C o n v e x m e a s u r e s o n l o c a l l y c o n v e x spaces
CHRISTEI~ BORELL
1. Introduction
T h e p u r p o s e of this p a p e r is n o t to give a c o m p l e t e t r e a t m e n t o f so-called c o n v e x m e a s u r e s b u t m e r e l y to p o i n t o u t a n e w t e c h n i q u e . I t will be p r o v e d t h a t a Gauss m e a s u r e is a c o n v e x measure, a n d using t h e i n e q u a l i t y (Is) below, we get v e r y n e a t proofs o f results a l r e a d y k n o w n for Gauss measures. On t h e o t h e r h a n d o u r results a p p l y to all c o n v e x measures.
T h r o u g h o u t this p a p e r E d e n o t e s a locally c o n v e x t i a u s d o r f f v e c t o r space, a n d ~ ( E ) is t h e set o f all B o r e l sets in E . G i v e n a B o r e l p r o b a b i l i t y m e a s u r e # on E we d e f i n e t h e i n n e r /t-measure # . ( A ) o f a subset A o f E b y
# . ( A ) = sup { # ( K ) I K c o m p a c t ~ A}.
W e shall s a y t h a t # is a R a d o n p r o b a b i l i t y m e a s u r e if # . ( A ) --~ #(A) for all A E ~ ( E ) . I t is k n o w n t h a t all B o r e l p r o b a b i l i t y measures on E are R a d o n p r o b a b i l i t y measures, if E is a Souslin space. (See e.g. [3, p. 132].) F u r t h e r , we d e f i n e M ~ ( u , v ) = (~u ~ + ( 1 - - ~)vS) 1/~, - - ~ < s < 0 , ~ m i n ( u , v ) , s ~ - - ~ , a n d u~v I ~, s ~-- O, for u, v > 0 . H e r e 0 ~ 0.
Definition 1.1. L e t # be a R a d o n p r o b a b i l i t y m e a s u r e on E a n d a s s u m e s E [ - - ~ , 0]. T h e n we shall s a y t h a t # belongs t o t h e class ~ , ( E ) if t h e i n e q u a l i t y
~ , ( ~ A + (1 - - ~)B) >_ M~(~(A), ~(B)) (L)
holds for all 0 < 2 < 1, a n d all A, B E ~ ( E ) .
A m e a s u r e belonging t o t h e class TX_+(E) will be called a c o n v e x m e a s u r e . B y definition, hA -~ (1 - - 2)B = {2x + (1 - - ~)ylx E A , y E B}. N o t e t h a t this set n e e d n o t be a Borel set e v e n if A a n d B are so. (See [7].) N o t e also t h a t ~,~(E) ~ ~ , ( E ) , if s 1 > s2, a n d t h e r e f o r e t h e class ~ _ + ( E ) is t h e largest one.
I n a r e c e n t p a p e r [4] t h e a u t h o r gives a c o m p l e t e d e s c r i p t i o n o f t h e classes 9X,(R")( - ~ < s < 0) a n d for t h e r e a d e r s c o n v e n i e n c y we r e c a p i t u l a t e t h e r e s u l t here
2 4 0 C I I R I S T E R B O R E L L
T~EORE~ 1.1. A Radon probability measure it on R n belongs to the class ~ ( R n) i f and only i f there are an integer k, 0 < k < n, a Radon probability measure on R k, absolutely continuous with respect to Lebesgue measure ink, and an affine m a p p i n g h: Rk--+ R n such that # = rh -1 and such that the function e,,k(dv/dmk)
i 8 c o n v e x , s
H e r e e~,k(u) = u -lj~, s = - - ~ , = u 1 ~, - - ~ < s < 0 , a n d ~ - - l o g u , s = 0 , for u _ ~ 0 , k > 0. W e recall t h a t a c o n t i n u o u s m a p p i n g h : E - - ~ F , for e a c h B o r e l p r o b a b i l i t y m e a s u r e r on E , i n d u c e s a Borel p r o b a b i l i t y m e a s u r e
~h -1 on F (a t o p o l o g i c a l space), d e f i n e d b y rh-l(C) = v(h-l(C)), w h e n C E ~ ( F ) . N o t e t h a t vh -~ is a R a d o n p r o b a b i l i t y m e a s u r e i f v is so.
T h e " i f " p a r t o f T h e o r e m 1.1 is r a t h e r e a s y to p r o v e a n d let us n o w b r i e f l y i n d i c a t e h o w t h i s c a n be done.
Proof of Theorem 1.1. U s i n g L e m m a 2.1 below, it is r e a d i l y seen t h a t we o n l y n e e d to consider t h e case w h e n # = fm~ a n d es,=(f) is a c o n v e x f u n c t i o n on R ~.
