On points of the approximate semicontinuity

ZESZYTY NAUKOWE WY ŻSZEJ SZKOŁ Y PEDAGOGICZNEJ w BYDGOSZCZY P r o b le m y Ma t ema t y c z n e 1985 z . 7 TOMASZ NATKANIEC WS P w B y d g o s z c z y ON POINTS OF THE A P P R O X IM A T E S E M IC O N T IN U IT Y Ws u s e t h e f o l l o w i n g n o t a t i o n . T h e s i g n I d e n o t e s t h e S '- i d e a l o f t h e M e a s u re z e r o s u b s e t s o f H, I f X f R I s me a s u r a b l e t h e n if £x> d e n o t e s t h e s e t o f a l l d e n s i t y p o i n t s o f X. F o r t h e f u n c t i o n f : R —* R t h e s i g n s a p - l l m i n f f (t ) and * x o p - 1 1 * su p f ( t ) d e n o te the a p p ro x im a te ly l o w e r and u p p e r t -*■ X l i m i t o f f a t x , r e s p e c t i v e l y . N o t i c e t h a t a p - l i m i n f f ( t ) = su p [ y : D * ( x , [ t : f ( t ) < . y } ) = 0^ and t - > x a p —1 1 * su p f ( t ) = i n f £ y : D * ( x , [ t : f ( t } > y } ) = 0 } , w h e re t e ^ X d ; c x , a ) = i ± » e u p ( - 5 - i ^ n . m ( J ) < i z , . n •♦co T h e s i g n s A ( f ) , S t f ) , S 1 C f ) d e n o t e t h e s e t s o f a l l p o i n t s BL A a t w h ic h f i s a p p r o x im a te c o n t in u o u s , u p p e r and l o w e r s e m i-c o n t ln u o u s , r e s p e c t iv e ly ! A ( f ) = j i t H : f Cx ) = a p - l l m I n f f ( t ) = a p - l i m su p f ( t ) } , 1 t - » x t + x S ( f ) = $ xfcR: f ( x ) ^ a p —11m su p f ( t > i , a t - » x T C f ) c {"x A R : f ( x ) > a p - l i m su p f (t > ^ , t -*x S 1 ( f ) = [ x C R s f t x ) < a p - l l m i n f f ( t ) j , * t + x T 1 ( f ) = f x £ R s f ( x ) < a p - l l m i n f f ( t ) } . a t x Z . G ra n d e sh ow ed i n t h e f o l l o w i n g f a o t s . FACT O. F o r e v e r y f u n o t i o n f : R —>■ R t h e s e t A ( f ) i s m e a s u r a b le . FACT I . F o r e v e r y f : R —*■ R we h a v e T C f) V T 1 ( f ) t 3 .

j

CL 6L FACT 2 . T h e s e t s Sa ( f ) - A ( f ) and S ^ C O - A ( f ) d o n o t c o n t a i n m e a s u r a b le s e t s o f t h e p o s i t i v e m e a s u r e . FACT 3 » L e t A ,B ,C sure s u b s e t s o f R s u ch t h a t

13 4 - CfcJ, - B S A and C £ A -B , - t h e r e e x i s t s a Gtf s e t D s u ch t h a t B = D -C , - t h e s e t A -B do n o t c o n t a i n a m e a s u r a b le s e t s o f p o s i t i v e m e a s u r e , - R-D i s t h e sum M U N, M and N a r e a F ~ s e t s , N6-3 and M S 'f ( M ) . Th en t h e r e e x i s t s a f u n c t i o n f : R — R su o h t h a t A ( f ) = B, S ( f ) = A , _a and T { f ) = C. _a My r e s u l t s a r e f o l l o w i n g { s e e [ 2 j ) . FACT U, (M A ) L e t A , C, A ” and C ' a r e s u b s e t s o f R s u ch t h a t ( i ) C u C ' 6 3 , ( i i ) B = A n A ' , ( i i i ) C £ A -B and c ' s a ' - B , ( i v ) t h e r e e x i s t s a G ^ s e t D s u c h t h a t B = D - ( C ^ C ; , ( y ) t h e s e t s A -B and a'- B do n o t c o n t a i n a m e a s u r a b le s e t s o f t h e p o s i t i v e m e a s u r e , ( v i ) t h e r e e x i s t s a s e q u e n c e (G ^ -) n ^ jj o f o p e n s e t s s u c h t h a t G . € G , D a H G and 0 , T ^ f{G ) - B i s n+1 n ’ nfcN n n e N ' n ' a s e t . 0 Then ( x ) t h e r e e x i s t s a f u n c t i o n f : R — R s u c h t h a t A ( f ) = B, S ( f ) = A, T ( f ) = C, s l ( f ) = A ' and T l ( f ) = C ' . a a a a

FACT 5 . (M A ) L e t A, B, C, A* and C ' a r e s u b s e t s o f R and t h e c o n d i t i o n s ( i ) - ( v ) and

( v i i ) R-D i s t h e sum M U N , M and N a r e a F^. s e t s , N t .3 and M S ( f CM )

h o l d . Th en t h e s t a t e m e n t { x ) h o ld s t o o .

