The Alexandroff-Le Cam theorem

S t r i c t l y s p e a k in g t h i s i s a th e o re m a b o u t weak c o n v e rg e n c e , b u t i t i s in c lu d e d i n t h i s c h a p t e r b e c a u s e th e p r o o f r e l i e s h e a v i l y upon r e s u l t s from S e c t io n 6 . The t r i c k n e e d e d i s s i m i l a r t o one u se d i n Example 6 . 1 .

L e t

X

be an a r b i t r a r y t o p o l o g i c a l s p a c e , and d e n o te th e cone o f b o u n d e d , c o n tin u o u s , n o n - n e g a tiv e f u n c ti o n s on

X

by

C

. Suppose a se q u e n c e {21 } o f p o s i t i v e l i n e a r f u n c t i o n a l s on

C

i s g iv e n , h a v in g th e p r o p e r t y t h a t th e se q u e n c e h ) } i s c o n v e rg e n t t o some r e a l num ber f o r e v e ry h €

C

. I t i s t r i v i a l t o show t h a t t h i s l i m i t d e f in e s a p o s i t i v e l i n e a r f u n c t i o n a l T on

C

by

T ( h) = lim T (h) f o r eac h h £

C .

n n-*°°

By v i r t u e o f Example 5 . 2 , eac h o f t h e s e l i n e a r f u n c t i o n a l s can be r e p r e s e n t e d as an i n t e g r a l w rt a f i n i t e l y a d d i t i v e B a ir e m easure w hich i s i n n e r r e g u l a r w rt th e p a v in g o f z e r o s e t s o f

X

. In a d d i t i o n , as Example 6 .3 h a s show n, i f th e l i n e a r f u n c t i o n a l s a r e a-sm o o th a t 0 th e n th e

representing measures can be extended to G-additive Baire measures. Now suppose that each of the T ’s is G-smooth at 0 . Then it might not be expected that anything could be deduced about the smoothness properties of T . Indeed any positive linear functional T can be expressed as the limit of a net of G-smooth functionals, so why should the statement for sequences be any different? But surprisingly enough, there is a difference!

THEOREM

7010

If each of the T

’s

is o-smooth at

0

then so n

is T . □

[The theorem is also true without each of the s being positive. We make this restriction to avoid undue complications, although our proof can be modified to cover the general case too.]

A version of this result was first proved by Alexandroff (1943) in a somewhat involved manner. Le Cam (1957) followed up with a sophisticated proof of the theorem in functional analytic form. The crux of his proof is a category argument similar to the one we shall give. By introducing the concept of a regular sequence of zero sets, Varadarajan (1965) was also able to further polish the result. His method is based on category techniques too.

A puzzling aspect of all these proofs is that measure theoretic ideas are needed to prove a purely functional analytic result. This has

parallels in other areas of functional analysis (cf, Dunford & Schwartz (1966, IV.6.4)), although Fremlin (1974) has shown that some of these can be circumvented. It would be nice if a non-measure theoretic proof could be given for Theorem 7.1, but so far the closest approach to this seems to be in Tortrat's (1971a, p. 289) modification of Varadarajan’s approach. Unfortunately his proof contains an error.

The method we use is based on a reworking of Tortrat’s, with the elimination of the use of regular sequences. We have been unable to completely avoid measure theoretic arguments though.

Proof of Theorem 7.1„ We shall prove that T is G-smooth at

0

wrt the paving of zero sets. If {Z^} is a decreasing sequence of zero sets with empty intersection, then there is a decreasing sequence

of

C

functions with 1 > h T O and Z = h ^(1) . Let g = hn . It

s u f f i c e s t o show t h a t T [ g ^ 0 as n 00 . [The r e s t o f th e argum ent u s e s no m easure t h e o r y . ] We p ro v e t h i s by m aking u se o f th e f a c t t h a t

\\g . ( l-g-A || 0 as n -*■ 00 , f o r eac h f i x e d k . [H ere | | . || d e n o te s th e

Yl K. u s u a l sup n o rm .] To s e e t h i s , n o t i c e t h a t on th e s e t { l- g^ > e} , yl/ Jc q < ( l - e ) w hich te n d s t o z e r o as n -+ 00 . L e t S = {h £ C : 0 < h < 1

}

. D e fin e a m e tr ic d on S v i a th e norm 00 v p ( f ) = I 2" I]/, [ l - g ) || ; fc=1 K t h a t i s , s e t d ( h ^, h^) = p .