A n a p p l i c a t i o n o f H S l d e r ' s i n e q u a l i t y (several t i m e s ) t h e n yields
n n n
f ( ~ x + (1 - - 2)y) ~ (Xa~ @ (1 - - X)bi) ~ M~(f(x) " H a , f ( y ) - ~ b~) ( ] . l )
1 1 1
for all x, y E R N, al . . . . , a , , b x . . . b ~ > 0, a n d 0 < ~ < 1. L e t ~ be t h e f a m i l y of all f i n i t e disjoint u n i o n s o f c o m p a c t n o n - d e g e n e r a t e d n - d i m e n s i o n M i n t e r v a l s , w i t h sides parallel to t h e c o o r d i n a t e axes, a n d w h i c h are i n c l u d e d in t h e i n t e r i o r of s u p p (f). I t is e n o u g h to s h o w t h e i n e q u a l i t y (L) w h e n A, B E ~ a n d 0 < ~ < 1.
L e t n ( A ) be t h e n u m b e r o f disjoint i n t e r v a l s d e f i n i n g A E ~. S u p p o s e f i r s t t h a t t h e i n e q u a l i t y (Is) is t r u e w h e n n ( A ) + n(B) ~ p, w h e r e p _~ 2 is a f i x e d n a t u r a l n u m b e r . W e will t h e n p r o v e t h a t t h e i n e q u a l i t y (Is) is still t r u e w h e n n ( A ) + n(B) -~
p @ 1. T o see t h i s we c a n a s s u m e t h a t n ( A ) > 2. W e t h e n choose a h y p e r p l a n e x~ = c, w i t h a n o r m a l v e c t o r p a r a l l e l to t h e i : t h basis v e c t o r , so t h a t n ( A ' ) < n ( A ) a n d n(A") < n ( A ) , i f A ' = A Cl {xi < c} a n d A" = A N {xi > c} r e s p e c t i v e l y .
Set
~,(A') --~ O~(A), N A " ) = (1 -- O)~(A), (1.2) w h e r e 0 < 0 < 1.
Since # << m,, we c a n f i n d a h y p e r p l a n e x~ ~ d so t h a t
tt(B') = O#(B), /~(B") = (1 - - O)tt(B), (1.3) w h e r e B ' - - B N { x ~ d } a n d B " = B f l { x ~ > _ d } .
U s i n g t h a t t h e i n e q u a l i t y (L) is t r u e w h e n e v e r A, B E ~ a n d n(A) + n(B) < p, we g e t
#(2A @ (1 - - ).)B) > #(;tA' + (1 - - ~t)B') U (2A" + (1 - - ;t)B") =
# ( 2 A ' + (1 - - 2)B') + #(2A" @ (1 - - 2)B") > M~(#(A'), tt(B')) + M~(#(A"), #(B")) OM~,(#(A), #(B)) + (1 - - O)M~(/t(A), #(B) = M~,(#(A), #(B)), 0 < ~ < 1.
C O N V E X M E A S U R E S O N L O C A L L Y C O N V E X S P A C E S 241 F r o m now on let 2, 0 < 2 < 1, be fixed. I t o n l y remains to be proved t h a t t h e inequality (I~) is t r u e w h e n A, B E ~ a n d n ( A ) = n ( B ) = 1. I f n o t there is a positive n u m b e r e such t h a t
#~(2A @ (1 -- 2)B) < M~(#(A), #(B)), (1.4)
where #, = ( f + o(e))mn, a n d
~o(e) = sup If(x) -- f(y)].
ix yl <~"
x, y e ~ t A + ( 1 2 ) B
Now let us divide A into two congruent intervals A ' = A N {x~ _< c} a n d A" = A N {x I > c}, respectively, a n d choose 0 so t h a t (1.2) holds. I n t h e n e x t step we choose d such t h a t ( 1 . 3 ) h o l d s w i t h B' = B g l { x I <_d} a n d B " = B fl {x 1 >_ d}. Using t h e same technique as above, we conclude t h a t the i n e q u a l i t y (1.4) c a n n o t be wrong for b o t h the pairs (A', B') a n d (A", B"). B y repetition we conclude t h a t
~ ( 2 C + (1 -- 2)D) < ~i~(~(C); ~(D))
where C ~ A, D _c B, a n d where diam (C) a n d d i a m (D) can be m a d e arbitrarily small. Setting
n t~
C = ] ~ [ u ~ - - ( 1 / 2 ) a , u ~ @ (1/2)a~], D = I I [ v - - (1/2)b~,v,-~ (1/2)b~], ( a , b ~ > 0),
1 1
we conclude t h a t there are points x E C , y E D , a n d z C 2 C + ( 1 - - 2)D such t h a t
n n n
(f(z) + ~o(s)) I--~ (2a~ § (1 -- 2)b,) < M~(f(x) - ~ a,,f(y) ~ b,).
