ZESZYTY NAUKOWE WYŻSZEJ SZKOŁY PEDAGOGICZNEJ W BYDGOSZCZY P ro b lem y M atem atyczne
1983
/84
z .5/6
WŁODZIMIERZ ŚLĘZAK WSP w B y d g o s z c z y
SOME CONTRIBUTIONS TO THE THEORY OF BOREL °C SELECTORS
L e t X and Y be t o p o l o g i c a l sp a c e s and l e t F b e a map fro m X to th e h y p e rs p a c e o f a l l nonem pty s u b s e ts o f Y. A c o n tin u o u s f u n c t io n f fro m X t o Y i s s a id t o b e a c e n tin u o u s s e l e c t i o n f o r F i f f ( j ) C F ( x ) f o r e v e r y x e. X. The n o t io n o f c o n tin u o u s s e l e o t i o n was in t r o d u c e d and s t u d ie d i n d e t a i l b y E. M ic h a e l, c f . [ l * » - l 6 ] and th e r e f e r e n c e s t h e r e , S e l e o t i o n th eorem s a r e i n o b v io u s way g e n e r a l i z a t i o n s o f e x t e n s io n th e o re m s . An e x t e n s io n p rob lem i s a s e l e c t i o n p rob lem i n w h ic h F ( x ) i s f o r e v e r y x e. X e i t h e r a s i n g l e t o n o r th e w h o le sp a c e Y . I n th e i n t r o d u c t i o n o f (1*»3 M ic h a e l p o in t e d o u t t h a t th e main r e s u l t o f h is s e l e c t i o n t h e o r y i s th e f a c t t h a t m ost o f th e known e x t e n s io n th eorem s can be s l i g h t l y changed and e s s e n t i a l l y g e n e r a l i z e d t o s u i t a b l e s e l e o t i o n th e o re m s . L e t oc be any c o u n t a b le o r d i n a l number. R e c a l l t h a t a m u lt if u n c t io n F : X —^ Y i s s a id t o be o f lo w e r c l a s s o r i f F ~ ( v ) : = [ i £ X s F (x )n V / 0 } i s a B o r e l s e t l n X o f a d d i t i v e c l a s s oc f o r ea ch open s e t V i n Y ( o f . T i l ] ) . I n [ i o ] th e f o l l o w i n g g e n e r a l p ro b lem i s fo r m u la t e d : U nder what a ssu m p tion s can th eorem s known f o r oc* = О be e x te n d e d t o a r b i t r a r y oC с SL ?
P a p e r Г*»] , s e e a ls o ГзЗ , c o n ta in s some f i n i t e - d im e n s io n a l B o r e l °C a n a lo g u e o f M i c h a e l 's fam ous theorem 3 . 1 ' ‘ ’ , s e e
P« 373 and
368 . The aim o f t h i s w ork i s t o im p ro v e
t h i s th eorem v i a s o - c a l l e d G a s t a in g r e p r e s e n t a t i o n o f F. We g i v e a l s o some r a t h e r t r i v i a l m o d i f i c a t i o n s o f th e r e s u l t s known i n th e c a s e o f m e a su ra b le s e l e c t o r s ( c f . f l j , Г2
} , f8
}F o llo w in g R o c k a f e l l a r f l 7 j we s a y ł . * " - ) ** * } a C a s ta in g r e p r e s e n t a t i o n o f F i f e a c h f A i s a s e l e c t o r f o r F , and ^ f ^ x ) , f
2
( x ) , . . » | I s d e n se i n F ( x ) f o r x ć. X. R e p r e s e n t a t io n s o f t h i s k in d a r e e x h i b i t e d b y C a s ta in g f l ] f o r th e c a s e w h ere th e m ea s u ra b le s t r u c t u r e on X i s t h a t o f a Radon m easure on l o c a l l y com pact s p a c e , and b y U im m elb erg C e l i n th e g e n e r a l s i t u a t i o n w h ere X i s an a b s t r a o t measu r a b l e s p a c e . The C a s ta in g r e p r e s e n t a t i o n f o r m u l t i f u n c t i o n o f lo w e r c l a s s ос a ls o e x i s t s b y v i r t u e o f [1 3 ] • S in c e th e th eorem k i n Г1
ЗЗ i s s t a t e d as a c o r r o l a r y t o some r a t h e r c o m p lic a t e d r e s u l t s , we g i v e h e r e th e d i r e c t p r o o f o f t h i s th eo re m . M o re o v e r , o u r th eorem1
