Chapter 0 Notation

CHAPTER 0

Notation 1

Notation from s e t theory The complement o f A i s denoted by [A o r more o f t e n A'. A

n Bc;

A A B i s t h e symmetric

some p r o p e r t y P i s denoted by { x

The n o t a t i o n A\B means

d i f f e r e n c e (A\B) U (B\A). The s e t o f a l l x E

E with

E

E : P ( x ) } o r , i f t h e r e i s no a m b i g u i t y , { x : P ( x ) l

o r simply { P I . The r e s t r i c t i o n o f a f u n c t i o n f t o a s e t A i s denoted by

flA.

S i m i l a r l y , ifd

G

i s a f a m i l y o f subsets, E l A i s t h e s e t o f t r a c e s on A o f elements o f e x p l i c i t l y elA = {B n A, B

E

:

&}.

2

C l o s u r e o f s e t s o f subsets

We sometimes use sentences o f t h e f o l l o w i n g f o r m : t h e f a m i l y E; i s c l o s e d under ( . . . ) ,

where t h e b r a c k e t s c o n t a i n s e t - t h e o r e t i c o p e r a t i o n symbols, sometimes

f o l l o w e d by t h e l e t t e r s f , c, a, m, which a b b r e v i a t e r e s p e c t i v e l y : f i n i t e , c o u n t a b l e , a r b i t r a r y , monotone. Two examples w i l l s u f f i c e t o c l a r i f y t h e i r meaning : " 8 c l o s e d under ( u f , n a ) "

%

means t h a t f i n i t e u n i o n s ( * ) o f elements o f 8 and a r b i t r a r y

i n t e r s e c t i o n s o f elements o f 8 s t i l l b e l o n g t o &;

"e i s

c l o s e d under (umc, ')I1 means

t h a t monotone c o u n t a b l e u n i o n s o f elements o f 8 ( i . e . u n i o n s o f i n c r e a s i n g sequences i n E ) s t i l l b e l o n g t o 8 and t h a t complements of elements o f

8

s t i l l belong

t o e . Sets o f subsets o r f u n c t i o n s a r e g e n e r a l l y denoted by c a p i t a l s c r i p t l e t t e r s . The c l o s u r e o f a f a m i l y o f subsets 8 under ( u c ) ( r e s p . ( n c ) ) i s denoted by ta (resp.g6)

-

t h i s n o t a t i o n i s c l a s s i c a l t o s e t t h e o r y . We w r i t e ( ( t ) a ) 6 =

ea6.

Lattice notation

3

L e t f and g be two r e a l - v a l u e d f u n c t i o n s . We w r i t e f v g and f and i n f ( f , g ) .

The n o t a t i o n f+ and f - has i t s c l a s s i c a l meaning : f

+

A

g f o r sup(f,g)

= f v 0, f - =

( - f ) v 0. More g e n e r a l l y ,

V,A

denate l e a s t upper and g r e a t e s t l o w e r bounds : f o r example, V

t h e a - f i e l d g e n e r a t e d by t h e u n i o n o f a f a m i l y o f a - f i e l d s 3, i s denoted by iTi.

( * ) Bourbaki i n c l u d e s under f i n i t e u n i o n s t h e "empty u n i o n " and s i m i l a r l y f o r i n t e r s e c t i o n s . We do n o t use t h i s convention.

PROBABILITIES

2-0

4

L i m i t s a l o n g R and M

+

The n o t a t i o n s

t means s

-+

t, s s t ; s f + t means s

+

t, s < t ; sn

+

t,

sn ++ti s used s i m i l a r l y f o r sequences ( s n ) , w i t h t h e a d d i t i o n a l meaninp t h a t ( s n )

+.The usual

i s i n c r e a s i n g . Obvious changes a r e r e q u i r e d i f J. appears i n s t e a d o f notations lim-,

lim inf

l i m inf,.

