Complex Measure, Dual Space of L p Space, Radon-Nikodym Theorem and Riesz Representation Theorems

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1 Complex Measure, Dual Spae of L p Spae, Rado-Nkodym Theorem ad Resz Represetato Theorems By Ng Tze Beg Our am s to show how to detfy the dual or ojugate spae of L p (,) ad C 0 (), the spae of otuous omplex futos o a loally ompat topologal spae, whh vash at fty There are some very useful ad bas results used I should meto the Lebesgue deomposto of a bouded postve measure wth respet to aother bouded postve measure Ths s aalogous to the Lebesgue deomposto of a reasg futo to sum of a absolutely otuous reasg futo, a reasg sgular futo ad a saltus type futo The Rado-Nkodym Theorem provdes the seed for the detfato of a bouded omplex lear futoal o a L p spae, through the Rado Nkodym dervatve The Rado-Nkodym dervatve s also the key to tegrato over omplex measure, through the polar deomposto of omplex measure Wth ths we a the represet a bouded omplex lear futoal o C 0 () by a Lebesgue tegral over a omplex measure The am s to fd ths measure ad show that t s uque Ths s the Resz Represetato Theorem for omplex measure The proof of the Rado-Nkodym Theorem uses a rual result Hlbert spae theory, more spefally that the bouded lear futoal of a Hlbert spae s determed by er produt wth a uque elemet of the Hlbert spae Ths s played out here by the Hlbert spae, L 2 (, ), where s a postve measure We have added the tegral represetato of otuous real lear futoal o the spae of bouded otuous real-valued futos o a ormal Hausdorff topologal spae We ed the artle wth a bref dsusso o Resz type represetato theorems for the topologal dual of the spae of bouded otuous futo o a ompletely regular Hausdorff spae Reall that f s a set ad M a -algebra o, the a postve measure o M s a outably addtve futo : M R mappg the -algebra M to the exteded postve real umbers, a real measure o M s a outably addtve futo : M R mappg the -algebra M to the real umbers ad a omplex measure o M s a outably addtve futo : M C mappg the -algebra M to the omplex umbers Hee, a real measure s a omplex measure but a postve measure s ot eessarly a real measure or a omplex measure

2 Suppose : M C s a omplex measure The for ay M, ( ) C ad so ( ) Our frst osderato s to fd a smallest postve measure suh that ( ) ( ) for all M Suppose s a outable dsjot olleto of sets M The M Hee, by outable addtvty, Ths meas s overget ad as the summato s depedet of the order of, the summato must be absolutely overget Theorem Let : M Cbe a omplex measure o the measure spae (, M ), where s a set ad M s a -algebra o Defe ( ) sup all parttos of The s a measure o M, alled the total varato measure of Note that for ay M, ( ) Proof Plaly, 0 We shall show that s outably addtve Take M Suppose F s a partto of by dsjot sets M We shall show that F ( ) We show that F ( ) as follows For eah teger, hoose t F a partto, j j G of F suh that The by defto of F, there exsts t G, j ( ( F ) j ) () 2

3 The G, j s a partto of Now, G, j, j double seres Therefore,, j G, ( ) j s a absolutely overget, j by defto of ( ), j G, j t by () for all t F It follows that F ( ) Next, we show that F ( ) Let j H j ad H be ay other partto of The for eah j, F H j j s a partto of F It follows that 3 F H Hj F Hj F Hj F Hj j j j j F Ths holds for ay partto H j of Therefore, that F ( ) j ( ) s a partto of F It follows ad so s outably addtve o M ad s therefore a postve measure o M Our ext result s a asserto that the total varato measure of a omplex measure s a fte postve measure Proposto 2 Let : M C be a omplex measure o the measure spae (, M ), where s a set ad M s a -algebra o The the total varato measure of,, s a fte postve measure We already kew that s a postve measure We oly eed to show that t s fte We shall eed the followg tehal lemma

4 Lemma 3 Let z, z 2,, zc The there exsts a subset S z, z2,, z suh that z 6 z S Proof Let w z The two les y x dvde the omplex plae to 4 quadrats, Q, Q2, Q3 ad Q 4as show the dagram below C y-axs Q 2 y = x Q 3 Q x-axs Q 4 y = x xame the pots eah quadrat I oe of the four quadrats, we must have that the sum of the modulus of the pots s greater tha or equal to 4 w Suppose t ours Q The zq z w 4 Observe that f z Q, the Rez z os z se 2 The 4 z Re z Rez z z z SQ 2 24 Q Q Q 6 4

5 Suppose where zq3 z w 4 The, smlarly, for z Q3, we get Thus 4 Rez z It follows that 2 Rez Rez zos, z Rez Rez z z z SQ3 Q3 Q3 Q3 Suppose zq2 z w 4 If z Q2, the Imz zos 0, where Hee, 4 Imz z os z, se 2 The 4 z Im z Imz z z z SQ Q2 Q2 Q2 6 Suppose Hee, zq4 z w 4 If z Q4, the Imz z os z, se 2 z z, where Imz Im os 0 The z Imz Imz z z z SQ4 Q4 Q4 Q4 So, we a take S to be oe of Q, Q2, Q3 ad Q 4terseto wth 4 z, z2,, z, whose sum s greater tha or equal to w Ths ompletes the proof of Lemma 4 3 Proof of Proposto 2 Ths s a proof by otradto Suppose there exsts B0M suh that B0 ( ) We shall show that we a deompose B0 A B, a dsjot uo wth A, B M, A ad B 5

6 Repeat ths proedure to B ad dutvely to B to get a sequee A, where, Aare parwse dsjot ad a sequee Bwth for all A teger Let C A The C M ad the olleto A ossts of parwse dsjot sets ad so by the outable addtvty of, A ( C) But the seres A aot overge absolutely as A But we kow that the seres must overge absolutely, se t s depedet of the order of the A Ths otradto shows that there does ot exst a member B0M wth B0 ( ) Suppose B0 ad so s a fte postve measure ( ) of B 0 suh that The for ay real umber t > 0, there exsts a partto t Ths mples that there exsts a teger N suh that t for all teger, the t Note that B0 teger N suh that ( ) So, we a take t6 ( B0) N t6 B0 Therefore, by Lemma 3, there exsts a subset S B0 6 6 S N N B t For f Hee, there exsts a N suh that,2,, 6

7 Now, let A The A B0 ad S dsjot It follows that 6 ( A) 6 6 ( A) 6 S, se are B Hee, A B 0 Let B B0 A The B B A A B 0 0 ( ) 0 Fally, se s a measure, A B B So, oe of or 0 B must equal Arrage for ths to be B ad the other to be A Reame, f eessary Thus, f ( ), the all B0 ad apply the above proess to B ad dutvely to B for >, to obta a olleto of dsjot sets, A ad a olleto of sets B wth 0 B of, for 0 Let C A B A B, A B A,, for, The C M ad by the outable addtvty A ( C) But the seres A aot overge absolutely as A otradtg that for a omplex measure, A must overge absolutely Thus ( ) It follows that for all M,, ( ) ( ) measure A ad so s a fte Corollary 4 Let : M C be a omplex measure o the measure spae (, M ), where s a set ad M s a -algebra o The { ( ) : M } s a bouded subset of the omplex plae Thus, every omplex measure s of bouded varato Proof Ths s beause ( ) ( ) Now, let M(M ) be the olleto of omplex measures o the -algebra M If, M(M ), defe ( ) ( ) ( ) for all M ad 7