Assume f o r th e moment two f a c t s , w hich w i l l b e p ro v e d l a t e r : (<S, d) i s a c o m p le te m e tr ic s p a c e ; and eac h o f th e maps h i—* T (7i)

(n = 1 , 2 , . . . ) i s c o n tin u o u s u n d e r t h i s m e tr ic to p o lo g y . Thus f* = {?2 £ 5 : |T (h ) - T ( h ) \ 5 e f o r a l l m > o} d e f in e s an n 1 1 n n+m 1 J i n c r e a s i n g se q u e n c e o f c l o s e d s u b s e t s o f S , whose u n io n c o v e rs S [ s in c e t h e c o n v e rg e n t se q u e n c e [ T^( h) } i s C au ch y ]. A p p ly in g th e B a ir e c a te g o r y th e o re m ( K e lle y (1 9 5 5 , p . 2 0 0 )) shows t h a t t h e r e e x i s t s an h £ S 9 6^ > 0 and an i n t e g e r n^ su ch t h a t : h £ S and d [ h , h^] < 6q

im ply t h a t h £ . As a c o n se q u e n c e : h £ S and

d [ h 9

h^) < 6 Q im ply

0

t h a t \T ( h ) - T ( h )| 5 e .

0

The i d e a now i s t o show so m e th in g l i k e dlfi^+g * < 6 Q e v e n t u a l l y , from w hich i t w ould f o llo w s t h a t | [g^] -T[g^\ | < 2e f o r a l l l a r g e

enough n . Hence lim su p T[ g ) < lim su p T [g ) + 2e = 2e . But t h e r e

_ _ Yl r. Yl

Yl Yl 0

i s no g u a r a n t e e - t h a t + g i s i n S [ t h i s i s th e fla w in T o r t r a t ' s p r o o f ] , so we m ust be a l i t t l e more c ir c u m s p e c t. The t r i c k i s t o c o n s id e r i n s t e a d th e two S f u n c t i o n s h = V q and h' = hn\ q

d K > h <) = p K - ' k

<j

= ) - p b j oo = J.?! 2’fe|lan^ 1' ^ 11 0 as n -*■ 00 , it follows that

|^n

[hn]-T

(ft ) I < e for large enough n . 0 Similarly Thus

d{\, K)

-

= ph0 a

gn)

-

p h n )

\Tn

~

£ for large enouSh

n

• Now observe that

h - h' - g

, so that

n

n ^ n

\Tn [gn)-T[gn)

I < 2e for large enough

n

.

It follows that limsup

T[g )

5

2c

, as before. This concludes the proof

n

of the a-smoothness of

T

> subject to the two assumptions made earlier. Consider the completeness of (s,

d)

first.

It is routine to verify that J is a metric on

S

. For the completeness, consider any ^-Cauchy sequence {fo } in

S

» i.e.

p[h -h ) ■+

0 as

n , m

°o .