g i v e s th e e x i s t e n c e o f den u m erab le seq u en ce o f s e l e c t i o n s p r o v id e d t h a t X i s a p e r f e c t l y n orm al t o p o l o g i c a l s p a o e , w h i l e i n fl 3 ] X i s assumed t o be a m e t r ic s p a c e . L e t us r e c a l l t h a t a n orm al t o p o l o g i c a l sp a c e i n w h ich ea c h c lo s e d s e t i s a G j i s c a l l e d p e r f e c t l y n o rm a l. The o r d i n a l sp a c e Го,_$2} w i t h t o p o lo g y g e n e r a t e d b y a l l s e t s o f th e fo rm £x : x > « c } and £ x : x < p } i s an exam ple o f p a ra c o m p a ct, h en ce a l s o n o rm a l, t o p o l o g i c a l sp a c e t h a t i s n o t p e r f e c t l y n o r m a l. Nam ely th e c lo s e d s e t i s n o t a G j . F o r , i f ( G ^ 1 = 1 , 2 , . . . } i s any c o u n t a b le c o l l e c t i o n o f open s e t s c o n t a in in g £2, , th en b eca u s e th e s e t s (<*',ß ) a r e a b a s i s , f o r e a c h i s 1,2
, . . . t h e r e e x i s t s an o r d i n a l such t h a t £oć^,.!j2] i s c o n t a in e d i n Ga » B e in g c o u n t a b le , th e c o l l e c t i o n £c< ^ i 1 = 1 , 2 , . . . } . has an u p p e r bound (3 < S l , mo £ } Q Gt • C l e a r l y fo ,£ 2 .] i s p a ra c o m p a c t. In d e e d , l e t £ , i ć I } b e any open c o v e r i n g . S in c e th e s e t s ( Л , р ) fo rm a b a s i s , d e f i n e f s [ o , J 2 ] —^ fO ,J L ] b y a s s o c i a t i n g w it h ea ch0
a f ( ß ) < ß such t h a t ( f ( ß ) , ß ] i s c o n t a in e d i n some U^, and s e t t i n g f ( o ) = 0 . By i n d u c t io n , cons t r u e t a sequ en ce (3Q =J2
. (3
, = f ( f t ) , . . . , ß k = f ( (3k - 1
) , . . . th en > ß j )> • • • and, s in c e e v e r y d e s c e n d in g sequ en ce o f o r d i n a l s i s f i n i t e , t h i s t e r m in a t e s w it h some •B ecau se th e p r o c e s s ca n n ot be c o n tin u e d , ß =0, and so (O ,SL~j
H ‘ n
i s c o n ta in e d i n ( ( 3 k _
1
»(lk 3 . C h o o s in g a c o n t a in in g71
ea ch ( ß f c ] and eome Э ^ 0 }, we have a f i n i t e su b-c o v e r in g { U± , UA , . . . , UA ^ ° o f ( Ui * * ^ » whi o h i s
о 1 n
c o n s e q u e n tly an open n e i g h b o u r h o o d - f in it e r e f in e m e n t . C on ver s e l y th e r e a d e r can v e r i f y t h a t th e su bspace [ o , J L ) i s p e r f e c t l y n o rm a l, but i t i s n o t p a ra c o m p a ct: th e open c o v e r i n g by s e t s [ 0 , o ć ), 0 < oc < Sh , has no open n e ig h b o u r h o o d - f in it e r e fin e m e n t . THEOREM 1. L e t X be a p e r f e c t l y .n orm al t o p o l o g i c a l sp a ce and l e t Z be a P o l i s h s p a o e . Suppose t h a t f : X — ? Z i s a m u lt if u n c t io n w it h c lo s e d v a lu e s * Then th e f o l l o w i n g c o n d it io n s a r e e q u iv a l e n t : a ) F i s o f lo w e r c l a s s ос , oc > 0 b ) t h e r e e x i s t B o r e l fu n c t io n s z * 1 = 1 , 2 , . . . such t h a t f d r ea ch x £ X we h ave F ( x ) = C l [ f A( x ) : i = 1 , 2 , . . . j P r o o f : ( a ) b . F ix a c o m p le te m e t r io d on Z. I t s u f f i c e s t o show t h a t f o r e v e r y £. > 0 t h e r e e x i s t B o r e l oC s e l e c t o r s Sn sX —^ Z f o r F such t h a t £gn ( x ) : n = 1 , 2 , . . * } i s an £ - n e t i n F ( x ) f o r ea ch x fc X . Once t h i s i s d o n e, к we can put t o g e t h e r th e fu n c t io n s { g ; n , k = 1 , 2 , . . . $ , t o g e t th e d e s ir e d r e s u l t , w h ere g a r e B o r e l oC s e l e c t o r s ic ^ i f o r F such th a t [ gn ( x ) : n = 1 , 2 , . '. . ^ i s a k” - n e t i n F ( x ) f o r ea c h xé X. S in c e Z i s s e p a r a b le , hence i t i s p o s s i b l e t o c o v e r Z b y open b a l l s K (z ^ , £ / 2 ) , 1 = 1 , 2 , , . . Put Х± : = { x £ X: F ( x ) n K ( z A , £ / 2 )/ 0 ] , 1 = 1 , 2 , . . . . N o te t h a t ea ch X^ i s o f B o r e l a d d i t i v e c la s s o r , i n v i r t u e o f th e p r o p e r t i e s o f F. F ix a B o r e l or s e l e c t o r h :X —>■ Z f o r F . The e x i s t e n c e o f such a s e l e o t o r i s en su re d by famous th eorem o f K u ra to w s k i and R y ll- N a r d z e w s k i, c f .