8 ( ) ( ) for all M ad ay C The ths makes M(M ) to a omplex lear spae wth orm gve by ( ) for M(M ) We verfy that s a orm o M(M ) For ay M, take a partto F of by dsjot sets M The ( )( F) ( F) ( F) ( ) ( ) Thus, by the defto of ( )( ) ( ) ( )( ) ( ) ( ), ( )( ) ( ) ( ) It follows that Plaly for ay omplex umber ad ay M(M ), ( ) ( ) Also 0 ( ) 0whh tur mples that ( ) 0 mples for all M, It follows that ( ) 0 for all M ad so 0 Thus, M(M ) s a ormed lear spae Whe s M(M ) a Baah spae? If s a Hausdorff topologal spae ad M s a -algebra otag the Borel sets of, we a assoate to eah omplex measure,, M(M ), a lear futoal o C( ) wth the sup orm, defed by ( f) fdfor f C( ), whe we a make sese of tegrato over a omplex measure (See Defto ) By Proposto 2, the total varato measure of s a bouded postve measure ad so s a bouded lear futoal However, we do ot kow f ths assoato s oe to oe ad ether do we kow f the assoato s oto Whe s a loally ompat Hausdorff topologal spae, we shall vestgate ths questo due ourse Theorem 20 (Resz Represetato Theorem) Now we look at the stuato of two measures, whh are basally depedet of oe aother, meag eah oe s o-zero o a set whh s dsjot from the set o whh the other s ozero We desrbe suh a stuato as follows Let be a Lebesgue measure o R Let L ( R, ) be the equvalee lasses of absolutely tegrable omplex futos o R wth the L ( ) orm, f fd R Now for a fxed f L ( R, ), defe ( ) fd for Lebesgue measurable set The s a measure Suppose ow we have two futos f, f2l( R, ) suh that ff2 0 Let A x: f( x) 0 The Let ( ) f d for =, 2 The we have, A A2, =, 2, ( ) f d f d( A) A 8

9 ( A) f d0ad 2 A 2 ( A) f d0 2 A 2 We abstrat the propertes of the two measures above the followg defto Defto 5 Let (, M ) be a measure spae, where M s a -algebra o Let be a postve measure o M ad be a omplex or postve measure o M (a) We say s absolutely otuous wth respet to ad wrte f M ad ( ) 0mples that ( ) 0 (b) We say s oetrated o A for some A M, f for all M, ( ) ( A) () Suppose s oetrated o A ad 2 s oetrated o A 2 wth A A2 The we say ad 2 are mutually sgular ad wrte 2 If s ay omplex measure oetrated o some set of -measure aero, the we wrte Note that f, the M ad ( ) 0 ( ) 0 Ths s beause for M ad ay partto, F, of by dsjot sets M, ( ) 0 ( F) 0 ( F) 0 ( ) 0 Proposto 6 Let (, M ) be a measure spae, where M s a -algebra o Let be a postve measure o M ad be a omplex or a real measure o M, f ad oly f, gve ay > 0, there exsts > 0, suh that for all M, ( ) ( ) Proof Suppose gve ay > 0, there exsts > 0, suh that for all M, ( ) ( ) Thus, there exsts 0 suh that ( ) ( ) If ( ) 0, the ( ) for all teger Hee, ( ) for all teger It follows that ( ) 0ad so ( ) 0 Ths meas 9

10 Coversely, suppose We shall prove that gve ay > 0, there exsts > 0, suh that for all M, ( ) ( ) We show ths by way of otradto Suppose there exsts a > 0 suh that for ay > 0, there exsts M, wth ( ) but ( ) So, takg, r a teger,there 2 r exsts r M, wth ( r ) 2 r but ( r ) Let Fr ad sr s F The F r M, for eah teger r, M ad r r 2 2 Fr s s r sr Hee, F r r sr for teger r It 2 follows that 0 If s a omplex measure, the s of bouded varato so that ( ) Se s a fte postve measure by Proposto 2, by the otuty from above property of a measure, Fr lm Fr r r Se F for teger r, we must have 0 r r r But 0 ad mples that ( ) 0 So, we have arrved at a otradto ad ths meas that gve ay > 0, there exsts > 0, suh that for all M, ( ) ( ) If s a real measure, the s a omplex measure ad we obta the same otradto as above for the overse We have the followg mmedate osequee of Defto 5 Lemma 7 Let (, M ) be a measure spae, where M s a -algebra o Let be a postve measure o M Suppose, ad 2 are omplex measures o M (a) s oetrated o A for all M, A mples ( ) 0 (b) If s oetrated o A, the so s ts total varato () If 2, the 2 0

11 (d) If ad 2, the 2 (e) If ad 2, the 2 (f) If, the (g) If ad 2, the 2 (h) If ad, the = 0 Proof (a) If s oetrated o A, A( ) ( A) 0 Coversely, suppose for all M, A mples ( ) 0 The for all M, ( ) AAAAA0 A Hee, s oetrated o A (b) Suppose s oetrated o A The for all M, ( ) A Take a partto, F, of by dsjot sets M The F F A Ase F A s a partto of A Se ths s true for ay partto, F, of by dsjot sets M, A Se A ad s a fte postve measure, A Therefore, A for ay M Hee, s oetrated o A () If 2, the s oetrated o A ad 2 s oetrated o A 2 for some A ad A 2 M wth A A2 By part (b), s oetrated o A ad 2 s oetrated o A 2 Hee, 2 (d) If ad 2, the s oetrated o A ad 2 s oetrated o A 2 for some A ad A 2 M suh that ( A) ( A2) 0 For all M, ( ) A ad ( ) A Now, 2 2 2

12 2 2 ad A ad A2 A are dsjot A A A A A ad belog to M Therefore, A A A A A A, se by part (a), A2 A 0 as ( ) Smlarly, we have 2 A A2 2 A A A, 2 Hee, for all M, 2A A2 2 Moreover, A A AA 0A A 0 Thus, (e) Suppose ad 2 The for ay M, 2 Therefore, ( ) ( ) ( ) 0 Ths meas 2 2 ( ) 0 ( ) ( ) 0 2 (f) We have already proved ths mmedately after Defto 5 (g) Suppose ad 2 Suppose 2 s oetrated o A 2 for some A 2 M wth ( A2) 0 It follows that for all M ad A 2 ( ) 0 As, ths mples that for all M ad A 2, ( ) 0 Thus s oetrated o some set the omplemet of A 2 beause for ay M ( ) A A A A A Se A2 A2, 2 (h) Suppose ad By part (g) Ths a oly happe f 0 We a verfy ths dretly mples that s oetrated o A for some A M wth ( A) 0 For ay M, But se ( A) 0, ( ) ( ) ( ) ( A A A) ( A) 0 ( A) 0ad as, ( A) 0 Therefore, ( ) 0 2