Then for each fixed

k

,

* ' n mJ

II (hn~hn) * ^-~gk) II * 0 as

1719 n *

00 *

(7,1)

Now for each

x £

X ,

g^(x)

1 0 as

k -*■

00 . This means that there is a

neighbourhood

N^

of

x

of the form { l - g ^ ^ > %} for some

k{x)

. Thus

sup \hn W - h mW \£ 2 | | ( ^ m) . ( i - £?fe(x)) || . ^ 1 X The s e q u e n c e {/z^} i s t h e r e f o r e u n if o r m ly c o n v e rg e n t on a n e ig h b o u rh o o d o f e a c h x ; so i t i s p o in tw is e c o n v e rg e n t t o some h £ S . So f o r e a c h f i x e d k , h [ l - g ^ j i s p o in tw is e c o n v e rg e n t t o th e c o n tin u o u s f u n c t i o n h.{±-Qfr) • B ut from ( 7 . 1 ) , th e se q u e n c e {/z^. (l- g ^ J : n - 1 , 2 , . . . } i s u n if o r m ly c o n v e r g e n t; th u s || [h^-h] . || -> 0 as n -> 00 , f o r eac h f i x e d k . Hence <i(/z^, /?) *> 0 , and so ( £ , d) i s c o m p le te .

F i n a l l y , f o r t h e c o n t i n u i t y o f th e map h i—► T^(?z) f o r e a c h f i x e d n , we n e e d t o u se t h e a -s m o o th n e s s o f eac h T . Choose k su ch t h a t n ^rS^k) < e anc^ a ^ > 0 so t h a t 2 ^ .6 < e . Then f o r /z^, £ S , ^2^ = < Ö imPl i e s t h a t II ^ l - ^ * H £ * Thus

< e . y i ) + 2Tn (3fe)

and th e r e q u i r e d c o n t i n u i t y f o ll o w s . □

8.

Further remarks

1. As S r in iv a s a n (1 9 5 5 ) re m a rk e d , th e i n n e r m easure a p p ro a c h to th e c o n s t r u c t i o n o f m e a su re s was lo n g c o n s id e r e d u n w o rk a b le . T h is i s q u i t e t r u e in one s e n s e . For by m e re ly m im icking c l a s s i c a l C a ra th e o d o ry o u t e r m easu re te c h n iq u e s we c a n n o t hope t o a r r i v e a t a c o u n ta b ly a d d i t i v e

e x t e n s i o n . Von Neumann (1 9 5 0 , p . 25) had a l r e a d y n o t i c e d t h i s ; he p o in t e d o u t t h a t t h e c o u n ta b le s u p e ra d d it'iv 'tty o f i n n e r m e asu re s c a n n o t b e u s e d as a s u b s t i t u t e f o r o u t e r m easure c o u n ta b le s u b a d d i t i v d t y. In d e e d t h e s u p e r ­ a d d i t i v i t y i s j u s t a co n se q u e n c e o f f i n i t e a d d i t i v i t y . To d e v e lo p th e th e o r y f o r i n n e r m e asu re s e f f e c t i v e l y g r e a t e r u se m ust be made o f th e

i n n e r r e g u l a r i t y and "sm o o th n e ss a t 0 " p r o p e r t i e s , a s we showed in S e c tio n 2 . T h is i s b o th th e s t r e n g t h and th e w eakness o f th e m eth o d , s i n c e i t means t h a t g r e a t e r s t r u c t u r e can be o b ta in e d b u t i n a l e s s g e n e r a l s e t t i n g th a n f o r o u t e r m e a su re s. More p r e c i s e l y , f o r f i n i t e m e asu re s th e two a p p ro a c h e s a r e v i r t u a l l y e q u i v a l e n t , b u t f o r th e i n f i n i t e c a s e some s o r t o f l o c a l f i n i t e n e s s i s n e e d e d b e f o r e i n n e r r e g u l a r i t y becom es a w o rk a b le p r o p o s i t i o n .

I n t e g r a l r e p r e s e n t a t i o n t h e o r e m s a r e t a i l o r made f o r s u c h a t h e o r y t h o u g h . 2. The R i e s z r e p r e s e n t a t i o n h a s a l o n g a n d f a s c i n a t i n g h i s t o r y w h ic h i s u s u a l l y t a k e n t o d a t e fro m R i e s z ' s i n i t i a l c o n t r i b u t i o n i n 1 9 0 9 ,

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