Г з ] ,
theorem s 3 and 11 . Suppose t h a t X^ i s nonem pty, and d e f i n e ”■“ * 2 ЬУ fo r m u la FA ( x ) = C l f F ( x ) o K ( z A , £ / 2 ) J . N o t io e t h a t i f G i s open i n Z , then F ~ ( g ) : = X± : Fa (x )a G 4 <t> } = { x £ X: C l f F ( x ) nK (z ^ ., £ / 2 ) n л G / 0 } = ( x &. X : F ( j ) o G i\ K (z ^ , £ /2 0 ]• i s o f B o r e l a d d i t i v e c l a s s o r i n X. S in c e X^ i s a ls o o f B o r e l a d d i t i v e c la s s in X, hence Fa(g) i s o f B o r e l a d d i t i v e c la s s oc' o f X^72
w it h th e In d u ce d t o p o lo g y . Ve can t h e r e f o r e a p p e a l on ce a g a in t o th e K u ra to w s k i and R y ll- N a r d z e w s k i theorem t o g e t a B o r e l
o f e e l e o t o r h^ : X^ Z f o r F^, T h e re e x i s t s e t s i , j c
1
,2
, . . . , w h ich a r e s im u lta n e o u s ly o f a d d i t i v e c la s s l e s s than oC and o f th e m u l t i p l i c a t i v e c l a s s l e s s than o t ,CO su ch t h a t f o r ea ch i d ,2
, . . . we h a ve X^ = • F o r j d , 2 , . . . d e f i n e h ^ IX —y Z as f o l l o w s on XA j on X - XAJ I f X^ i s em p ty, s e t h ^ j= h f o r e v e r y J d , 2 , . . . . I n e i t h e r c a s e , as i s e a s y t o c h e c k , th e fu n c t io n s h±J 9 p a r e B o r e l OC s e l e c t o r s f o r F« To c o n c lu d e th e p r o o f o f t h i s Ite m , we o la im t h a t f o r ea ch x ć X , ( h ^ j ( x ) ; i , J d ,2 , . . . J i s an 1 - n e t i n F ( x ) . F o r , i f z <u F ( x ) , we can f i n d an i Q sucht h a t z £ K ( z ^ , £ /
2
) , h en ce F ( x ) n K ( z A , £ , /2
) i s nonem pty and x t X^ ,° S o t h e r e i s an Jo such tR a t z t X ^ j . ц f o l l o w s t h a t h± j ( x ) = h ± ( x ) é F i ( x ) c C l К ( z ± , 1/S ) . C o n s e q u e n tly , d £ z , ° h . . ( x ) ) i d ( z ° z . )+ d ( z . » ° h . , ( x ) ) ^ . € .0 0
о о о о Thus U î = { h ^j I i ,j =1
, 2, . . . k =1
, 2 ,, , , u ^ ,. . .J i s as r e q u ir e d . b a . L e t G be an open s e t i n Z and Ü * ( f l t f 2 , . . . i . Then F ~ ( z ) s = { x s F ( x ) n G 0 0 } = = ( x s ГС1 U (x )J Л О / f t ] = { x : и ( х ) л G 0 0 } = OO = \ x : u ^ x J f e G f o r some u ^ u }= [ x : un ( x ) « G } = = ^ u~ 1
( g ) i s o f B o r e l a d d i t i v e c l a s s °C , so t h a t F i s n= i n o f lo w e r c la s s oC .Now, l e t Z be a l i n e a r s p a c e . The o on vex h u l l and c lo s e d c o n v e x h u l l o f a s e t В С Z a r e d e n o te d b y co n v В and C l con v B, r e s p e c t i v e l y . The f o l l o w i n g th eorem i s an e a s y g e n e r a l i z a t i o n o f theorem from f l
3 » s e c t i o n
6 :
THEOREM 2. L e t X be a p e r f e o t l y n orm al t o p o l o g i c a l sp ace and l e t Z be a s e p a r a b le F r e c h e t sp a c e and F :X —*> Z a m u lt if u n c t io n o f lo w e r o la s s <X w it h c lo s e d v a lu e s . Then th e
73 m u lt if u n o t io n s с » п т F , C l oon v F d e f in e d b y x i— e e n v F ( x ) , x (- ♦ C l fc o n v F ( x ) j a r e a la o o f l o v e r o l a s i oC • M o re o v e r , Z n e ed o n l y b e a s e p a r a b le v e t r i о l o o a l l y o o n vex s p a c e , I f ea ob F ( x ) I s assumed t o b e o o m p le te . P r o o f : By th eorem 1 t h e r e i s a o o u n ta b le c o l l e c t i o n U ж l u ^ , « e f B o r e l =C s e l e o t o r s f o r F such t h a t F ( x ) e C l U ( x ) f o r a l l x t X . L e t Q be th e s e t o f a l l se q u en o es (q ^ , q 2 , . • • • • • ) o f n o n - n e g a t iv e r a t i o n a j numbers
suoh t h a t a l l b u t f i n i t e l y many 4 ^ * a r e 0 and ^ q ^ s l , The s e t ^ Q. i s c l e a r l y c o u n ta b le and so i s
V ! в ! ^ 1 q n * Un ! ( q 1 * 4 2 4 n ) £ Q i *
V i s a o o u n t a b l e ^ e o l l e o t i o n o f B o r e l oc fu n c t io n s such th a t C l V ( x ) : s 4n ' un ^ * ^ ’ ( 4 1» q 2 >*>* ) t Q . } = C l co n v U (x)e = oon v U ( x ) f o r a l l x £. X. H en ce, a g a in a p p ly in g Theorem 1, c o n v F and C l co n v F a r e o f l o v e r c l a s s « - .
