By C. Constantinescu

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Extra info for Duality in Measure Theory

Example text

EM . = 0} , be a r e p r e s e n t a t i o n exists a unique We set element of of (x,M) C (y), . 1 a) is ~ f ~ is w an which proves v'f ° isomorphism of vector a) and b) . c) and d) lattices. Its are o b v i o u s . e) The e q u i v a l e n c e u~gL loc (v~) is o b v i o u s . For any A~R we have (~-p) (A) = < ] A . ~ , < > : ~ £ d ( ( u ] A) (vul) = v -I ((~). 1 a), b), e) . 4 The assertion Theorem 5 . I . 5 surable (gl) real &i in ~,~M follows Let I on ~I Z ~ Then t h e r e exists Mp such that from Thoerem we d e n o t e ~ by ~((g (Y,u,v) of set ~((g ~ : (X,M) = O.

5 and t h e r e f o r e we h a v e ~ 0 f o r any defined The map : ~i(~} belongs ~P(~) of the m a p LP(u)~L~ which ~ lattices. 1 surjectivity LP(u) < ~ {~lu£M} a) ~ 0 is ~I (~) (~l (u))n C Mpp a fundamental solid is a f u n d a m e n t a l for any ~ . 8). of C o n t i n u o u s real functions there exists ~6M such that on for a n y = ~ f~K(Y). Remark iff Then Y ~ > 0. Let ~ {~I~M} to see t h a t Let (Y,u,v) is d e f i n e d with compact ~ M pp such be a r e p r e s e n t a on the set carrier K(Y) and t h e r e f o r e = Sfd(v~) It is e a s y to see t h a t the a b o v e Hence difficult ~ ~ 0 l~I

3. Topologie Y Y' = /]xd(V-]l) by u (Y,u,v) and (Pro- = ~]ydl is an o p e n d e n s e In p a r t i c u l a r G~n~rale Y' set of is an o p e n and c l o s e d set of We set N : = {u6M I f ~ i l o c ( V ) } . Y which is a h y p e r s t o n i a n Ch i, § ii, ex 22 a). b I) f£i +. 2 c) i of U , N is a f u n d a m e n t a l is an o p e n Y for any (n,~)g~xN and IY ~ = u therefore V (u(fAn]x) n~]~ for a n y ~ ~N. 2 e) (f) Y') exists s u p t u ["f f n ] x ) ) "'' (Y) n~ on a d e n s e in set of Y'. 3.