1 1 1
B y choosing C a n d D small enough, we have f(z) @ og(e) ~_f(2x ~ (1 -- 2)y), a n d we have got a n i n e q u a l i t y opposite to (1.1). This contradiction proves t h a t t h e i n e q u a l i t y (L) m u s t be t r u e when A, B E ~ a n d n(A) 4- n(B) = 2. This proves the "if" p a r t of Theorem 1.1.
2. Characterization of the classes ~ l t s ( E ) ( - - oo < s < 0)
Suppose ~1 . . . G E E', t h e topological dual of E. W e shall write #h -1 =
w h e n /~1. 9 Z ~ '
h(x) = (~l(x), . .., G(x)), x E E. (2.1)
THEOIiEM 2.1. A Radon probability measure # on E belongs to the class ~J~s(E) if and only if #21...~n E ~J~(R n) for all ~1 . . . G E E', and all positive integers n.
242 C t t l : t l S T E I ~ B O I ~ E L L
W e n e e d
LE~MA 2.1. Let E and F be locally convex Hausdorff vector spaces and let h: E --> 2' be a continuous linear mapping. Then #h -1 E ?~(F), i f # E ~ ( E ) .
Proof of Lemma 2.1. W e h a v e
Xh-~(C) § (1 - - ~)h-~(D) = h-~[2(C N h(E)) § (1 - - X)(D N h(E))] (2.2) for all 0 < 2 < 1 , a n d all C, D c F .
U s i n g t h a t (#h-1),(C)~ tt,(h-l(C)), C E F, t h e p r o o f of L e m m a 2.1 follows a t once f r o m t h e i n e q u a l i t y (L).
Proof of Theorem 2.1. L e m m a 2.1 p r o v e s t h e " o n l y i f " p a r t . T o p r o v e t h e "if"
p a r t let h be o f t h e f o r m (2.1). T h e a s s u m p t i o n s a n d t h e i d e n t i t y (2.2) t h e n yield
#,(,~h-~(C) § (1 - - 2)h-~(D)) ~ M~(#(h-~(C)), #(h-~(D))) (2.3) for all 0 < ~ < 1, a n d all C, D E ~ ( R " ) .
N o w choose 2, E [0, 1], ~nd c o m p a c t sets A a n d B in E . H o l d i n g ~ , A , a n d B fixed, we shall p r o v e t h e i n e q u a l i t y (I,), w h i c h will p r o v e t h e t h e o r e m . To this e n d let 0 be a n o p e n set c o n t a i n i n g ~A + (1 - - ~)B. Since 2A § (1 - - ~)B is c o m p a c t t h e r e is a n o p e n c o n v e x n e i g h b o u r h o o d V of t h e origin so t h a t
O p e r a + ( 1 - - ~ ) B + 2V.
F u r t h e r , choose Xl . . . . , x~ (~ _/1 a n d y1 . . . . , y~ E B such t h a t
nt n
O (x,§ V) 2 A , U ( y j § V) ~__B, (2.4)
1 1
a n d set
F = U (~x, § (~ - X)yj + 17).
i , ]
N o t e t h a t 0 2 F . F o r each z ~ F , i C { 1 , . . . , m } , a n d j E ( 1 . . . n}, t h e H a h n - B a n a c h s e p a r a t i o n t h e o r e m gives us a kqz C R, a n d a ~q~ E E ' such t h a t
z ~ ~x, + (1 - ~)yj + ~,jol([ki~., § ~ [ ) , ~jo ([~jo, + ~ [ ) 2 17. -1 (2.5) Set
- 1
"~z = U (~Xi -~ ( 1 - - ~)yj ~- ~ijz ([]Cijz, -~ 0 0 D ) "
i , j
Clearly, F = f'lz ~ F Fz,
/~(F) ---- i n f #(F,~ N . . . N F ~ ) . (2.6)
z 1 . . . Z p ~ F p E Z +
a n d since # is a R a d o n p r o b a b i l i t y m e a s u r e we h a v e
F u r t h e r m o r e , it holds t h a t
CON~EX ~ E A S U ~ E S ON LOCALLY CONVEX SPACES 243
~,j i,j,r
= ' ~ [ U (Xl -Jr- f~ ~jzXr([~ijZr, ~- 00[)2 -t- (1 - - ~ ) [ U (Yj -~- ~ ~Zlr(El~ijZr , -~ 0 0 [ ) ] .
i i,j,r j i,j.r
t I e r e t h e r i g h t - h a n d side is o f t h e f o r m
2h-a(C) + (1 - - 2)h-~(D)
C, D E 1t ~'~, a n d h ~ (E') m"e. T h e e q u a t i o n s (2.3)--(2.6) t h e r e f o r e for suitable
y i e l d
tt(O) > Ms(#(A), #(B)).