THEOREM 3 . L e t X b e a p e r f e o t l y n orm a l t o p o l o g i c a l sp ace and Z a s e p a r a b le Banaoh a p a c e . L e t f : X —► Z b e a B o r e l « " map and r : X —+■ R a B o r e l oc f u n c t io n w it h non nega t i v e r e a l v a l u e s . Then th e m u lt if u n c t io n d e f in e d b y fo rm u la
F ( x ) s К ( f ( x ) , r ( x ) := [ z é Z i l| f ( x ) - ж 11 4 * "(x )$ i s o f l o v e r d a s s aC .
P r o o f . L e t ( Z j f Z g , . . . ) b e a den se sequ en ce i n th e u n it b a l l o f Z . Put un ( x ) s a f ( x ) + r ( x ) - * n , n s 1 , 2 , . . . . Then ea ch un i s c l e a r l y o f B o r e l c l a s s oC and v e h ave an e q u a lit y
F ( x ) : = К ( f ( x ) , r ( x ) ) = C l { u n ( x ) ; n s 1 , 2 , . . } By th eorem 1, F i s o f l o v e r d a s s cC ,
U s in g th eorem 1 v e a r e a b le t o p r o v e th e f o l l o v i n g s u p e r p o s i t i o n th eorem f o r m u lt ifu n c t io n a o f l o v e r c la s s o( :
THEOREM U. L e t X , Y and Z b e s e p a r a b le m e t r ic sp a ces and f : X x Y — > Z a c o n tin u o u s f u n c t i o n . L e t f : X — ^ Y be a m u l t i f u n c t i o n o f l o v e r c la s s or > 0 v i t h c o m p le te v a lu e s . The th e m u l t i f u n c t i o n G:X Z d e f in e d by fo rm u la
I k P r o o f j A p p ly in g th e C a s t a i n g 's r e p r e s e n t a t i o n , l e t U b e a c o u n t a b le s e t o f B o r e l cxT s e l e c t o r s f o r F su ch t h a t C l U ( x ) = F ( x ) f o r e a c h x & X . L e t В b e an op en s u b s e t o f Z. Then we h a ve g ” ( b ) = ( x t f ( { x } x F ( x ) ) ЛВ / 0 } ? e [ x : f ( U } x C l U ( i ) ) f l B / = | X 3 i ; f ( x , y ) t B f o r some y t C l Ü ( x ) ^ = { x : f ( x , u ( x ) ) ć B f o r some u €.U ^s { * * f ( x , u ( x ) ) e B } . So i t rem a in s t o show t h a t ^ x : f ( x , u ( x ) )feB i s a member o f B o r e l a d d i t i v e o la s s o f . By s e p a r a b i l i t y o f X x Y th e map Ьц : X — > X x Y d e f i n e d b y h ^ (x ) = ( x , u ( x ' ) i s o l e a r l y o f B o r e l o l a s s oc . Then £x : « ( * ) ć B^ = b ” ł ( f * * ł ( B ) ) b e lo n g s t o B o r e l a d d i t i v e o l a s s o f , яо t h a t G i s o f lo w e r o la s s ос . Now, l e t Z be a F r e c h e t s p a c e . I f К i s a c l o s e d , c o n v e x s u b s e t o f Z , th en a s u p p o r t in g s e t o f К i s , b y d e f i n i t i o n , ( s e e [ l k ] ) , a c l o s e d , c o n v e x , p r o p e r s u b s e t S o f К ( s may e v e n be a s i n g l e t o n ) such t h a t i f an i n t e r i o r p o in t o f a segm ent i n К b e lo n g s t o S, th en th e w h o le segm ent i s c o n ta in e d i n S . The s e t I ( k ) o f a l l e le m e n ts o f К w h ich a r e n o t i n any s u p p o r t in g s e t o f К w i l l b e c a l l e d th e i n s i d e o f K. The f a m i l y D (Z )= [ bC Z : B s oon v В and В з 1 ( C l B ) } in t r o d u c e d i n i s s e e m in g ly th e a d e q u a te r a n g e sp a c e f o r th e C e d e r - L e v i th eo re m . The r e a d e r i s r e f e r e d t o th e e x e e l l e n t m onograph (*5 3 f o r th e s tu d y o f p r o p e r t i e s o f o o n vex s e t s . P r o p o s i t i o n 1. ( E v e r y c o n v e x s u b s e t К o f F r e c h e t sp a c e Z w h ich i s e i t h e r o l o s e d , o r has an i n t e r i o r p o in t o r i s f i n i t e - d i m e n s i o n a l , b e lo n g s to D (z) . P г о о f . I f К i s c lo s e d , t h i s i s o b v io u s . I f К has an i n t e r i o r p o i n t , and i f z €. ( C l К ) - К (com p lem en t o f К i n C l К ) th e n th e Hahn-Banach th eorem g u a r a n te e s th e e x i s t e n c e o f c l o s e d h y p e r p la n e HC Z w h ic h s u p p o r ts C l К a t z , bu t d o e s n o t o o n t a in C l K ; c l e a r l y H n C l К i s a s u p p o r t in g s e t o f C l К and h ence z b e lo n g s no t o l ( c i К ) . F i n a l l y , i f К i s f i n i t e d im e n s io n a l, th e n К has an i n t e r i o r p o in t