5

n-

f o r l i m i t s a l o n g M w i l l be w r i t t e n s i m p l y as l i m n ,

Integration theory The word measure w i t h o u t f u r t h e r qua1 i f i c a t i o n always means " p o s i t i v e (1) c o u n t a b l y a d d i t i v e s e t f u n c t i o n on an a b s t r a c t measurable space". We do n o t adhere t o t h e c o n v e n t i o n t h a t a l l measures c o n s i d e r e d a r e a - f i n i t e : t h i s would c o s t us t o o much g e n e r a l i t y i n p o t e n t i a l t h e o r y b u t w e ' l l c o n s i d e r o n l y c o u n t a b l e sums of bounded measures (no d i f f i c u l t i e s r e l a t i n g t o F u b i n i ' s theorem, f o r example,can'arise

with,

such measures). The n o t a t i o n c l a s s i c a l meaning.

X

IILJII

t

-

,'JL p - , l i i l = LJ + i-~ has i t s denotes t h e t o t a l mass <1u1.1> o f 1-1 (sometimes i n f i n i t e ) . The

v i-~ = sup(X,u),X

A

p = inf(A,p),

i n t e g r a l o f a f u n c t i o n f w i t h r e s p e c t t o a measure 1-1 i s denoted by o f t e n abridged i n t o j f p ;

5 denotes a Radon-Nikodym d e n s i t y ,

i

f(x)i-I(dx)(')

:

w i t h o u t " d " . However

when a measure p on R appears as t h e d e r i v a t i v e o f an i n c r e a s i n g f u n c t i o n F, we use t h e s t a n d a r d n o t a t i o n w i t h "d" f o r S t i e l t j e s i n t e g r a l s [ f ( x ) d f ( x ) , and i n p a r t i c u l a r J

I f ( x ) d x i f F ( x ) = x. I f 1-1 i s a p r o b a b i l i t y law, we o f t e n w r i t e E C f I f o r

( e s p e c i a l l y when A i s a c o m p l i c a t e d e v e n t ) .

6

I

f p and E C f , A l f o r j A f P

F u n c t i o n spaces IfE i s a t o p o l o g i c a l space,

e(E),

Cb(E), C c ( E ) ,

e0 (E)

denote t h e spaces o f

r e a l - v a l u e d f u n c t i o n s which a r e r e s p e c t i v e l y c o n t i n u o u s , bounded and c o n t i n u o u s , c o n t i n u o u s w i t h compact s u p p o r t , c o n t i n u o u s and t e n d i n g t o 0 a t i n f i n i t y ( t h e l a t t e r when E i s l o c a l l y compact). A d j o i n i n g a

+

t o t h i s n o t a t i o n (C+(E) and

SO

on) enables

m

us t o denote t h e c o r r e s p o n d i n g cones o f p o s i t i v e f u n c t i o n s . As u s u a l e C ( E ) denotes t h e space o f i n f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s w i t h compact s u p p o r t sumed t o be a rnanifeld, u s u a l l y Rn).

(E

t h e n i s as-

If (E,E) i s a measurable space, t h e n o t a t i o n T ( 6 ) ( r e s p . b ( E ) ) denotes t h e space o f 6-measurable ( r e s p . bounded 6'-measurable)

r e a l f u n c t i o n s . Spaces o f measures

a r e used o n l y on a Hausdorff t o p o l o g i c a l space E : d ( E ) , d ( E ) t h e n a r e t h e cones o f bounded ( r e s p . a r b i t r a r y ) Radon measures on

E and Wb(E), R ( E ) a r e t h e v e c t o r spaces

generated by t h e s e cones.

(1) P o s i t i v e means 20, a c c o r d i n g t o European use. ( 2 ) A l s o p ( f ) , .

AND POTENTIAL

3-0

7

Topol o g i c a l spaces A t o p o l o g i c a l space E i s s e p a r a b l e i f i t c o n t a i n s a c o u n t a b l e dense s e t . I t i s

c a l l e d P o l i s h i f t h e r e e x i s t s a d i s t a n c e c o m p a t i b l e w i t h i t s t o p o l o g y under which E i s complete and s e p a r a b l e . L o c a l l y compact spaces w i t h c o u n t a b l e base a r e c a l l e d

LCC spaces. We w r i t e 1 . s . c .

