEM A.
Let (X,S,m) be a Lebesgue p ro b a b ility space, and p be a p ro b a b ility measure equ ivalen t to m . For sub-o-algebras A ,8 c S ,
Ip(8|A) - I n(B|A) ♦ I(p,m|Av8) - I(p,m|A) . (3.8)
Proof.
Combine equations (3 .4 ) and ( 3 .7 ),and use D e fin itio n 3.4.
□
We now In vestig a te how these expressions are transformed by a n on -sin gu lar endomorphism T o f (X,S,m) .
The stra igh tforw a rd proof o f the fo llo w in g lemma Is omitted.
3.6. LEMMA.
Let T be a n on -sin gu la r endomorphism o f (X,S,m ) , and l e t A be a sub-o-algebra o f S . For every measurable function f :X -*-F f o r which f » T 1s m -1ntegrable, we have
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Em("f«T |T_1 A) - E , ( f | A ) . T . mT
(3.9)
3.7. Remark.
I f A 1s a sub-o-algebra o f S , and T 1s a non-singular endomorphism, the atoms o f T'^A can be chosen to s a tis fy
[ x ] .1 - T_1[TxL . x e X , (3.10)
T 'A a
and th is together w ith equation (3.3 ) and Lemma 3.6, y ie ld s
(m T '^ T x “
t f V
•
( 3 *n )
f o r m -a.e. x < X .
3.8. LEMMA.
Let A,B be sub-a-algebras o f S , and p be a p ro b a b ility measure on (X ,S ) equivalent to m , then
a) I(p,m|T’ 1A) ■ I(pT"\m T~^ |A)*T
b) I_ (T* 18|T’ 1A) - I ,(8 | A )«T .
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r-1
V ^ J T - T I T''* 1 mT 1 dmT 1 as required.b) The follo w in g calcu lation proves the re s u lt.
Im(T ‘ 18|T*1A ) (x) T-1A ■ - lo g in ( [ x ] , , ) ; by D e fin itio n 3.2 x T 'AvT 'b T ^ A -1 ■ - lo g mx (T CTx]^vg) j from equation (3.10)
• - log (mT ^ ^ ( [ T x ] ^ ) ; using equation (3.11)
■ I _t(® IA)(Tx) mT
fo r m-a.e. x < X .
□
A COHOMOLOGY.
We sh a ll now describe a cohomology on a c e rta in equivalence re la tio n o f «-a lg e b ra s. This cohomology Is fundamental to the properties o f the Information cocycle th a t we sh a ll la te r In v e s tig a te .
3.9. NOTATION.
Let (X,$,m) be a Lebesgue p ro b a b ility space, and le t fl(S) denote the c o lle c tio n o f a ll sub-o-algebras o f S .
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3.10. DEFINITION.
Two o-algebras A ,8 e n(S) are ca lle d I-re la te d (denoted by A j'B ) 1f
V A | B ) ♦ I m(8|A) < - ro-a.e. .
3.11. PROPOSITION.
The re la tio n ^ does not depend on the choice o f m w ith in It s equivalence cla ss. Furthermore, f 1s an equivalence re la tio n on n(S) .
Proof.
The f i r s t assertion follo w s from Lenina 3.5. Symmetry and r e f le x l v lt y o f f are Immediate, and I t s t r a n s it iv it y 1s an easy consequence o f Remark 3 .3 .
□ A general recipe fo r con stru ctin g cohomologies from equivalence re la tio n s 1s contained In [FAM], and th is con stru ction can, o f course, be applied to the p a ir ( n ( S ) ,f ) . Rather than go In to th is 1n any g e n e r a lity , we sh a ll Instead consider one p a r tic u la r example.
3.12. NOTATION.
L e t (J(X,m,R) denote the a d d itive group (under polntwlse a d d itio n ) o f m-equlvalence classes o f measurable real valued functions on X . For every Integer n a 1 , define R ^ c n (S )n+^ by
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R(n) » {(A 0...An) < n (S )n+1 ; f Ak fo r k - 1... n> ,
and put ■ n(S) . Let C’ 1 ■ U (X,m Jl) and denote by Cn , n 2 0 , the group o f a ll maps from R ^ In to U(X,m,F) w ith polntwlse addition as composition. The group Cn 1s calle d the n -th cochain group. For n 2 0 we have coboundary operators i n_j : Cn"^ + Cn given by
( sn - l f ) i A 0*, , , , A n) " k^ " 1 )k f(A 0, , , ' ,Ak ...
V
’ where ^ Indicates the term to be om itted. As usual one puts z " : - ker a , Bn : - J C " ' 1) and Hn : - Zn/Bn fo r n 2 0 , and c a lls these groups the group o f n-oooyolaa, n-ooboundariaa and the n -th cohomology group, re s p e c tive ly. Our reason fo r Introducing these cohomology groups 1s th a t they provide the framework fo r the follow ing observation which 1s c ru cia l to the re s t o f th is section.Define, fo r every p ro b a b ility measure p % m , a function Kp : R^1) U(X,mJR) by
Kp(A ,B ): - I p(A|8) - I p(8|A) (3.12)
fo r every A.8 < n (S ) with A | 8 .
3.13. THEOREM.
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p ro b a b ility measure p on (X ,S ) w ith p m we have
Kp(A,8) ♦ Kp(8,C) - Kp(A,C) .
In other words, K « Z1 . P
Proof.
Kp(A,8) + Kp(8,C) - Kp(A ,C )
- Ip(A|8) + I p(C|Av8) - I p(C|8) - I p(A|8vC)
♦ I p(8|C) + I p(A|8vC) - I p(A|C) - I p(8|AvC)
+ I p(C|A) ♦ I p(8|AvC) - I p(8|A) - I p(C|Av8)
- I I (Av8vC|P) - I_(Av8vC|P)
P c{A ,8 ,C } P P
■ 0 .
Note that th is computation 1s v a lid since, fo r
A .8 .C c Q(S) , A | 8 | C » A v 8 | C .
a
He now show th a t Kp does not depend e s s e n tia lly on the p w ith in the equivalence class o f m .
(3.13)
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3.14. PROPOSITION.
For every A ,B c a (S ) w ith A ^ 8 ,
Kp(A.B) • K JA .B ) ♦ I(p.m|A) - I(p,m|8) . (3.14)
In other words, Kp and are cohomologous, t . e .
P roof.
Equation (3.14) follow s from Lemma 3.5. To see th a t the map sending the p a ir (A ,8) « R ^ to I(p,m|A) - I(p,m|8) is a coboundary, ju s t note that i t is the Image under o f the map th a t sends A c R ^
to I(p,m|A) . Q
3.15. Remark.
In passing we derive two more expressions which w i l l be useful la t e r . Using equation (3 .7 ), Kp can be w ritte n In terms o f conditional measures as
dp*
K (A ,8)(x)
- log- i ( x )
(3.15)
dpx
f o r m-a.e. x « X . Equation (3.15) gives another "q u ick" proof o f the cocycle equation (3.13).
For the next statement we assume T to be a n on -sin gu lar endo morphism o f (X,S,m) . Using Lemma 3.8 (p a rt b) together w ith equations