Since 0 is a n a r b i t r a r y o p e n set c o n t a i n i n g 2A + (1 - - 2)B, we get t h e i n e q u a l i t y (I,). This p r o v e s T h e o r e m 2.1.
A R a d o n p r o b a b i l i t y m e a s u r e # on E is said t o be a Gauss measure, if #~
is a Gauss m e a s u r e on R for all ~ C E ' . T h e set of all Gauss measures on E will b e d e n o t e d b y ~ ( E ) . Assuming t h a t # is a Gauss measure, it is well k n o w n t h a t
#~1--. r E ffr ") for all ~1 . . . . , ~n E E ' , a n d all positive integers n.
T h e o r e m s 1.1 a n d 2.1 t h u s give COROLLARY 2.1. ~ ( E ) C_ ~YJ~0(E).
I n particular, t h e W i e n e r m e a s u r e W on C[0, 1], e q u i p p e d w i t h t h e u n i f o r m t o p o l o g y , satisfies t h e i n e q u a l i t y (I0).
N o w a d a y s t h e r e is no real p r o b l e m to c o n s t r u c t R a d o n p r o b a b i l i t y measures o n o u r m o s t i m p o r t a n t v e c t o r spaces. W e recall t h e f u n d a m e n t a l t h e o r e m s due t o K o l m o g o r o v , Minlos, a n d Sazonov. (See e.g. [16], [15], a n d [21].) T h e so-called R a d o n i f y i n g mappings, i n t r o d u c e d b y L. S c h w a r t z (see e.g. [2]), also m a k e i m p o r t a n t c o n t r i b u t i o n s t o this area.
To check t h a t a given m e a s u r e # belongs to ~J~(E), it is n o t n e c e s s a r y to p r o v e t h a t #sl...en belongs t o g)~s(R n) for all ~ 1 . . . ~n E E ' , a n d all n E Z + . R a t h e r t h a n giving a general t h e o r e m , we will illustrate this in a few examples. Suppose tt is a l~adon p r o b a b i l i t y m e a s u r e on RZ+, e q u i p p e d w i t h t h e p r o d u c t t o p o l o g y , a n d let H , : RZ+ - + R ~ be t h e n a t u r a l p r o j e c t i o n . W e claim t h a t # C ~J~,(RZ+), i f
/~H21 C
g2~(R n) for all n C Z+. T o see this, lete,,(x)
= xn, x = {xn}[, a n d n o t e t h a t each b o u n d e d linear f u n c t i o n a l on Rz+ is a f i n i t e linear c o m b i n a t i o n o f t h e % Therefore, given $1 . . . . , ~n EE',
t h e r e are m E Z+, a n d a linear m a p p i n g h: R m - + R ~ such t h a t_ h-1 ( # H 2 l ) h -1.
~ 1 . . " ~n - - [~1... ~,~ ="
Using L e m m a 2.1, we conclude t h a t /~1 ..-~, E ~J~(Rn). This p r o v e s t h e assertion.
N o w let # be a R a d o n p r o b a b i l i t y m e a s u r e on a separable H i l b e r t space H , a n d let {e~}~ ~ be an o r t h o n o r m a l basis in H . W e claim t h a t # E ~J~(E), if
244 C HI:~I S T E I : t B OI:~ELL
# ~ . . . . ~ ~ Tdt~(R ~) for all p o s i t i v e i n t e g e r s n. As a b o v e , we conclude t h a t ffYt~(R~), i f e a c h ~ is a f i n i t e l i n e a r c o m b i n a t i o n of t h e G. H e n c e it /~1- 9 9 t~
suffices t o p r o v e t h a t t h e class ~Jt~(R") is w e a k l y closed.
TheOREM 2.2. L e t E be metrizable ~) a n d let #~, n ~ N, be R a d o n p r o b a b i l i t y m e a s u r e s on E such that #~ ~ #o, as n - + -~ ~ . T h e n #o ~ ~t~(E), i f ~ ~ ~J~(E) f o r each n ~ Z+.
Proof. L e t d b e a t r a n s l a t i o n - i n v a r i a n t m e t r i c on E. F o r e > 0, a n d C c E , let C~ = {x]d(x, C ) ~ e}, a n d let q0~ b e a c o n t i n u o u s f u n c t i o n = 1 on C, a n d
= 0 on E ~ C . N o w choose 2 C [ 0 , 1 ] , a n d c o m p a c t s u b s e t s A a n d B o f E . W e h a v e
M 2 ( t t ( A ) , # ~ ( B ) ) >
s[jVA
#n, V , #,, , n E Z+.B y l e t t i n g n - - ~ - ~ ~ , a n d m a k i n g use of t r i v i a l inequalities, we g e t /~0((~A + (1 - - ~)B):~) ~_ M~(~o(A), #0(B)).