REMARK 0 . N one o f t h e i m p l i c a t i o n s { v i ) C v i i ) and ( v i i ) ( y i ) ho I d s .

Ve c o n s i d e r t h e f o l l o w i n g e x a m p le s .

EXAMPLE 0 . L e t C £ ^ 0 , 1 > b e t h e C a n t o r ' a s e t s u c h t h a t C £ .t ! and P b e t h e s e t o f a l l b i r a t e r a l l i m i t p o i n t s o f

I

C . S ln o e t h e s e t C - P i s d e n s e i n C and P i s o f t h e s e c o n d c a t e g o r y i n C, P i s n o t Fg. s e t .

L e t us assum e t h a t B = R -P . T h en t h e c o n d i t i o n ( v i ) d o e s n o t h o l d . I f D i s a G ^ s e t and B £= D t h e n R-D P and R-D & J . Thu s f o r M = 0 and N = R-D we h a v e R-D = M U N . So t h e c o n d i t i o n ( v i i ) h o l d s . EXAMPLE 1 . L e t C b e t h e C a n t o r 's s e t s u c h t h a t C 4- ^ and C £ < 2 , 3 > , c ' b e t h e s e t o f a l l b i r a t e r a l l i m i t p o i n t s o f C and C 11 = C - c ' . I t i s c l e a r t h a t * f ( C ) £ C ' . I f M £ C ' i s a F<y s e t t h e n N = C-M i s a Gj- s e t and N i s d e n s e i n C . H en ce N i s r e s i d u a l i n C. S u p p o se t h a t N i s a F - . s e t . Th en N = F and £> X l & J N n a r e c l o s e d and p a i r w i s e d i s j o i n t s e t s [ 3 ] , S in c e N i s o f t h e s e c o n d c a t e g o r y i n C, t h e r e e x i s t s Fn w h ic h i s o f t h e s e c o n d c a t e g o r y i n C. S in c e Fq i s c l o s e d , t h e r e e x i s t s an o p e n i n t e r v a l I s u ch t h a t 0 0 C () I £ F ^ . H en ce Fn ^-^ •

A ssu m e that B - D - R -C . T h e n the c o n d it io n (v ii) d o e s n o t h o ld .

L e t Gn = D f o r n = 1 , 2 , . . . . N o t i c e t h a t B = 0* H en ce t h e c o n d i t i o n ( v i ) h o l d s . REMARK 1. T h e r e e x i s t s a s e t B and t h e r e e x i s t s a f u n c t i o n f : R — ^ R s u c h t h a t A ( f ) = B and t h e c o n d i t i o n s ( v i ) and ( v i i ) do n o t h o l d . We c o n s i d e r t h e f o l l o w i n g e x a m p le . EXAMPLE 2 . L e t P b e t h e s e t d e f i n e d i n E x a m p le 0 ,C b e t h e C a n t o r 's s e t fr o m E x a m p le 1 and B = R . ( C U P ) . L e t g : R -j> R b e a f u n c t i o n s u c h t h a t A ( g ) = R -P an d h : R - » R b e a f u n c t i o n su ch t h a t A ( h ) = R -C . L e t us d e f i n e a f u n c t i o n f : R — > R a s f o l l o w s f ( x ) = g ( x ) + h (x ). I t i s e a s y t o show t h a t A ( f ) = B and t h e c o n d i t i o n s ( v i ) and

( v i i ) d o n o t h o l d . PROBLEMS ( 1 ) L e t us assum e t h a t f o r A , a' , B , C, c' S R t h e c o n d i t i o n s ( i ) - ( v ) h o l d . D oes t h e n t h e s t a t e m e n t ( x ) h o l d ? (2 ) I s t h e r e f o r e v e r y f u n c t i o n f :R —* R a G^. s e t D su ch t h a t A ( f ) = D - ( T ( f ) u T ^ f ) ) ? >

ft

ft

1 3 9

136 REFERENCES [ 1 ] G ra n d e Z . , Q u e lq u e s r e ma r q u e s s u r l a s e m ic o n t i n u i t e su p a r i e u r e , Fund. M a th . C X X V / l / l985/ [2 ] N a tk a n i e c T . , Zbi o r y p u n k tó w c i ą g ł y c h i p ó ł c i ą g ł o ś c i f u n k c j i r z e c z y w i s t y c h , d o c t o r ' s t h e s i s [ 3 ] Si e r p iń s k i W. , S u r u n e p r o p r i e t e d e s en s e a b l e s F^_ l i n e a i r e s , Fh n d. M a th . 14 ( 19 2 9 )