13 Theorem 8 Let (, M ) be a measure spae, where M s a -algebra o Suppose ad are two postve bouded measures o M The (a) (The Lebesgue Deomposto Theorem) There s a uque par of measures, a (the absolutely otuous part of wth respet to ) ad s (the sgular part of wth respet to ) suh that, a s where a ad s Moreover, a s ad both measures are postve (b) (Rado-Nkodym Theorem) There s a futo h L (, ) suh that ( ) a hd for all M ad h s almost everywhere uque wth respet to Here L f f f d (, ) : C ; s measurable ad Not all measure spaes are bouded measure spaes, for example, the Lebesgue measure o R s ot bouded But R s a outable uo of sets of fte Lebesgue measure We say a measure spae (, M, ), where s a postve measure, s -fte or s a -fte postve measure f every set M s at most a outable uo of sets wth fte -measure Remarks After provg ths theorem, we shall mmedately exted to the ase where s a postve ad -fte measure (for example, whe s the Lebesgue k measure o R ) ad s a omplex measure 2 It s helpful to thk of as a Lebesgue measure o [0,] 3 Obvously, f s defed by ( ) fd for M ad a fxed f L (, ), the The pot of the part (b) of the theorem (Rado Nkodym Theorem) s that the overse s also true 3

14 4 The Rado Nkodym Theorem s ofte abbrevated to d hd a d d a or h ad h s alled the Rado Nkodym dervatve of a wth respet to 5 The uqueess part of the theorem may be prove easly as follows Suppose we have a s a s, where a, a ad a, a s Let va a s s The by property (e) Lemma 7, The a a s s v a a ad s v by property (d) Lemma 7 It follows the by s property (h) Lemma 7 that v 0 Cosequetly, a a ad s s I part (b) of the theorem (Rado Nkodym Theorem), that h s almost everywhere uque wth respet to, s dedued as follows Suppose h s aother futo L (, ) suh that ( ) hd for all M The 0 ( ) ( ) hd hd hh d a a a for all M Therefore, hh 0almost everywhere wth respet to ad so h h almost everywhere wth respet to We shall eed the followg tehal lemma for the proof of Theorem 8 Lemma 9 If s a bouded postve measure o the measure spae (, M ) ad f L (, ) s suh that ( ) fd, for all M wth ( ) 0, the 0 f almost everywhere wth respet to That s to say, f all the averages of f over all M belog to the ut dsk, the almost all values of f belog to the ut dsk Proof Let B z: z be the ut dsk Take z 0 outsde the ut dsk B ad a real umber r suh that 0r z0 Let B0 z: zz0 r The B B0 4

15 B B0 r z0 Let f ( B) The 0 M, se f L (, ) We shall show that Assumg ths s true for B 0, the t s true for ay ope dsk the omplemet of B As B s ope ad so s a outable uo of suh dsks, t follows that f B 0 as f B s a outable uo of sets of measure zero Hee, 0 f almost everywhere wth respet to Now we show that 0 0 Suppose o the otrary that fd z f z d f z d ( ) ( ) ( ) The ( 0 0) rd r But fd z z fd z ( ) ( ) beause ( 0 0) fd Ths meas r z0 Ths otradts that r z0 0 0 Hee, Remark I the proof of Lemma 9, we a use ay losed dsk plae of the losed ut dsk B 5

16 Lemma 9* If s a bouded postve measure o the measure spae (, M ) ad f L (, ) s suh that ( ) fd, 2 2 for all M wth ( ) 0, the to 0 f almost everywhere wth respet 2 2 For the proof, we may replae B by B z: z 2 2 Take z 0 outsde the dsk B, that s z0 Take 0r z0 Let B0 z: zz0 r Let 2 2 f ( B) 0 0 f B The as above we a show that 0 0 It follows that ad so 0 f 2 2 Suppose o the otrary that 0 0 fd z f z d f z d ( ) ( ) ( ) The ( 0 0) rd r But fd z fd z z fd ( ) ( ) ( ) Hee, r z0 z0 2 2 Ths otradts that 0 r z Hee, For the proof of Theorem 8, we shall use a very useful property of a Hlbert spae 6

17 Let H be a Hlbert spae wth er produt x, y satsfyg the followg propertes: () x, x 0; x, x 0x 0; (2) x, y y, x, the omplex ojugate of y, x ; (3) xy, z x, z y, z ; (4) x, y x, y If the orm of a Baah spae V arses from a er produt, the t s alled a Hlbert spae More presely, a er produt o a (real or omplex) lear spae V s a salar valued futo o V V, whose value o (x, y) V V s deoted by x, y ad the futo satsfes the followg propertes: () x, x 0; x, x 0x 0; (2) x, y y, x, the omplex ojugate of y, x ; (3) xy, z x, z y, z ; (4) x, y x, y The orm o H s gve by x x, x for x H We have the Shwarz Iequalty for er produt: x, y x y for all x, y H Wth respet to the metr assoated wth the orm, H s a Baah spae, e, a omplete metr spae Defe for a fxed y H, the lear futoal, y: HC, gve by y( x) x, y for all x H The y s a bouded (omplex) lear futoal As y( x) x, y x y, y( y) y, y y y, y y y( x) y sup : x0 x y ad o aout of 7

18 The bas result Hlbert spae theory s that the overse s also true Suppose :H C s a bouded lear futoal, the yfor some y H It s ths way that we set up the ojugate lear sometry Suppose H * s the olleto of all bouded omplex lear futoal o H The assoato of the bouded omplex lear futoal wth y as y: : H* H gve by ( ) y, where y, s a lear sometry preservg orm Note that ( ) yfor ay omplex salar Note also that ( ) y The proof of ths result s depedet of measure theory We brefly gve the proof here If = 0, the take y = 0 ad plaly, y Suppose 0 Let N xh: ( x) 0 H It s easly see that N s a losed subspae of H As H s omplete, N beg a losed subspae of H, s omplete The orthogoal omplemet of N must ota a ozero g We may hoose g suh that ( g) The ( x) 0 mples that x N ad so x, g 0 For eah x H, ad so x( x) g ( x) ( x) ( g) ( x) ( x) 0 x( x) g N Hee x( x) g, g 0 It follows that x, g ( x) g, g 0 Ths meas 2 ( x) g x, g for all x H ad as 2 g 0, g ( x) x, g for all x H 2 g Thus, f y, the ( x) x, y for all x H Note that y s uque 2 g For f y H s suh that ( x) x, y for all x H, the x, yy 0for all x H Hee, 2 Therefore, y y y y y y, y y 0 8

19 Proof of Theorem 8 Let = + Se ad are bouded postve measures, s a bouded postve measure o M The for ay M measurable futo f: C, f d fd fd () vdetly, () s true f f for M It the follows that () holds for measurable smple futos se ay smple futo s a omplex lear ombato of measurable haraterst futos If f s real valued, oegatve ad measurable, the there exsts a reasg sequee of oegatve measurable smple futos s suh that s f potwse o The we have sd sd sd The applyg the Lebesgue Mootoe Covergee Theorem, we have fd lm s dlm s dlm s d fd fd Now f f L (, ), e, f d, the Re f ad Im f are measurable ad Re f, Re f, Im f ad Imf Re, Re, Im, Im f f f f f are all measurable ad tegrable, se Se () holds for o-egatve real valued measurable futos, () holds for f L (, ) Moreover, f L (, ) ad f L (, ) Coversely, suppose f L (, ) adf L (, ) The Refd Refd Refd ad Imfd Imfd Imfd It follows that f L (, ) f d fd fd Now we take the Hlbert spae, 2 C, H L 2 (, ) f: ; f s measurable ad f d wth er produt, f g f gd ad orm 2 f f d 2, ad For the proof that H s a Hlbert spae, see Theorem of Covex Futo, L p Spaes, Spae of Cotuous Futos, Lus s Theorem 2 9