I f ~ t e n d s to zero, we o b t a i n t h e i n e q u a l i t y (I,) a n d t h e p r o o f is clear
F i n a l l y we shall s a y a f e w w o r d s a b o u t t h e s p a c e 1310, 1], e q u i p p e d w i t h t h e s u p - n o r m t o p o l o g y . G i v e n a R a d o n p r o b a b i l i t y m e a s u r e # on 1310, 1], we h a v e E~YJ~,(R ~) for all 0 < tl ~ t2 ~ ~ t, ~ 1,
t h a t # E ~J~(13[0, 1]), if # , h ~ . . . ~ - - . . . .
a n d all p o s i t i v e i n t e g e r s n. H e r e ~, is t h e D i r a c m e a s u r e a t t h e p o i n t t. F o r e x a m p l e , s e t S~ = ( ~ @ W)h -1, w h e r e h(O, x) = Ox, 0 > 0, x E C[0, 1], a n d w h e r e a~ is t h e p r o b a b i l i t y d i s t r i b u t i o n of a r e a l - v a l u e d r a n d o m v a r i a b l e X, such t h a t
~ / X 2 h a s a c h i - s q u a r e d i s t r i b u t i o n w i t h ~ degrees of f r e e d o m . A s t r a i g h t - f o r w a r d 9 E ~ ~/~(R ~) a n d so S~ E g2_~/~(13[0, 1]).
c o m p u t a t i o n shows t h a t (S~)%.. %
T h e m e a s u r e S~ will be called a S t u d e n t m e a s u r e on 1310, 1], a n d in t h e special case a = t we h a v e t h e C a u e h y m e a s u r e on 1310, 1]. I t is n o t h a r d to p r o v e t h a t S ~ IV, w h e n ~ - + q - co.
3. Integrability for certain functions of seminorms
F o r a long t i m e it w a s a n o p e n q u e s t i o n w h e t h e r t h e n o r m in a s e p a r a b l e B a n a c h s p a c e E m u s t be L P - i n t e g r a b l e for e a c h 1 ~ p ~ ~), w i t h r e s p e c t to a n a r b i t r a r y Gauss m e a s u r e . T h e q u e s t i o n w a s solved in t h e a f f i r m a t i v e b y V a k h a n i a in case E - - I q, 1 ~ q ~ ~- ~ . (See e.g. [3, p. 184].) L a t e r , a m u c h s t r o n g e r r e s u l t w a s
2) This condition can easily be omitted.
CONVEX MEASURES O~ LOCALLY CO:NVEX SI~ACES 245 given b y Fernique [811). Given a Gauss measure # on E, which now m a y be a r b i t r a r y , a n d a /t-measurable seminorm 9, which is finite a.e. [#], F e r n i q u e proves t h a t exp (s~02) C L~(/~), i f ~ > 0 is small enough. Other, less precise results, m a y be f o u n d in [17] a n d [3, p. 187].
W e have
THEOREM 3.1. Let tt C ?~(E) and assume that q~ is a re-measurable seminorm on E, which is finite a.e. [#]. Then in case s ~ O, the function exp (sq~) C Ll(/t), if ~ ~ 0 is small enough, and in case s ~ O, the function q~e Le(#) for all O ~ p ~ -- 1Is.
We need
LEMMA 3.1. Let [~ E ~;~(E) and let A be a convex, symmetric about the origin. Assume #(A) - - 0 ~ 1/2. Then
in case s = O : # . ( E ~ t A ) < 0 \ ~ - ] , t ~ l ,
in case -- + < s < 0: # , ( E ~ t A ) ~ / t - ~ 1 [ ( 1 - - 0 ) ' - - 0 " ]
( ~
#-measurable, subset of E,
1/s
+ 0 ~ / , t > _ l .
Proof of Lemma 3.1. We have
2 t - - 1
E ~ A 2 ~ - - I ( E ~ t A ) + t ~ A , t _ ~ l . (3.1) 2 t - - 1
I n fact, assume a' -- x @ ~ a", where a , ' a" E A. This yields t + 1
(t + I a' t - - l~ ) x = t \ ~ - + (--a") E t A , which proves (3.1).
Now let -- Go < s < 0. The i n e q u a l i t y (I,) gives
2 t - - 1
S ( E ~ A) ~ (t + 1) y , ( W ~ tA) + : t - ~ i #'(A),
a n d a n easy c o m p u t a t i o n yields t h e desired estimate. The ease s = 0 can be t r e a t e d similarly.
