Z E S Z Y T Y N A U K O W E W Y Ż S Z E J S Z K O Ł Y P E D A G O G I C Z N E J w B Y D G O S Z C Z Y P r o b l e m y M a t e m a t y o z n e 1985 z. 7 W A C Ł A W A T E M P C Z Y K W Ł A D Y S Ł A W W I L C Z Y Ń S K I U n i w e r s y t e t Łó d z k i O N S O M E G E O M E T R I C A L C H A R A C T E R I Z A T I O N O F S I N G U L A R N O R M E D M E A S U R E S \
Let X b e a v e c t o r a p a o e ( r e a l o r c o m p l e x ) and let К be a s u b s e t o f X h a v i n g at l e a s t two p o i nts. W e s h a l l s a y that two d i f f e r e n t p o i n t s p < , P 2 в K are a n t i p o d a l in К ( o r s i m p l y a n t i p o d a l ) if a n d o n l y i f f o r e v e r y ” « x,, é. К and f o r e v e r y r e a l n u m b e r t the e q u a l i t y t < p 1 - P 2 ) = 3t1 ~ X 2 i m p l i e s jt|£1 . It is e a s y to p r o v e that f o r e v e r y p a i r x^ , x 2 o f d i f f e r e n t p o i n t s b e l o n g i n g to the c o m p a c t set К i n H a u s d o r f f t o p o l o g i o a l vector^ s p a c e X t h e r e e x i s t s a p a i r o f a n t i p o d a l p o i n t s p ^ P j j f e K a n d a r e a l n u m b e r t, Itlźi s u c h that t ( P t - P 2) = x, - x 2 . L e t (X, Д ) be a n y m e a s u r a b l e s p a c e a n d ji,\J n o n n e g a t i v e m e a s u r e s d e f i n e d o n this s p a c e a n d n o r m e d b y the c o n d i t i o n yi(X)= ^(Х)я 1. U s i n g the J o r d a n d e c o m p o s i t i o n t h e o r e m w e c a n p r o v e the n e x t T H E O R E M 1. T w o n o r m e d m e a s u r e s yi,V d e f i n e d on X are a n t i p o d a l i f a n d o n l y if | yi - V | (X)= 2, w h e n j yi - V | ( X ) m e a n s the t o tal v a r i a t i o n of a s i g n e d m e a s u r e s yi - on X. F r o m this t h e o r e m a n d H a h n d e c o m p o s i t i o n t h e o r e m we c a n o b t a i n a s i m p l e g e o m e t r i c a l c h a r a c t e r i z a t i o n o f a n t i p o d a l m e a s u r e s o n X. T H E O R E M 2. T w o n o r m e d m e a s u r e s }>»V? d e f i n e d o n X are a n t i p o d a l i f a n d o n l y i f t h e y a r e s i n g u l a r . Let u s o o n s l d e r th e c l a s s K X » А ЛТг of m e a s u r a b l e г f 4 t r
sp a c e s a n d the f a m i l i e s ^yi } iç I o f n o r m e d m e a s u r e s Г r t r ^ f
d e f i n e d o n X. Put u = (?) P » = ® V • U s i n g t h e o r e m 2
154
it is easy to prove
THEOREM 3. If there exists
f Qt Г
such that
, о
О
Jo
are antipodal then the product measures
p,i> are antipodal.
If /"* is a finite set then the above theorem can be
reversed. Using the Lebesgue-Radon-Nikodym theorem we can prove
n
n
THEOREM
The measures p
= ® p. ,
yj, ф
defined on
k
=1k
=1n
n
a product(P
measurable spaces and normed by the
c o n d i t i o n ~ = 1 * lc=1»2 »***n * a r ® a n t i p o d » 1 a n d o n l y if t h ere e x i s t s a n a t u r a l n u m b e r к , 1 ^ к £ n s u c h
о
о
that the measures
p^
,
are antipodal.
о
о
W e s h a l l c o n s t r u c t the e x a m p l e s h o w i n g that th e t h e o r e m 3
can not be reversed if p , a r e the product measures on the
product of infinitely many measurable spaces. Suppose that
X = < 0, 1 > ,
are Borel subsets of X . Put for all n « N
n
n
n
and for each Borel subset
EC< 0,1>
(1
i fcE
(|a °»rd En^i
2’ j? ’ •••'}' k
v 'E b 'o 1 * . ’ I ^ - j o } = 0
•oo
It easy to see that \? « u
. Let u = <25 u ,
Q
- Cap
J
n " in
I
n
<n
r\
„ ,
n
= 1