20 We defe a omplex lear futoal :H C by ( f) f d Note that ths s well defed By Hölders Iequalty (Theorem 0, Covex Futo, L p Spaes, Spae of Cotuous Futos, Lus s Theorem), f d f f d f d ( ) f d,, 2, 2, as ( ) ( ) ( ), se ad are bouded measures ad 2 2 f d for f H Therefore, postve Hee, fd exsts Moreover,, ( f) fd f d f d fd f d f d se s se s postve ad f d f d f d 2 2 f d f 2, 2 2 ( ) ( ), Hee, 2 ( ) ad s a bouded omplex lear futoal Se H s a Hlbert spae, there exsts g H, g s uque almost everywhere wth respet to suh that ( f) f, g for all f H That s, fd f gd (*) We shall ext show that g s real ad uque almost everywhere wth respet to ad that 0g Now, for M, 0 d ( ) Substtutef (*), we get 0 d gd ( ) ( ) Hee, f we take M suh that ( ) 0, the 0 gd gd ( ) ( ) 20

21 Therefore, Im 0 Hee, Imgd 0for all M Therefore, ( ) gd Img Img 0almost everywhere wth respet to Thus, g s real almost everywhere wth respet to ad for all M suh that ( ) 0, Ths meas 0 ( ) gd ( ) gd Therefore, by Lemma 9*, 2 2 g Se g s 2 2 real almost everywhere wth respet to, 0g almost everywhere wth respet to We ow redefe g to take the value 0, where g s ot real ad where 0 g( x) does ot hold Ths futo s obvously equal to g almost everywhere wth respet to We shall ow assume that g s real, 0 g( x) ad If f H L 2 (, ), the as = +, ad are bouded g H L 2 (, ) 2 postve measures, f, gh L(, ) ad by the Hölders Iequalty, gf L(, ) ad ( g) fd fd g fd ( f) g fd f gd g f d f gd, as f gd f gd f gd Hee, we have for all f H L 2 (, ), ( g) fd f gd (**) 2 2 Note that f, g H L(, ) f, g L(, ) f g L(, ) Let Ax:0 g( x) ad S x: g( x) The A ad S are measurable, A S ad AS Let ( a ) A ad ( ) s S for all M The plaly, a s oetrated o A ad s s oetrated o S Thus a s Put f S (**),we get 2

22 ( g) Sd Sgd Se A S, g 0o S ad 0o A, we have the 0 ( g) d gd gd d ( S) S S S S It follows that s S Note that g s bouded ad so g s bouded for ay teger Se ( ), g H L 2 (, ) get, e, Now, puttg 2 2 ( )( ) ( ) 2 f ( gg g ) g g g g d g g g gd, 2 ( ) ( ) (**), we g d g g g d (***) If x S, the g( x) so that g ( x) 0 If x A, the 0 g( x) ad so g ( x) ց 0 o A Therefore, by the Lebesgue Mootoe Covergee Theorem,, as, ( g ) d A( g ) d Ad ( A) a( ) for ay M 2 The tegrad o the rght had sde of (***), gg g, reases mootoally to some futo h potwse ad h s o-egatve So, as h s a potwse lmt of a reasg sequee of o-egatve measurable futos, h s measurable ad by the Lebesgue Mootoe Covergee Theorem, 2 ( ) g g g d hd ր as Therefore, ( ) hd for ay M a If =, the ( ) hd hd a hd Hee, h L (, ) Hee, a ad as a( ) hd 0 ( a ) A ( A ), Plaly, f () = 0, the 22

23 Ths ompletes the proof of Theorem 8 We ow dsuss the varous extesos of Theorem 8 Varous xtesos of the Rado Nkodym Theorem () To where s postve ad bouded but s postve ad -fte For k k example, may be the Lebesgue measure o Ror R as Ror R s -ompat k for Ror R s a outable uo of sets of fte Lebesgue measure We wrte, where We may suppose that the outable famly are parwse dsjot If ot, we may replae by 23 The apply the theorem to eah We get, a,, s, ad h eah defed o xted the defto to trvally Defe value to be zero o The sple together so that h( x) h( x) f x Se 0h, 0h As ( ), h L (, ) (2) To whe s postve ad bouded but s real Wrte, ad The ad are postve 2 2 ad bouded measures Applyg the theorem to the postve ad egatve parts of, we get a, s, h ad,, a s h The let a a a ad s s s The a s As a ad a, by property (e), a a a Also, as s ad s, s s s by property (d) Thus, by property (g), a s Now, for M, ( ) a h d, ( ) a h d so that ( ) ( ) ( ) h h d Let hh h a a a (3) To whe s postve ad bouded ad s omplex Let R be the real part of ad I be the magary part of Now apply part (2) ad spled together smlarly (4) To whe s real but s postve ad -fte

24 Wrte, ad The ad are postve 2 2 ad bouded measures Apply exteso () to ad to get,, ad 24 a s h a, s, h The sple together as (2) as follows Let a a a ad s s s The a s As a ad a, by property (e), a a a Also, as s ad s, s s s by property (d) Thus, by property (g), a s Now, for M, a ( ) h d, a ( ) h d so that a( ) a ( ) a ( ) h h d Let hh h (5) To whe s omplex but s postve ad -fte Wrte Re Im The Re adim are real measures Apply exteso (4) to Re adim separately ad the sple together (6) To whe both ad are postve ad -fte Wrte, where are parwse dsjot,, for eah teger Applyg (), we get a,, s, ad h eah defed o The sple together as () We obta h( x) h( x) olude that h L(, ), we a oly assert that loally L (, ) for x But we aot h L(, ), that s, h s (7) To whe s omplex ad s a bouded postve measure We eed to defe fd for omplex measure Oe defed, the exteso s mmedate If s a omplex measure o the measure spae (, M ), the ts total varato measure,, by Proposto 2, s a bouded or fte postve measure Note that for M, ( ) 0 ( ) 0 Followg Defto 5, we say s absolutely otuous wth respet to f ( ) 0 ( ) 0 Ths s equvalet to s absolutely otuous wth respet to Thus, for a omplex measure, Smlarly, we say, e, f s oetrated o A wth ( A) 0 Ths s equvalet to Thus, f s a bouded postve measure ad s a omplex measure, the by Theorem 8, a s, where a, s ad there exsts h L,

25 suh that a ( ) hd Thus, a ad s By the polar deomposto of (see Theorem 0 below), there exsts h L, suh that h ad d hd Therefore,, ( ) a hd h hh d hh hd gd where g hh The futo g L, hh L se h s bouded ad so (, ) (See Defto below) Some applatos of the Rado-Nkodym Theorem Observe that for a omplex umber z, we a wrte, where e For z ze a omplex futo, f, we a smlarly wrte f f h, where h For a omplex matrx A, we a wrte A UR, where U s utary ad R s postve sem-defte Hermta For a omplex measure, we a wrte, as we shall show later, whereh d hd, All these represetatos are kow as polar deomposto aalogy wth the polar represetato of omplex umbers The Polar Deomposto of a Complex Measure Theorem 0 (Polar Deomposto) If s a omplex measure o the measure spae (, M ), the there exsts a measurable omplex futo h: Csuh that h L,, h ad d hd More presely, for ay M, ( ) hd Proof Plaly, By Proposto 2, s a bouded postve measure Therefore, by the Rado-Nkodym Theorem (Theorem 8, part (b)), there exsts 25