Proof of Theorem 3.1. W e only consider the case -- 0o < s < 0. B y definition,
l) S e e a l s o L a n d a u & S h e p p , O n t h e s u p r e m u m o f a G a u s s i a n p r o c e s s , S a n k h y a , S e r . A . 32, 3 6 9 - - 3 7 8 ( 1 9 7 t ) .
246 C ~ I R I S T E R B O R E L L
oo
0
Since ~o < -t- oe a.e. [~], t h e r e is a 2 > 0 such t h a t #{xl~(x ) < ~t} > 1/2. L e m m a 3.1 implies t h a t #{xIq)(x ) > ~tt} = O(tl/s), t--~ + o% a n d this p r o v e s t h e t h e o r e m .
4. A z e r o - o n e l a w for c o n v e x m e a s u r e s
L e t /~ be a Gauss m e a s u r e on E, a n d assume t h a t G is a n a d d i t i v e ,
# - m e a s u r a b l e , s u b g r o u p o f E. T h e n K a l l i a n p u r [11] 1) p r o v e s t h a t G is #-trivial, t h a t is #(G) = 0 o r 1. K a l l i a n p u r ' s p r o o f has b e e n simplified b y L e P a g e [14J.
T h e r e are several o t h e r p a p e r s in t h e l i t e r a t u r e , p r o v i n g less precise zero-one laws, a n d t h e i n t e r e s t e d r e a d e r m a y consult [5], [20], a n d [10].
W e h a v e
THEOREm 4.1. Let # be a convex measure on E and G an additive subgroup of E. Then # . ( G ) = O or 1.
P r o @ Suppose # . ( G ) > 0, a n d let K 0 be a c o m p a c t subset o f G such t h a t
#(K0) > 0. Set K = K 0 U ( - - K0), a n d let H be t h e least a d d i t i v e s u b g r o u p o f E c o n t a i n i n g K , t h a t is
H = (.J ( K + K + . . . q - K ) .
n E Z + ~ ,,"
n t e r m s
W e will p r o v e t h a t # , ( G ) ---- 1 a n d t h e r e f o r e it suffices to p r o v e # ( H ) ---- 1. S u p p o s e t o t h e c o n t r a r y t h a t # ( H ) < 1, a n d choose e > 0 so t h a t
s < min (1 - - #(H), #(K)).
F u r t h e r , choose a c o m p a c t subset L o f E ~ H such t h a t
#(L) > 1 - - # ( H ) - - e.
N o w o b s e r v e t h a t
E \ ( H U L ) ~ - - { E \ [ H U ( ( n - - 1 ) K + n L ) ] } +
(1)
1 - - K, n e Z + .- - n
This r e l a t i o n can be p r o v e d in t h e same m a n n e r as (3.1). Using t h e i n e q u a l i t y ( I _ ~ ) , we get
# ( E ~ , ( H U L)) > rain (#(E ~ [ g tJ ((n - - 1)K + nL)]), #(K)).
1) See also footnote 1), p. 245.
C O I ~ V E X :IVIEASLrRES O:N L O C A L L Y C O I ~ V E X S P A C E S 2 4 7
B u t
H e n c e
be(E ~ ( H U L)) ---- 1 -- be(H) -- be(L) < s < be(K).
s > be(E ~ ( H U L)) ~ be(E ~ [ H U ((n - - 1)K -}- nL)]), n CZ+.
This yields
be((n-- 1 ) K - ~ n L ) ~_ 1 - - be(H) - - s ~ O, n E Z + . (4.1) N o w choose a c o m p a c t subset A o f E such t h a t
be(E ~ A) < 89 - - be(H) - - s). (4.2)
Since K a n d L are c o m p a c t sets, a n d 0 ~ K ~ L, we can f i n d a positive i n t e g e r n so t h a t
(n - - 1)K -~ n L c E ~ A . (4.3)
Clearly, (4.1)--(4.3) give a c o n t r a d i c t i o n . H e n c e be(H)--~ 1, which p r o v e s t h e t h e o r e m .
G i v e n a c o n v e x m e a s u r e be, a n d a n additive, be-measurable, s u b g r o u p G o f E, it is i n t e r e s t i n g t o k n o w w h e t h e r G is o f p r o b a b i l i t y zero or one. R a t h e r t h a n giving a general t h e o r e m of this kind, we p r e f e r to i l l u s t r a t e t h e m e t h o d in a simple
c a s e .
L e t D o ---- [0, 1], a n d D n ~-- {(t, u)lt, u ~_ O, t ~- n u ~_ 1}, n E Z+. F o r
x E R [~ set Aox(t ) = x(t), t E D o, Z]ix(t , U) : x ( t -~- U) - - x(t), (t, U) E D~, a n d A~x(t, u) = A~(An_ax(', u))(t, u), (t, u) E Dn, n ~ 2.