7Ï w it h r e s p e c t t o th e s m a lle s t l i n e a r v a r i e t y Z ^ Z c o n t a in in g K. Hence К t ü ( Z 1 ) С D ( z ) . P r o p o s i t i o n 2 . ( I 'l U } ) I f ( t l f t 2 , . . . J i a a donee s u b s e t o f a nonem pty, c l o s e d , c o n v e x , s e p a r a b le s u b s e t К o f a F r e o h è t sp a c e Z , and i f * i ” 4 1 / Z i = S + - a x ( 1 , d f t j t ) ) ’ i = 1 ' 2 » — oo w here d i s an i n v a r i a n t ne t r i o on Z , th en z : = 2 Jz^ b e lo n g s t o th e I n s i d e I (k) o f K. P r o o f . Suppose z 4 i ( k ) . Then t h e r e e x i s t s a s u p p o r t in g s e t S C к such t h a t z t S , N ov f o r e v e r y i = 1 , 2 , . . . , z i s e i t h e r an i n t e r i o r p o i n t o f a segm ent i n К one o f w hose end p o in t s i s z ^ , o r e l s e z s z ^ . I n e i t h e r c a s e we must h a ve
S . B u t, f o r e v e r y 1 = 1 , 2 , . . . th e p o in t z^ i a e i t h e r an i n t e r i o r p o in t o f th e segm ent
[ t ^ t ^ := ( t é Z : t = a t 1 + ( l - a ) t A ; O i’ a ^ l J o r e l s e z ^ = t^ , so i n e i t h e r o a s e we must h ave t^ é S . But { *
1
* *2
’ * * * ^ i s d en se i n K., and s in c e S i s c l o s e d , t h i sf i n a l l y im p l i e s t h a t SsK, w h ic h i s im p o s s ib le .
Ve arm now i n a p o s i t i o n t o s t a t e and p r o v e o u r main s e l e o t i o n th eo re m : THEOREM 5 . L e t X b e a p e r f e o t l y n orm al t o p o l o g i c a l sp a c e and Z a s e p a r a b le Freohüit s p a c e . I f F :X —>■ Z i s a m u l t i f u n c t i o n o f lo w e r c l a s s o r , o r ^ 0 , whose v a lu e s a r e i n D ( z ) , th en F has a B o r e l oC s e l e c t o r . REMARK 1. I n c a s e oc =0 we must o b s e r v e t h a t X i s e o u n t a b ly paracom pac t . P r o o f . D e fin e C l F :X —*■ Z b y fo r m u la ( C l F ) ( x ) = C l [ f ( x ) ] j what we must f i n d i s a B o r e l oc s e l e c t o r f : X — Z such
t h a t f ( * ) è l ( c i F ( x ) ) f o r e v e r y x € X. O b v io u s ly C1F i s a ls o o f lo w e r c l a s s °C , i n v i r t u e o f o u r theorem 2 • Th us, i n v i r t u e o f th e th eorem 1, C l F has a C a s ta in g r e p r e s e n t a t i o n | f j , f 2 , * . . ^ . Now, l e t \
76
fA (x)- f,(x)
g ^ ( x ) = f , ( x ) + ■ - . » 1 = 1 » 2 , , ( i , d ( Г ± ( х ) , f , ( x ) ) w here d d e n o te an I n v a r i a n t m e t r ic on Z . °°m
. i
\ Put f ( x ) m 21 2 g ^ f x ) . By P r o p o s i t i o n 2 we h ave j=1 J f ( x ) é l ( C l F (x) ) c- F (x) s in c e F ( x ) b e lo n g s to D ( z ) . On th e o t h e r hand, s in o e g j , j = 1 , 2 , . . . a r e o f B o r e l c l a s s où and s in c e th e s e r i e s d e f i n i n g f : X —► Z c o n v e r g e s a lm o s tu n ifo r m ly on X, i t f o l l o w s t h a t £ i s a l s o o f B o r e l c l a s s
oC , and thus has a l l th e r e q u ir e d p r o p e r t i i e .
REMARK 2 « Some p a r t i c u l a r v a lu e s o f F may f a i l t o be c o n v e x , o f . Г
1 5
]REMARK 3 . I t seems t o be p o s s i b l e t o fo r m u la t e th e a b ove theorem 5 I n th e same g e n e r a l fra m ew ork as th e one i n w h ich th e s e l e c t i o n th eorem o f M a g e r l f l 2 j i s fo r m u la t e d , i . e . i n th e fram ew ork o f (k , a )- p a r a c o m p a c t p a ved s p a c e s and а-c o n v e x h u l l - o p e r a t o r s on к -bou nd ed u n ifo r m s p a c e s .