26 h L, suh that ( ) hd Rado-Nkodym Theorem, dsussed above) We may prove t dretly here (We are usg the exteso (3) of Note that Re ad Im Therefore, by Theorem 8, there exst ad h Im L, h L Re, Therefore, where Re Re ( ) suh that for all M, h Red ad Im ( ) Im ( ) Re ( ) Im ( ) Re IM h d h h d hd 26, IM Note that h s measurable ad h L, h h h Now we show that h almost everywhere wth respet to Frstly, we show that h almost everywhere wth respet to Let Ax: h( x) r, for 0 < r < Let A be a partto of A by dsjot sets M The ( A) hd ( ) ( ) A hd r d r A r A A A Therefore, ( A) r ( A) for 0 < r < Hee, ( A) 0for 0 < r < Ths mples that h almost everywhere wth respet to Next, we show that h almost everywhere wth respet to Suppose M ad ( ) 0 As ( ), hd ( ) ( ) Therefore, hd ( ) hd ( ) ( ) It follows the by Lemma 9, that 0 h almost everywhere wth respet to Hee, h almost everywhere wth respet to Thus, x: h 0

27 Redefe h so that o ths set all x x : h, h(x) = The we have ( ) h x for We shall ow proeed to defe tegrato over a omplex measure For a measurable smple futo, a omplex umber for,, where s a s measurable ad as sd a( ) a hd a d a hd a hd shd If f s real valued, o-egatve ad measurable, the there exsts a reasg sequee of measurable o-egatve smple futos s suh that s ր f We a wrte h = Re h + Im h, RehReh Reh ad Imh Imh Imh It the follows from the Lebesgue Mootoe Covergee Theorem that Re ր f Reh d, s Reh d Re s h d ր f h d, Im ր f Imh d, ad s Imh d Im s h d ր f h d So for a o-egatve futo f, we say fd exsts f Re f hd ad f Im hd exst, e, fd fhd For a real value measurable futo, we a wrte f f f ad defe fd f d f d ad fd fd fd fally for measurable omplex futo f, Re Im Defto I summary, we may defe for f a omplex measurable futo, a omplex measure, fd fhd 27

28 So, f L (, ), f ad oly f, Ref hd ad Im oly f, fd f hd, f ad If ad are omplex measures, the relato, fd f d fd, (*) holds wheever f s a bouded measurable futo or whe f L, ad f L, Plaly, (*) holds for f a measurable haraterst futo Ths s beause ( ) ( ) ( ) d d d Hee, (*) s true for measurable smple futos The (*) holds for ay bouded measurable futo f vdetly, f f L,, the f L, ad so (*) holds ad f L, We may defe omplex measure by usg ay fxed omplex futo f L,, where s a postve measure o M Proposto 2 Suppose s a postve measure o the measure spae (, M ) ad f L f C f d (, ) : ; That s, f ( ) fd, the ( ) If d fd, the d f d f d Proof It s easy to show that s a omplex measure o M By Theorem 0 (Polar Deomposto of Complex Measure), there exsts a measurable futo, h: C, suh that h L,, h ad d hd More presely, for ay M, By hypothess, ( ) ( ) hd fd Now for a haraterst futo d fd Therefore, for a measurable smple measurable, ( ) 28, where s

29 futo s, sd sfd It follows that for a bouded measurable futo g fd Se h s a bouded measurable futo, g, gd hd h fd Hee, for ay measurable M, Smlarly, by usg ( ) hd, we get hd hhd d ( ) 0 hd h fd It follows that ( ) 0for all M Therefore, h f 0 almost h fd everywhere wth respet to Therefore, h f h f f almost everywhere wth respet to Hee, ( ) f d for ay M Ths ompletes the proof Now we use Theorem 0 for a real measure o (, M ) Theorem 3 Hah-Jorda Deomposto Theorem Let be a real measure o (, M ) 2 (Jorda) Wrte ad measure of The (Hah) 2, ad That s, A A wth A A ad for ay M, ( ) ( A ) ad ( ) ( A ), where s the total varato are bouded postve measure Proof By Theorem 0 (Polar Deomposto), there exsts a measurable futo, h: C, suh that h L,, h ad d hd More presely, for ay M, ( ) hd 29

30 Se s real, we may assume that h s real Hee, h Let A x : h( x) ad A x: h( x) The A ad A are measurable Plaly, A A Se ad s a bouded postve measure, by Proposto 2, ad are bouded postve measures o M ad Now, If M, Smlarly, h( x), xa h 2 0, xa ad 0, xa h 2 h( x), x A ( ) d ( ) ( ) 2 hd hd d A 2 A A ( ) d ( ) ( ) 2 hd hd d A 2 A A Therefore, s oetrated o A ad s oetrated o A ad so Next, we show that the Jorda Deomposto s optmal the followg sese Corollary 4 Let be a real measure o (, M ) Suppose 2, where ad 2are postve measures The ad 2 Proof Reall that ad For ay M, ( ) ( ) ( ) ( ) ( ) ( ) ( ) Therefore, for ay partto,, of by dsjot sets M, ( ) ( ) ( ) ( ) ( ) 2 2 Hee, by defto of, ( ) ( ) 2( ) for all M Now, 2 so that 2 so that 2 2 Observe that ( ) ( ) ( ) ( ) ( ) 2( ) 30

31 2 ( ) ( ) ( ) ( ) 2 ( ) 2 Thus, ( ) ( ) for all M Ths meas Now, 2 0ad so 2 p The Dual Spae or Cojugate Spae of L (, ) Suppose (, M ) s a measure spae ad : M R s a postve measure p Suppose f L (, ) q ad g L(, ), where p ad q are ojugate des suh that The we have the Hölders equalty (see p q Theorem 0, Covex Futo, L p Spaes, Spae of Cotuous Futos, Lus s Theorem), f g f g, p, q,, where h, s the (, ) Therefore, f g L (, ) L orm gve by h h d, for p p Defe : (, ) C by ( f) f gd The for ay f L (, ) g L ( f ) f gd f gd g f g q, p, g p Hee, g s a bouded omplex lear futoal o L (, ) ad the orm of ths lear futoal, Reall that for a lear futoal :V g q, g o a orm spae V wth orm, ( x) sup : xv, x0 x C We vestgate f the overse s true Is ay bouded omplex lear futoal : p q L (, ) C expressble as gfor some g L(, )? Oe ase s lear Take pq 2ad we kow L 2 (, ) s a Hlbert pae wth er produt, f g f gd By a o-measure theoret argumet, f :H C s a bouded lear futoal o a Hlbert spae H, the there exsts y H suh that ( x) x, y for all x H AsH L 2 (, ) s a Hlbert spae, for 3

32 2 a lear futoal : L(, ) C, there exsts a futog L 2 (, ), uque almost everywhere wth respet to, suh that g Oe ase s false Ths s the ase whe p For L (, ), the aswer s false beause L (, ) does ot fursh all bouded lear futoals o L (, ) (See xample 256, Prple of Real Aalyss, Alprats ad Burkshaw) If p, the aswer s always yes However, we wll prove ths together wth the ase p = wth the addtoal hypothess that the measure be -fte Subsequetly, we shall prove the ase for p, wthout the -fteess odto o the measure Oe ase s usually yes, exept very bg spaes (e, where ope sets are ot -fte), for p = We ote that for g L (, ), we a defe a bouded omplex lear futoal : (, ) C by ( f) f gdfor f L (, ) g L g Ths s beause se f L (, ), there exsts a set B of -measure zero suh that ( ), f x f for all x B so that f g f g almost everywhere wth, respet to ad so f g L (, ) Thus, ( f) f gd f gd g f d g, f gd g f,,, Therefore, by the defto of g, g, g Hee, g s a bouded omplex lear futoal Next, we shall show that g, Se g s measurable, there exsts a measurable futo h suh that h ad g g ghso that hg g Ideed, we a defe h as follows Let The B0 ad V B0 are measurable se g s measurable B0 x : g( x) 0 z Let : C{0} C be defed by ( z) The s otuous o C {0} Let z g g B 0 The g s measurable ad g 0 Defe h g, e, 32

