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By C. Constantinescu
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Additional resources for Duality in Measure Theory
Example text
I (U) and call OF M E A S U R E S on Mp i. S t r u c t u r e For any DUALS Let u~M fundamental . solid We set := {~gM p Iu&M(~ ) } £1 (~) ~£[i (u) (£1oc(U)) (~[oc(~)) fSdu := <~,~> (~-u largest : R > u-integrable (locally ~-in- we set , ~ , A I > <]A. 1 any be d e n o t e d ~6Mp by fi ~, ~6~i(~)~ for any there = ffd~ we h a v e := < ~ o c / ~ ) l ~ . ~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) .
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. ~ be a B o r e l mea- and f o r any f a m i l y map ~ I) > Mp such that f o r any (~l ) IGI and f o r any f a m i l y in the set {~@((~i)16i ) ~ ~((~$i)16i)} is nowhere dense.
F I ) )6L IGI For a n y f a m i l y (f) in we h a v e L and f $((f ) i Let ($i)i~i tion of tions (X,M) f on space of be a family in Y ~((~1)~i)~i which (Y) ~ = f . 9 a) there exists a unique f ~ C (Y) is a nowhere dense set. 1 b) there exists a unique We want to show that 6 6£M p such does not depend on the re- of be another r e p r e s e n t a t i o n Mp set. 8) M(y). '$ a) we get ~n = for any (~'n)o9 n~M p . ~ ) tel ) = ~ ~I ) and t h e r e f o r e {~$'~m((~{ -] = ~ Hence ) ~i)} : { (~'6')o~$ # <0((G'~ ) a i ) o $ } ^, ({u ~' # m((~'~ {~' = # ~((~ and t h e r e f o r e ) ei)}.