REMARK k e Ve may ask w h e th e r o r n o t th e a l g e b r a i c s t r u c t u r e on Z i s e s s e n t i a l i n C e d e r - L e v y th e o re m . Ve g i v e t o fo r m u la t e some op en p rob lem i n t h i s d i r e c t i o n ( c f . f l 9 ] )
L e t Y 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 . A s u b s e t B e. Y i s s a id t o be r e l a t i v e l y q u a s ic lo s e d , i f B ^ I n t c ß C l B, w here I n t c i В d e n o te s i n t e r i o r o p e r a t io n w i t h r e s p e c t t o th e t o p o lo g y in d u c e d on su b sp a ce C l В o f th e sp a c e Y . The f o l l o w i n g may b e a p r o m is in g p rogra m :
P rob lem : l e t X b e a p e r f e c t l y n orm a l s p a c e and l e t Y be a P o l i s h s p a c e . Assume t h a t F :X Y i s a m u l t i f u n c t i o n o f lo w e r c l a s s oc , o f > 0 , w it h r e l a t i v e l y q u a s ic lo s e d v a l u e s . I s i t t r u e , t h a t t h i s m u l t i f u n c t i o n p o s s e s s e s a lw a y s a B o r e l oc- s e l e c t o r ? REMARK 5 . (a b o u t r e l a t i v e t o p o l o g y ) I n " T o p o lo g y " b y J . D u n g u n d ji, p . 77g and a l s o i n " I n t r o d u c t i o n t o t o p o l o g y " b y H. P a tk ow sk a , p . 2120 th e f o l l o w i n g f a l s e fo r m u la i s e r r o r o u s l y s t a t e d : I n t g A = В Л I n t A .
77 ( l n B e r e l c e n s e ) s u b s e t À С X and l e t Y be an a r b i t r a r y I n f i n i t e - d i m e n s i o n a l Banaoh s p a c e * Then t h e r e e x i s t s a l o v e r s e m lo o n tin u o u s m u l t i f u n c t io n F : X -«► Y w it h co n ve x and d i s j o i n t v a lu e s b u t w it h no B o r e l m ea su ra b le s e l e c t o r . P r o o f : L e t ( « ^ , 1 ć I ) b e some Hamel b a s is f o r Y . F ix some In d e x 1 and put H : = . © T Span e . , w here
о J Ć. J J v
J a l - » and Span d e n o te h e r e l i n e a r h u l l o p e r a t o r . H ext o b s e r v e t h a t H i s a d en se l i n e a r su bspace o f Y , and thus i s o o n v e x . L e t us d e f i n e F :X —^ Y b y fo r m u la
(
H i f x 4, A -H + e ^ i f x £ X - A . О I t i s e a s y t o c h eck t h a t F :X ~ Y i s lo w e r s e m ic o n tin u o u s . In d e e d , f o r e a c h open b a l l B ( y , r ) = у + в ( 0 , r ) we have F” ( B ( y , r ) ) = X i n v i r t u e o f d e n s i t y o f v a lu e s o f F, b o th H and H + e^ . L e t fs X —*■ Y b e an a r b i t r a r y s e l e c t o r f o r o u r m u lt if u n c t io n F . Ve h a ve ( f ) “ 1 ( R - £ o } ) = X - A w h ile R- ^ 0 } i s o p en . S in c e Х- X i s n o t B o r e l m e a s u ra b le , hence th e a b ove e q u a l i t y m eans, t h a t f i s n o t w e a k ly B o r e l m e a su ra b le and thus i t i s n o t s t r o n g l y B o r e l m e a s u ra b le . O b v io u s ly we h a ve H + e^ - co n v (H + e^ ) and we o b s e r v e t h a t th e i n t e r s e c t i o n (H °+ e^ ) л H i s em pty. Thus o u r th eorem i s p r o v e d .REMARK 6 . I n th e c a s e when o u r Banach sp ace Y i s a l s o s e p a r a b le th e v a lu e s o f F i n th e a b ove theorem may be ch oosen t o be an Fg. - s e t s . I n d e e d , l e t H i n fo r m u la / £ j b e a den se su bspace o f Y spanned b y a c o u n t a b le f a m i l l y
с ■> 00
* S in c e H = п^1 Hn ' w h ere H1 := Span e , H := H Span e , and each H i s c lo s e d as a f i n i t e
-n n-1 v—•' n ' n
- d im e n s io n a l su bspace o f Y , hence we o b t a in t h a t H b e lo n g s t o th e p a v in g o f ^ .- s u b s e t s o f Y . A m o d i f i c a t i o n o f t h i s c o n s t r u c t io n w e re u sed i n my p a p e r (i 8J t o s o l v i n g some open p rob lem p o sed b y J . C e d e r and S . L e v i i n i n c o n n e c tio n w i t h h is f i n i t e - d i m e n s i o n a l v e r s i o n o f o u r th eorem 5. N o te
a l i o , t h a t o u r m u lt if u n c t io n F g l vo n b y fo rm u la / £ / i o n o t o f th e f o r a F s g” ' f o r some s in g le v a l u e d f u n c t io n g t F ( x ) — whor e th e im age F ( x ) i e d e f i n e d in o b v io u s way ae F ( x ) t s U F ( x ) . X £ X REMARK 7 » T h e r e e x i s t e a n o n e e p a r a b le p r a h i l b e r t sp a c e Y and a m u lt i f u n c t io n F f r o n th e r e a l l i n e R t o th e o n e - d in e n - s i o n a l open o o n v e x s u b s e ts o f Y euch t h a t F i s i n lo w e r c l a s s 1t b u t h a v in g no any B o r e l m ea su ra b le s e l e c t o r . In d e e d , d e f i n e Y