(u.s.c.)

t o abbreviate t h e expression lower (upper) semi-conti-

nuous. O r d i n a l s and t r a n s f i n i t e i n d u c t i o n

8

We c o n s i d e r arguments by t r a n s f i n i t e i n d u c t i o n as e x t r e m e l y c o n v e n i e n t and i n t u i t i v e and we use them f r e e l y . The f o l l o w i n g s e c t i o n c o n t a i n s e v e r y t h i n g o u r r e a d e r may need t o u n d e r s t a n d o u r p r o o f s . L e t ( J , s ) be some non-countable we1 1 - o r d e r e d

s e t w i t h a l a s t element

E.

Such a s e t " e x i s t s " : w e l l - o r d e r any u n - c o u n t a b l e s e t (axiom o f c h o i c e ) and a d j o i n E

t o i t i f necessary. F o r a l l a

(i.e.

5

E

J , l e t Ca be t h e s e t o f B

J such t h a t B

E

<

a

a, B # a).

The s e t o f a such t h a t Ca i s u n - c o u n t a b l e c o n t a i n s

E

by h y p o t h e s i s , i t i s t h e r e -

f o r e non-empty and hence has a s m a l l e s t element j o . L e t I = C . w i t h t h e o r d e r J induced by J . The o r d e r e d s e t I has t h e two f o l l o w i n g p r o p e r t i & : (1) i t i s w e l l - o r d e r e d and n o n - c o u n t a b l e ; (2) f o r a l l i

E

I , Ci

i s countable.

I t can be shown t h a t t h e s e two p r o p e r t i e s c h a r a c t e r i z e t h e o r d e r e d s e t I up t o

isomorphism, and i n p a r t i c u l a r t h a t I i s i s o m o r p h i c t o t h e s t a n d a r d s e t o f a l l count a b l e o r d i n a l s , b u t we do n o t need t o know t h a t . Having chosen I once and f o r a l l , we adopt t h e f o l l o w i n g t e r m i n o l o g y :

- t h e elements o f I a r e c a l l e d ( c o u n t a b l e ) o r d i n a l s ;

-

given a

E

I , i t i s v e r y easy t o see t h a t t h e s e t o f a l l B E I such t h a t CY < B i s We denote by a + 1 i t s s m a l l e s t element

non-empty ( o t h e r w i s e I would be c o u n t a b l e ! ) .

and we c a l l i t t h e successor o f a and a t h e p r e d e c e s s o r o f a determined as t h e l a r g e s t element o f { B : R

-

< IY.

+

1 ( i t i s uniquely

+ l} ;

t h e s m a l l e s t element o f I i s denoted by 0, we w r i t e q u i t e n a t u r a l l y 0

1+ 1

=

2, 2 + 1

=

3,

...

+

1 = 1,

The l e a s t upper bound o f t h e " i n t e g e r s " t h u s c o n s t r u c t e d i s

o f t e n denoted by w ;

- an o r d i n a l o f t h e f o r m a

+

1 i s o f t e n c a l l e d a f i r s t k i n d o r d i n a l and an o r d i n a l

# 0 w i t h o u t a predecessor a second k i n d o r d i n a l ( o r a l i m i t o r d i n a l ) . We now group i n a s i n g l e s t a t e m e n t some l e s s o b v i o u s p r o p e r t i e s o f I.

(1) R e c a l l t h a t a t o t a l l y o r d e r e d s e t i s w e l l - o r d e r e d i f e v e r y non-empty subset c o n t a i n s a s m a l l e s t element.

4-0

PROBABILITIES

LEMMA. a ) For every ordinal a

E

I , a # 0 , t h e r e e x i s t s a s t r i c t l y increasing mapping

R i such t h a t f ( 0 ) = 0 , f ( a ) = 1 . f of t h e i n t e r v a l C0,al of I W b ) Conversely, f o r every increasing mapping f of t h e whole of I t h e r e e x i s t s an ordinal y such t h a t f ( 8 ) = f ( y ) f o r a l l 8

2

R~

k

y.

c ) For every l i m i t ordinal a, t h e r e e x i s t s an increasing sequence of o r d i n a l s a , < a such t h a t a = supnan.