W e h a v e
T H E O R E M 4.2..Let
be C ~s(C[0, 1]), T h e n
where
(~ be a a-finite p o s i t i v e B o r e l m e a s u r e on Dn, a n d let 0 ~ s > - - l / p , where l ~ p ~ + ~) is a f i x e d real n u m b e r .
be{x E C[0, 1]
IA,~x e
LP(~)} = 1iff e e
L~(~),e(')=(flAox(.)Iedbe(x)) lIP .
S h e p p [18, Section 19] p r o v e s t h e special case n = 0, a << ml, p = 2, be ~-- W, a n d V a r b e r g [22, T h e o r e m 3] t h e special case n = 0, a < < m l , p = 2, be e ~ ( c [ 0 , 1]).
Proof. F u b i n i ' s t h e o r e m gives
f ePd~ = f ( f lAox]'d~) dbe(x).
(4.4)2 4 8 C I I R I STEll, B O R E L 5
T h e " i f " p a r t is trivial. To p r o v e t h e o t h e r direction, set
(f T
q~(x) = [A~x[Pd(~ , x E 1310, 1].
T h e n ~ is a # - m e a s u r a b l e s e m i n o r m , a n d qv < -~ oo a.e. [#], b y a s s u m p t i o n . T h e o r e m 3.1 n o w tells us t h a t ~ E LP(#), a n d (4.4) gives t h e result.
F o r e x a m p l e , let S~ b e t h e S t u d e n t m e a s u r e i n t r o d u c e d a t t h e v e r y e n d o f S e c t i o n 2, a n d a s s u m e ~ > p, n > 1. T h e n a s t r a i g h t - f o r w a r d c o m p u t a t i o n shows t h a t S~{x C {J[0, 1][A,~x C LP(Itl-~dtdu)} = 1 if a n d o n l y if a < p/2 @ 1.
N o w let # C g J ~ ( R Z + ) , 0 > s > - - 1/19, w h e r e 1 < p < q- oo is a f i x e d real n u m b e r , a n d l e t 0k >_ 0, /c E Z+. T h e s a m e t e c h n i q u e as in t h e p r o o f of T h e o r e m 4.2. t h e n yields
#{x I ~ Okixk] p < @ oo} = 1 i f f ~ Ok f ]xkl'd#(x) < + oo. (4.5)
1 J
1
F o r i n s t a n c e , let 2k _> 0, S: l 2 --> l 2, b y s e t t i n g
k E Z+, X 2 k < q- 0% a n d d e f i n e a linear o p e r a t o r
S{x~}
= {~k~k}, { ~ } e l~.B y S a z o n o v ' s t h e o r e m [16, p. 160], t h e r e is a Gauss m e a s u r e # on l 2 such t h a t fei<X'y>d~(x) e - <sy' y>, Y E 12 .
U s i n g (4.5), we conclude t h a t
~{m/~
Oklxkl p <§ oo}
= 1 i f f ~ 0k2~/2 < @ oo.F i n a l l y we shall discuss H S l d e r conditions. L e t o~: ]0, 1 ] - ~ ] 0 , - ~ oo[ b e a c o n t i n u o u s f u n c t i o n . W e shall s a y t h a t a c o n t i n u o u s f u n c t i o n x: [0, 1] --~ I I b e l o n g s to H,(m), n > 1, if
]A.x(t, u) l
s u p < ~- oo.
(,, u) e b . o ( u )
THEORE)% 4.3. Let tt, p, and e be as in Theorem 4.2. Then
e(t, u)
sup ~(u)
- - ~ o o implies #(H~(co)) = O.Proof. L e t y be a n o n n e g a t i v e L e b e s g u e i n t e g r a b l e f u n c t i o n on D , ,
f a,,x(t, _~) ~ 11/~
~(x) = o(u) y(t, u)dt du] , x c
0[0, ]].
a n d set
COINVEX M E A S U R E S Ol~ L O C A L L Y CO:NVEX SPACES 2 4 9
Suppose #(H~(~o))> 0, t h a t is # ( H , ( o ~ ) ) = 1. T h e n ~s < + c~ a.e. [#], a n d T h e o r e m 3.1 gives 9 E LP(#). H e n c e , b y F u b i n i ' s t h e o r e m ,
Since y ~ 0 is a n a r b i t r a r y L e b e s g u e i n t e g r a b l e function, we h a v e
e s s s u p (o~(u))-l~(t, u) < § co.
T h e o r e m 3.1 implies t h a t ~ is continuous, a n d t h e r e f o r e we h a v e a c o n t r a d i c t i o n . H e n c e #(H~(o~)) = 0, which p r o v e s t h e t h e o r e m .
T h e converse o f T h e o r e m 4.3 is, of course, wrong. T a k e # = W, ~(u) -~ u ~/2, a n d n : 1. I t is well k n o w n t h a t W(HI(u~/2)) : O.