1
= { h i R — R i supp h1
= { x J h ( x ) ^ 0 } i s f i n i t e } . th en < g| h > : = J Z g ( x ) * h ( x ) i s a s c a l a r p ro d u c t i n Y . X fc. R Decompose R i n t o 2 d i s j o i n t n o n n e a s u ra b le s u b s e ts : R = А О В and put(
i e 6- Y : g ( t ) < 0 } g £ Y : g ( t ) > 0 and supp and supp g ж g * { t } } i f i f t £ At f c B .S ee ex am p le 2 o f f l 8J f o r th e p r o o f , t h a t F i s as r e q u ir e d . P a p e r
Гб J
c o n ta in s many o t h e r s i n t e r e s t i n g c o u n te r e x a m p le s . THEOREM 7 , L e t С d e n o te th e sp a o e o f com p lex num bers, and R th e sp aoe o f r e a l num bers. T h e r e e x i s t s a lo w e r s e m ic o n tin i ous m u lt if u n c t io n F : R —> С w it h a r o w is e c o n n e c te d v a lu e s , bu t h a v in g no any B o r e l m e a su ra b le s e l e c t o r . P r o o f : L e t S 1 d e n o te th e u n it c i r c l e i n th e com p lex p la n e . S in c e th e c l a s s o f a l l B o r e l m ea su ra b le fu n c t io n s fro m R t o S has th e з а м c a r d i n a l i t y as R i t i s c l e a r t h a t we can ch o o se a f u n c t i o n g : R —> S such t h a t th e g ra p h o f f i n t e r s e c t s th e g ra p h o f ea ch B o r e l m ea su ra b le fu n c t io n fro m R t o S^, Put F ( x ) : = s' - j _ g ( x ) } and o b s e r v i th a t F ~ (u ) i s em pty o r th e w h o le sp ace R w h en e ve r U i s open i n C, Thus F i s lo w e r s e m ic o n tim io u s . O b v io u s ly th e v a lu e s o f F a ra op en а г о я . We p r o v e t h a t F i s w ith o u t any B o r e l s e l e o t o r . Assume ad absurdum t h a t f t R —> S i s some B o r e l s e l e c t o r f o r F , T h e r e i s a p o in t x <c R suohО
t h a t f ( x o )= в ( х о ) h en ce l ( x o ) n o t b e lo n g s t o F ( x o )ss = S 1 " { g ( x Q) } = S 1 - t f ( x o ) J . T h is c o n t r a d i c t i o n f i n i s h o u r
79
• r g U M n t ,
THEOREM 8 , T h e r e e x i s t a a lo w e r s e m ic o n tin u o u s m u l t i f u n c t i o n F fro m th e r e a l l i n e R t o th e co m p lex p la n e С w i t h open a r c s as v a lu e s and G g r a ph b u t w it h no B o r e l 1 s e l e c t o r . P r o o f : ( c f *
Г
18
-3 » e x . k ) R e c e n t ly Z. Grande e x h i b i t e d a B e r e l 2 f u n c t i o n g : f o , 2 Го, 21Г) w h ich i n t e r s e c t s e a c h B o r e l 1 f u n c t io n fr o m fo ,2 1 T ) i n t o Го,2 1Г), s e e [7
] * H is c o n s t r u c t io n b a s e d upon K a n t o r o w ic z u n i v e r s a l f u n c t io n К f o r B o r e l 1 f u n c t io n s , n am ely g ( x ) : = K ( x , x ) . By l i f t i n g th e o re m , we may a s s e r t t h a t x f—у g ( x ) : = e x p i * g (x ') = a o o s g ( x ) + i s i n g f x j e s " 1 i s a fu n o t io n fr o m (о , 2 T ) i n t o th e u n i t c i r c l e S 1 , i n t e r s e c t i n g ea ch B o r e l 1 fu n c t io n f :Г о ,2 7 Г ) — ^ S 1 . L e t us d e f i n e o u r m u l t i f u n c t i o n F :R —£■ С b y fo r m u la F ( x ) ï * S 1 " { e ( [ * 3 H * w h ere f x j i s th e c l a s s o f x 6 R i n th e q u o t ie n t g ro u p R / 2 ÎT Z = f o , 2 l T ) . I t i s e a s y t o eh eok as i n th eorem 7 t h a t F i s lo w e r se m io o n tin u o u s w i t h op en a r c s as v a lu e s b u t a d m it t in g no B o r e l 1 s e l e c t o r . Л M o r e o v e r , s in c e g b e lo n g s a l s o t o th e secon d B o r e l c l a s s and th u s Gr g i s s ( ( x , y ) : y = g ( x ) } i s a Fg^ s u b s e t o f th e c y l i n d e r [ o , 2 ) x S 1, henoe Gr F : = j ( x , y ) : y é F (x )^ = [ o , 2 f i ) x S 1 - Gr g i s а s u b s e t o f t h i s o y l i n d e r and th u s o f th e w h o le p r o d u c t s p a c e R x S 1 . T h is c o m p le te s o u r argu m en t.From th eorem s 7 and 8 i t f o l l o w s t h a t th e r a n g e sp a c e in o u r th eorem 5 ca n n o t b e g e n e r a l i z e d i n v a r io u s w a y s. A n o th e r i n t e r e s t i n g q u e s t io n s i s w h e th e r o f n o t th e p o lis h n e s s o f Z i s e s s e n t i a l i n th eorem 1. Ve u se th e exam ple 5 o f f l 8 j to a n sw er th e a b o v e p ro b le m ,
THEOREM 9 « Assum ing th e continuum h y p o t h e s is , t h e r e e x i s t m e t r ic s p a c e s X and Z and a m u lt if u n c t io n F: X —*• Z w it h nonem pty c lo s e d v a lu e s such t h a t F i s i n f i r s t lo w e r o l a s s , b u t F has no B o r e l s e l e c t o r .