33 h( x) g( x) ( g( x) ( x)) The futo h s measurable beause s otuous o C {0} I partular, h Moreover, g( x) g( x) g( x), xv g( x) ( x) g( x) g( x) B( x) 0 g( x) 0, xb0 B0 ( ) ( ) ( ) g( x) g xhx g x Let f( x) h( x) Plaly, f s measurable ad f ad so t s essetally bouded ad f Therefore,, ( ) g f f gd hgd gd g, Hee, g( f) g g f Therefore,,,, g, g sup g( f): f, f L (, ), g Lkewse, f g L (, ), the the omplex lear futoal as g: L(, ) C defed by ( f) f gdfor f L (, ) s a bouded lear futoal as g,,,, ( ) g f f gd f gd f g d g f so that g, g For ay M, g( ) gd gd ad g( ) g ( ), g gd Hee, for ( ) 0, ( ) radus g, we olude that gd Cosequetly, Therefore,, g g, g g The by Lemma 9, wth losed dsk of g g almost everywhere wth respet to g q Now, we assume that p Take g L(, ), where We have g p q already show that g, We ow show by a smlar argumet as above q, 33

34 that g q, g p p( q) q f g Take g ad so q f g h The f s measurable Moreover, p q p f d g d It follows that f L (, ) Now, ( f ) ( g h ) g h gd g d g g q q q q q p q p q p q g d g d f d g d g f q, p, Hee, g( f) g f Thus, f g 0, q, p, q, by defto of g ad that g g, q, g g q, If g 0,, the g = 0 almost everywhere wth respet q to Hee, g 0ad so 0 I summary, we have the followg, result g g q Theorem 5 Suppose (, M ) s a measure spae ad : M R s a postve measure Suppose p ad qare ojugate des suh that p q q The for ay g L(, ), the omplex lear futoal, C, defed by ( f) f gd, s a bouded omplex lear : p g L (, ) futoal suh that g q, g g For a measure spae wth a -fte measure we have the followg represetato of bouded omplex lear futoal Theorem 6 Suppose (, M ) s a measure spae ad : M R s a -fte postve measure Let p ad q be suh that Suppose p q p : L (, ) Cs a bouded omplex lear futoal The there exsts a q uque g L(, ) suh that, g ( f) f gd ( f) p for all f L (, ) Moreover, g g q, More presely, the dual spae of p p q L (, ), L (, ) *, s sometr somorph wth L(, ), uder a Baah spae somorphsm preservg orm 34

35 Proof The proof s dffult ad we shall do t two steps The uqueess part s easy ad we shall dspose of ths presetly q Suppose gad g L(, ) are suh that they both satsfy the oluso of the theorem Take ayf for ay M wth ( ) The 0 ( f) ( f) gd g d gg d gg d It follows that for all M, 0 wth respet to By Theorem 5, q, g g g d Hee, g g almost everywhere We shall use the futoal to defe a measure o M We shall use the Rado Nkodym Theorem Frst of all, f 0, the we a just take g to be zero almost everywhere wth respet to So we assume that 0 Step We osder the speal ase whe s a fte postve measure, e, () < p For ay M, plaly belogs to L (, ), se d ( ) ( ) We defe a measure o M by ( ) ( ) for M We hek that ths defes a measure o M Trvally, ( ) 0 Plaly s ftely addtve, for ad 2M wth 2, ( ) ( ) ( ) Thus, by duto, we obta that, f s ay fte olleto of dsjot measurable sets M, the ( ) Now we show that s 35

36 outably addtve Suppose members of M Let A s a dsjot uo of outably fte The p A, Ad p A as, ( ) 0 by the otuty from below property of the measure So, se s bouded ad so s otuous, ( ) ( ) as, beause A A A 0 A as A Hee, ( ) ( ) ( ) ( ) ( ) A Ths proves that s a omplex measure 36 p p Therefore, If M ad ( ) 0, the 0almost everywhere wth respet to Hee, ( ) ( ) (0) 0 Thus,,,e, s absolutely otuous wth respet to Therefore, by the Rado Nkodym Theorem (Theorem 8 xteso (3) for postve fte ad omplex ), a s, a, s ad there exsts a measurable futo g L (, ) suh that ( ) gd Hee, by Lemma 7 (e) ad so as s, s 0, by Lemma 7 (h) s a Therefore, a ad ( ) It follows that, for ay M, gd ( ) ( ) gd gd p We shall exted ths equalty to arbtrary f L (, ), a ( f) f gd (*) We have just show that (*) s true for measurable haraterst futos Therefore, (*) s true for measurable smple futos We the lam that (*) holds for every f L (, ) Note that (*) holds for o-egatve measurable

37 bouded futo, beause f f s bouded ad o-egatve, the there exsts a reasg sequee of measurable smple o-egatve futos s suh that s ր f Beause f s bouded, s ր f uformly (See Theorem 7) As ( s ) s gd, by the Lebesgue Domated Covergee Theorem, lm ( s ) lm s gd f gd Note that as ad so lm ( s ) ( f) Thus ( f ) f gd 37 s ր f uformly, sf 0, ad so (*) s true for a bouded measurable o-egatve futo If f s a bouded measurable real valued futo, the we a wrte f f f, where f ad f are bouded oegatve measurable futos ad so by learty, (*) holds for bouded measurable real valued futo Fally, f f s a bouded measurable omplex futo, the wrte f Ref Imf, where Re f ad Imf are bouded real valued measurable futos Therefore, by learty (*) holds for ay bouded measurable futos If f L (, ), the there exsts measurable subset B of suh that ( ), f x f f x xb 0, xb ( ), for all xb ad ( B) 0 Letf ( x) The f f almost everywhere wth respet to ad f s bouded ad measurable Therefore, ( ) ( ) q Now we shall show that g L(, ) We osder the ase p = f f fgd f gd For ay M, gd ( ) ( ) Therefore, for ( ) 0,, ( ) gd Therefore, by Lemma 9, wth losed dsk of radus, g almost everywhere wth respet to Hee, g L (, ) ad, Now for the ase < p < g As show the proof of Theorem 5, there exsts a measurable futo h suh that hg g ad h For eah teger, let x: g( x) ad q f hg p follows that f L(, ) Plaly, f s measurable ad bouded ad so f L (, ) It p