5. The support of a convex measure
T h e s u p p o r t supp (#) o f a R a d o n p r o b a b i l i t y m e a s u r e # is, b y definition, t h e least closed set w h i c h carries t h e t o t a l mass one. I t is well k n o w n t h a t t h e s u p p o r t o f a Gauss m e a s u r e on E is a closed linear s u b v a r i e t y , at least if t h e space E is n o t t o o c o m p l i c a t e d . (See [9], [12], a n d [13].)
W e h a v e
THEOREIV[ 5.1. Let # be a convex measure on E and assume that
supp (#~) ~- singleton set or R, all ~ E E'. (5.1)
Then supp (#) is a closed linear subvariety of E. Especially, it holds that
supp (#) ~- N { H I H e 3}, (5.2)
where ~ is the f a m i l y of all closed hyperplanes in E with #-measure one.
Proof. T h e i n e q u a l i t y ( I _ ~ ) implies t h a t supp (#) is convex. F u r t h e r m o r e , we
h a v e
supp (#) c
n{HI e 3}
since /~ is a l~andon p r o b a b i l i t y measure.
Suppose
Xo C [N { H [ H E 3}] ~ supp (S). (5.3)
T h e H a h n - B a n a c h s e p a r a t i o n t h e o r e m gives us a ~ C E ' such t h a t k: = sup ~(supp (~)) < ~(x0).
B u t #~(]k, ~- oo[) ~-- #(~-l(]k, -~- oo[)) = 0, a n d t h e r e f o r e (5.1) says t h a t ~ is constant: = l a.e. [#]. Clearly, 1 ~ k a n d ~-1({l}) C 3- F r o m (5.3) we h a v e
250 cRmSTE~ BorEaL
} ( X 0 ) - l, which implies t h a t 1 > / c . This contradiction proves (5.2) and the theorem.
Note that a Gauss measure /~ satisfies (5.1).
6. Some properties of convolutions
We shall conclude this paper with a brief discussion of convolutions. First, we give a generalization of the important inequMity due to Anderson [1]. (See also
[19].)
A measure # is said to be symmetric if #(A) ---- # ( - - A) for all A e ~(E).
TI~EO~EM 6.1. Let /~ be a symmetric and convex measure on E, and let f , g be nonnegative Borel functions such that the sets
{xlf(x) >_ t} and {xlg(x) ~_ t}
are convex and symmetric about the origin for all t ~ O. Then ( f , g)~(2x) ~ ( f * g),Ax), ]~t ~ 1, x e E, where
( f . g),(x) ~-
ff(x-y)g(y)d~(y),
x E E .Proof. W i t h o u t loss of generality it can be assumed t h a t
f
and g are characteristic functions of convex sets A and B, respectively, b o t h symmetric about the origin. Then(A + ~x) n B 2 - - ~ [(A -- x) n B] + - ~ [(A + x) n Z], L~I <-- 1, x e E.
I-~enee
# ( ( A + ~ x ) A B ) > # ( ( A + x ) P ~ ) , ]~l ~ 1 , x e E , which proves the theorem.
THEOREM 6.2. Let # C ~)~o(E) and v C 9~o(F). Then the product measure
# Q v E Tf~o(E | F). Especially, the convolution # * ~ E ?)~o(E) in case F -~ E.
Note here t h a t u (~ v extends to a R a d o n probability measure on E Q F, equipped with the product topology. (See [3, p. 94].)
Proof. The product measure # Q v clearly satisfies the inequality (I0) when A and B are rectangles in E | F. The first assertion therefore follows from Theorem
CONVEX ~EASURES ON LOCALLY CO~VEX SPACES 251
2.1. a n d [4, Cor. 3.1]. T h e s e c o n d a s s e r t i o n follows f r o m t h e f i r s t a n d L e m m a 2.1, s i n c e # * v --~ (/~ if9 v)h -1, w h e r e h ( x , y ) = x -ff y, x, y E E . T h i s p r o v e s t h e t h e o r e m . F r o m T h e o r e m 6.2 i t is n o t h a r d t o p r o v e t h a t ( f . g ) . is a l o g a r i t h m i c a l l y c o n c a v e f u n c t i o n o n E , w h e n e v e r # C ~ 0 ( E ) , a n d f , g a r e n o n n e g a t i v e , l o g a - r i t h m i c a l l y c o n c a v e , B o r e l m e a s u r a b l e f u n c t i o n s . T h i s e x t e n d s [6], w h i c h p r o v e s t h e r e s u l t w h e n E = R" a n d # = m , .
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Received December 6, 1973 Christer Borell
D e p a r t m e n t of Mathematics Sysslomansgatan 8
Uppsala, Sweden