Proof : a ) Let A be t h e s e t of o r d i n a l s 4 # 0 such t h a t no such mapping f of C0,81 i n t o R e x i s t s . I f A i s non-empty, i t has a smallest element a. Clearly a cannot have a predecessor and a i s n o t 0 . Hence a i s a l i m i t o r d i n a l . For every 8 < a, l e t f B be a s t r i c t l y increasing mapping of C0,81 i n t o R such t h a t f 8 ( 0 ) = 0 , f ( a ) = 1 B and l e t g be t h e mapping of C0,al i n t o R equal t o f a on C0,Bl and t o 1 on 18,al. B Since t h e r e a r e countably many p < a, t h e r e e x i s t numbers E a l l s t r i c t l y positive,

8'

such t h a t C B ~ 8= 1. Then t h e function 1 E g s a t i s f i e s t h e statement on CO,al, 888 contradicting t h e d e f i n i t i o n of a. Hence A i s empty.

1

b) Let A = s u p f ( 8 ) . For a l l n l e t a n be such t h a t f ( a n ) > A - ; ( f ( a n ) > n B i f A = + m ) and l e t y = supnan. Then f ( R ) = f ( y ) = A f o r a l l 8 2 y . c ) F i n a l l y , l e t a be a l i m i t ordinal and l e t f be a s t r i c t l y increasing mapping take an In follows

I t suffices t o t o be t h e s m a l l e s t ordinal B such t h a t f ( 8 ) > c - 1. n na'ive language, t h e " p r i n c i p l e of t r a n s f i n i t e induction" can be s t a t e d as : l e t P(a) be a "property of an ordinal a''( i n o t h e r words, a subset A of I :

o f C0,al i n t o a bounded i n t e r v a l of R . Let c = supp,,.f(B).

A ) , such t h a t (1) i f P i s true f o r a, P i s true f o r a + 1, ( 2 ) i f B i s a l i m i t ordinal and P i s t r u e f o r every ordinal for B. P(a) i s t r u e f o r a i f and only i f a

E

ct <

8, P i s t r u e

(3) P i s true for 0. Then P i s true for a l l a

E

I . This i s obvious : Let A be the s e t of a l l a f o r which

P(a) i s not t r u e . I f A i s non-empty, A has a f i r s t element which has no predecessor

(l), i s not a l i m i t ordinal ( 2 ) and i s n o t 0 ( 3 ) . This i s absurd, hence A must be empty. This " p r i n c i p l e " a p p l i e s of course t o every well-ordered s e t and not only t o I . T r a n s f i n i t e induction can be used i n a s l i g h t l y d i f f e r e n t way, t o construct a function f on the set of a l l countable o r d i n a l s . One then a p p l i e s the above argument, P(a) being t h e property " t h e r e e x i s t s on the i n t e r v a l C0,alone and only one function f such t h a t . . . " , and t h e conclusion being " t h e r e e x i s t s on t h e whole of I one and only one function f such t h a t . . . " . Usually t h e dots . . . represent induction r u l e s , defining f ( a ) when f ( B ) i s known f o r a l l < a. 9

F i n a l l y , a word about t h e ( r e s t r i c t e d ) "continuum hypothesis". I t i s known t h a t t h e following axiom:

,

5-0

AND POTENTIAL

I has t h e power o f t h e continuum ( i n o t h e r words, t h a t i t i s p o s s i b l e t o "enumerate" by means o f I t h e p o i n t s o f R, t h e sequences of i n t e g e r s , ...) i s independent o f t h e usual axioms o f s e t t h e o r y . U n t i l now, t h e a d o p t i o n o r r e j e c t i o n o f t h i s axiom has been s i m p l y a m a t t e r o f t a s t e , w i t h no r e a l l y u s e f u l r e s u l t o f a n a l y s i s depending on i t . We s h a l l see below s e v e r a l v e r y b e a u t i f u l consequences o f t h e continuum h y p o t h e s i s (due t o Mokobodzki), which l e a d us t o a d o p t i t i n t h i s book w i t h t h e same s t a n d i n g as t h e axiom o f c h o i c e . See however t h e comments.