38 Now, f p ( q) p g 0 o q g o, Puttg f (*), we get q q ( f ) f gd g hgd g d We also have, ( q) p p q p ( f ) f g d g d Hee, q p q g d g d p, q Lettg teds to, we get q q g d q q Ths s beause g ր g mootoally ad so by the Lebesgue Mootoe Covergee Theorem, lm q q q g d lm g d g d q It follows that g L(, ) Now we shall show that g We reall that the olleto, S s: C ; ss a smple measurable futo wth x: s( x) 0 s p p dese L (, ) the L (, ) metr (See Proposto 6, Covex Futo, L p Soaes, Spae of Cotuous Futos, Lus s Theorem) Se ( ), every smple measurable futo s S We have already show that (*) holds for all smple measurable futos ad that meas ad g agrees o S As p ad gare both otuous o L (, ), g g q, Step 2 g By Theorem 5, Now we move o to the ase whe the measure spae (, M, ) s -fte We may assume that ( ) for eah, a dsjot uo of measurable sets wth Let Y The Y ( ) p Note that for ay M, we a defe : L (, ) C by 38

39 p ( f) ( f) for f L (, ) It s easy to see that s a omplex lear futoal p satsfes ( f) ( f) f f for all f L (, ) p, p, Therefore, p Take = Let : L(, ) C be defed as above by ( f) ( f) ad p we have Note that f L (, ), where (, M ) s the submeasure spae of (, M, ) Cosder defed by ɶ h( x), x, h( x) 0, x ɶ ( h) ( hɶ ) p : L (, ) C ɶ p, for ay hl (, ) ad h ɶ : C s gve by Plaly, h ɶ : C s -measurable Obvously ɶ s a lear futoal Moreover, as oted above, ɶ ( ) ( ɶ) ɶ h h h h p, p, It follows that ɶ ad ɶ p s a bouded lear futoal o L (, ) As ( ) q g L(, ) ad, by what we have just proved for fte measure, there exsts p suh that for ay (, ) h L, ɶ ( h) ( hɶ ) hgd g q, q, ɶ g, where g : C s a measurable exteso of g to defed by g( x), x, g ( x) Thus, 0, x ( f) f ɶ f, where f s osdered as a futo o, f g d f g d f g d p Therefore, for all f L (, ), 39

40 ( f ) f g d f g f g p,, p, q q, Hee, g q, p Now, osder : L(, ) C defed by ( f) f Y The ( ) f Y f f f f f gd p Let : L ( Y, ) 2 f gd ( g g g ) fd C be defed by Y p ( f) ( f) for f L ( Y, ) the obvous exteso of f to by defg f( x) f( x) Y, whe x Y ad ad f s f( x) 0 whe x Y s obvously a omplex lear futoal ad for all p f L ( Y, ), Y ( f) ( g g g ) fd ( g g g ) fd 2 2 Y Y Y f f g g g fd Thus, ( ) ( ) ( ) 2 f g g g f g g g 2, 2 p, q, p Y q, Y Therefore, s a bouded lear futoal ad g g g 2 q, Now ( ) ( ) so that It f f f f f Y Y p, p, Y follows that g g2 g q, I partular, q, g for eah teger Let gg g2 g g Note that ths s well defed For ay x, x for some teger so thatg ( x) 0for ad ( ) g x g( x) for k As g s measurable, g s measurable k 40

41 Observe that by defto of by Fatou s Lemma, Therefore, g, g( x) g ( x) lmf g ( x) q q q q lm f lm f g d g d g g2 g q, Hee, q q g g d q, Now ( f) f f Y q It follows that g L(, ) as p For all f L (, ), gfd gfd gfd g fd lm g g2 gfd lmy ff Hee, by Theorem 5, q, Y g Now, we osder the ase p = The preedg argumet apples to the ase p =, yeldg, : L(, ) C, ɶ : L(, ) C, g L (, ) suh that ɶ ( ) ( ɶ h h) hgd for h L (, ), ɶ g g : L( Y, Y ),, C, wth ( f) ( f) for We also have: L(, ) C, f L ( Y, ) ad Y 2 We have ( f) f f f gd ( g g g ) fd Y also dedued that ( f) ( gg2 g) fd gg2 g f gg2 g f,,,, Y so that s a bouded lear futoal ad so g g g Se, we dedue as before that g g g 2 exsts a measurable set B suh that ( B ) 0ad g ( x) g ( x) g ( x) 2 for all 2, x B, Hee, there 4

42 Let x B, B B The B 0ad B B B B Therefore, for all g ( x) g ( x) g ( x) 2 for all teger Hee, lm g( x) g2( x) g( x) g( x) for all x B so that g, Ths mples that g L (, ), f f gd ad, 42 g Theorem 7 Suppose (, M, ) s a measure spae ad f: C s a measurable futo If f s tegrable wth respet to, the the set x: f( x) 0s -fte p It follows easly that for ay f L (, ), p <, the set x: f( x) 0s - fte Proof The futo f s tegrable meas that f s measurable ad fd Partto (0, ) by ad 0 f,,, Let f, for teger Plaly, s measurable for 0 < Se fd, fd ( ) fd Note that for teger 0 ad so ( ) for x: f( x) 0 x: f ( x) 0 teger 0 As 0 p If f L (, ), the, x: f( x) 0s -fte p f d ad as x: f( x) 0 x: f p ( x) 0 0 follows that x: f( x) 0s -fte Here we ote that, t p f d for eah teger 0 ad the same argumet apples to gve the same oluso p ( ) We ow show that Theorem 6, for p, we may drop the odto that the measure be -fte

43 Theorem 8 Suppose (, M ) s a measure spae ad : M R s a postve measure Let p ad q be suh that p Suppose : L (, ) p q Cs a q bouded omplex lear futoal The there exsts a uque g L(, ) suh that, g ( f) f gd ( f) p for all f L (, ) Moreover, g g q, More presely, the dual spae of p p q L (, ), L (, ) *, s sometr somorph wth L(, ), uder a Baah spae somorphsm preservg orm Proof Let S be the olleto of -fte measurable subsets of M That s, S = { M : s -fte} Now for eah S, by Theorem 6, there exsts a uque g vashg outsde p of suh that for ay f L (, ) ad f vashg outsde of, suh that Ths s beause defed by ( f ) f g d f g d p : L (, ) C, ( f ) f, for f L p (, ) f x x 0, x, where ( ), f( x), s a bouded omplex lear futoal Se f: C s M measurable, f s q M measurable Atually, Theorem 6 gves a uque g L(, ) suh that p ( f) f g d for f L (, ) Note that : g C s M measurable ad so the exteso ( ), g ( x) s M measurable Hee, g x x 0, x ( f) p f g d Moreover, for f L (, ), 43

44 p p f d f d f d so that p f L (, ) B, the g g For eah S, defe B p We ote that f almost everywhere wth respet to o B by uqueess ( ) q g d ( f ) f f f p, Now, for f L p (, ) p,, Hee By Theorem 6, g g q, q, It follows that ( ): S s bouded above by q Let sup ( ): S The there exsts a sequee of -fte measurable sets S suh that ( ) Let H Plaly, H s -fte ad so H S ad as H for H, H Let g g H g L q (, ) gh( x), xh 0, xh Therefore, Note that f s ay set of -fte measure ad otas H, the by uqueess, g g almost everywhere o H wth respet to O the other had, ad H q q q q g d g d g d ( ) g d g d H q q q g d g d g d H H q q q q g d g d g d g d H H H H q q ad so g d 0 Ths mples that g 0 almost everywhere o H H Thus, g g almost everywhere o wth respet to p Now we take ay futo f L (, ) Let Gx: f ( x) 0 The G s measurable ad -fte by Theorem 7 Therefore, G H s -fte Let f f The 44