P r o o f : ( o f . [
9J ,
Г18З )
L e tSi
d e n o te th e s m a lle s t c o u n t a b le o r d i n a l . L e t Z = ^oC: oç < SŁ J d e n o te th e s e t o f a l l c o u n t a b le o r d i n a l s w it h th e d i s c r e t e t o p o lo g y , w h ile th e r e a l80 l i n * R= : X i s in q u ir e d t o g e t h e r w it h u s u a l e u c lid e a n t o p o l o g y . A rra n g e a l l p o in t s o f X i n a t r a n e f i n i t e sequ en ce o f t y p e 5Ü and d e f i n e th e m u lt if u n c t io n F :X —* Z b y fo rm u la X 3 x ^ I ß : f i £ ос } f e r < *'< Л -E v id e n t ly F i s o f f i r s t lo w e r o l a s s . L e t f :X —^ Z b e any s e l e o t o r f o r F . S in c e f ( X«. ) -^ *<“ f o r e v e r y c*r< SI , ea ch f i b e r f _ 1 ( Ç ) * 3 ^ * ^ c o u n t a b le . S in c e th e f i b e r s a r e p a ir w is e d i s j o i n t and s in c e th e f a m i l y o f a l l s u b s e ts o f a l l f i b e r s has c a r d i n a l i t y g r e a t e r th an th e s e f o f a l l B o r e l s e t s , i t f o l l o w s t h a t t h e r e e x i s t s some s u b s e ts i , o f Z so t h a t f _ 1 (Z ) = U i e c o t B—m e a s u r a b le . S in c e Z i s b o th Q j a ; ° open and c lS s e d i n th e d i s c r e t e sp a c e Z, t h i s o o m p le te a th e argu m en t.
A cknow ledgem ent : I w ou ld l i k e t o a c k n o w le d g e many h e l p f u l c o n v e r s a t io n s and s u g g e s t io n s b y P r o f e s s o r Z. G ran de. REFERENCES [ 1 ] C a s t a in g C ., Sur l e s m u l t i - a p p l i c a t i o n s m e a s u r a b le s , Revue F r a n ç a is e d ' I n fo r m a t iq u e e t de R e ch e ro h e O pera t i o n e l l e s , 1 ( 1 9 6 7 ) , 91-126[2] C a s t a in g C . , V a la d ie r M ., C onvex a n a l y s i s and m ea s u ra b le m u lt ifu n c t i o n s , U n i v e r s i t é d es S o ie n o e s e t T e o h n iq u e s du L a n g u ed oc, M o n t p e l l i e r (1 9 7 5 ) [
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9 ] Ś lę z a k W. , Some rem arks a b ou t C e d e r - L e v i th eorem ,P r o c e e d in g s o f R e a l Func t i o n s C o n fe r e n c e h e ld i n B y d g o s z c z ,
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859-903
PRZYCZYNEK DO TE O R II SELEKTORÓW BORELOWSKIEJ KLASY ALFA
S t r e s z c z e n ia Głównym w yn ikiem p r a c y j e s t n a s t ę p u ją c e u o g ó l n ie n ie t w ie r d z e n ia C ed era i L e v i e go : k a żd a m u lt ifu r ik c ja F :X —^ Z , g d z i e X j e s t d o s k o n a le norm alną p r z e s t r z e n i ą t o p o l o g i c z n ą , a Z ośrodkow ą p r z e s t r z e n i ą F r e c h è t a , p r z y jmu ją c a s w o je w a r t o ś c i z o k r e ś lo n e j r o d z in y D/Z/ i b ę d ą c a d o l n e j k l a s y a l f a , p o s ia d a b o r e lo w s k i s e l e k t o r k l a s y a l f a . L i c zn e k o n t r - p r z y k ł a d y z a m ie s z c z o n e w p r a c y w s k a z u ją , ż e p r z y j ę t e o g r a n i c z e n ia na w a r t o ś c i F n i e mogą b y ć o s ł a b i o n e .