45 ( f) ( f) fg d fg d fgd fgd p Thus, we have show that for ay f L (, ) 6, we a dedue that g q,, ( f) fgd As Theorem Ths ompletes the proof However, for p =, we may ot relax the -fteess odto for the measure Theorem 6 For there s a example of a measure spae (, M, ) wth ot -fte ad a bouded lear futoal,, o L (, ) suh that there does ot exst g L (, ) satsfyg ( f) fgd The ext theorem s a result, whh we have used, about approxmato of measurable o-egatve futo by smple measurable futos Theorem 9 Suppose f: R s a o-egatve measurable futo, where (, M ) s a measure spae The there exsts a reasg sequee of measurable smple futos ( s ) overgg potwse to f If f s bouded, the ( s ) overges uformly to f Proof We ostrut the sequee ( s ) as follows For eah teger, dvde the terval [0, ] to 2 sub-tervals of legth 2 f, 2 2 Let s, 2, F 2,,2,, 2, F f [, ) ad Se f s measurable, the sets, ad F are measurable 45

46 j Note that,, j, j, where or j2 O the set,, 2 2 ( ) j j s x takes o the value whe x s, j ad the value whe x s, j Observe also that F f f f F f [, ) [, ),, ad f,, : 2 to ( )2 Thus, o the set set f, F, s ( ) x, s ( ) x to Therefore, s s takes o the value + whe x s, j ad o the takes o values, whe s ( x ) s defed ad s equal Se f( x), take a teger N suh that N > f (x), the for all N, s ( ) x N ad so the sequee s potwse overgee Moreover, for eah teger > f (x), f (x) les, for some suh that 2 ad so 2 2 ( ) ( ) s x f x Furthermore, s( x) f( x) Hee lm s ( ) ( ) x f x 2 Now, suppose f s bouded suh that 0 f K ad K Frst of all, ote that F for all teger K For ay teger > K, f, 2 2, f 2 K 2 Ths meas for 0 f K, we effetvely partto the terval [0, K] to 2 K sub-tervals eah of legth 2 Observe that se f( x) K, for ay teger N K, N > f (x) for all x, ad so for all N, s ( ) x Nfor all x ad so the sequee s uformly bouded Moreover, for eah teger N, f (x) les, for some suh that so that, s( x) f( x) for all x Hee, for all N ad for all x, 2 f( x) s( x) f( x) Ths meas that ( s ) overges uformly to f 2 46

47 The Resz Represetato Theorem - The Complex Verso I Postve Borel Measure ad Resz Represetato Theorem, we represet a postve (omplex) lear futoal, : C( ) C, where s a loally ompat Hausdorff topologal spae ad C( ) s the spae of otuous omplex futos o wth ompat support wth the uform orm, by ( f) f d, for some postve measure, whh s almost regular ad omplete o a - algebra M otag all the Borel sets of There was o questo of beg otuous, e, bouded Atually, some ases, wth addtoal odto o, t s true that s postve mples that s bouded Note that C( ) s edowed wth the uform sup orm If the represetg measure satsfes ( ), the for ay f C( ) u u ( f ) fd f d f d f ( ) Reall that f sup f( x): x u,, se f s otuous wth ompat support It follows that ( ) ad so s bouded Ths meas that f s ompat, by Theorem (Resz Represetato Theorem) of Postve Borel Measure ad Resz Represetato Theorem, the represetg measure s fte ad so the postve omplex lear futoal s bouded ad so s otuous If we spealze to postve real lear futoal : C, R( ) R, where C, R( ) s the spae of otuous real valued futo o wth ompat support, the as a osequee of the represetato theorem, s a bouded real lear futoal f the represetg measure s fte But we would eed some addtoal odto, for example whe s ompat, to obta a fte represetg measure However, a real lear futoal o the ormed lear spae CR ( ) wth the sup orm s otuous f ad oly f t s bouded Whe s ompat ad Hausdorff, a real lear futoal o the ormed lear spae CR ( ) a be represeted by a regular fte real Borel measure expressble as 47

48 the dfferee of two regular fte postve measures (See Theorem 3, Fte Borel Measure ad Resz Represetato Theorem) Now we wat to osder ay bouded omplex lear futoal : C( ) ad represet as ( f) C fd for some omplex measure o a -algebra M otag all the Borel sets of Se : C( ) C s bouded, we a exted to the ompleto of C( ), u, e, C ( ), 0 u the spae of otuous omplex futos o whh vashes at fty (See Proposto 25, Covex Futos, L p Spaes, Spae of Cotuous Futos, Lus s Theorem) Hee, we mght as well osder the represetato of bouded omplex lear futoal : C0( ) C o C ( ), 0 u Theorem 20 Resz Represetato Theorem - The Complex Verso Let be a loally ompat Hausdorff topologal spae ad : C0( ) C a bouded omplex lear futoal o C ( ) 0 wth the uform sup orm The there exsts a uque regular omplex Borel measure suh that ( f) fd Moreover, ( ) That s to say, the dual spae or ojugate spae of C, 0 ( ) C ( ) * M( ) 0, where M ( ) s the olleto of all regular Borel omplex measures wth orm gve by ( ) ad " " here meas Baah spae somorphsm preservg orm Reall that a omplex measure s regular f s regular as a postve measure s fte f s fte as a postve measure If [0,], the C [0,] * C [0,] * s the spae of all regular omplex Borel measures o [0, ] 0 Before we prove the theorem, we preset a tehal result oerg the regularty of the sum of regular omplex measures 48

49 Proposto 2 Suppose s a topologal spae ad (, M ) s a measure spae, where M s a -algebra otag all the Borel sets of Suppose ad 2 are two regular omplex Borel measures The 2 s also a regular omplex Borel measure Proof Plaly, 2s a omplex Borel measure The measures ad 2 are regular meas that ad 2 are regular We show that 2 s er regular s er regular mples that for ay M, gve > 0, there exsts ompat K suh that K ( ) ( ) That s to say, ( K) ( ) ( K) () Smlarly, as 2 s er regular, for ay M, gve > 0, there exsts ompat K 2 suh that ( K ) ( ) ( K ) (2) Let K K K2 The K s ompat adk ( K) ( K) ( K) Hee, ( ) 2 ( K) ( ) Ths mples that 2 2 ( ) sup ( K), Kompat ad K Thus, for ay M, 2( ) sup 2( K), Kompat ad K follows that 2 s er regular It We ow show that 2 s outer regular ad 2 are both outer regular Ths meas for ay M, gve > 0, there exsts a ope set V suh that ( V) ( ) Therefore, ( V ) Smlarly, there exsts a ope set V 2 suh that 2( V2 ) Let V V V2 The V s ope ad 49

50 V Therefore, 2 ( V ) ( V ) 2( V ) 2 Hee, ( V) ( V ) ( ) ( ) It follows that 2( ) f 2( V), V ope ad V As ths holds for ay M, 2 s outer regular Therefore, 2 s regular ad so 2s regular Proof of Theorem 20 We prove the uqueess part of the theorem Suppose ad 2 are two regular omplex Borel measures satsfyg the oluso of the theorem The, where ( f) ( f) fd fd fd 2 By Proposto 2, 2 2 s also a regular omplex Borel measure By Theorem 0, there exsts a measurable omplex futo h: Csuh that h L,, h ad d hd That s, for ay M, ad for ay f C ( ) 0 ( ) hd, fd fhd We shall show that ( ) 0 Oe we have show ths, the se for all M, ( ) ( ) ( ) 0, ( ) 0 ad so = 2 Now, ( ) d hhd hhd fd hhd fhd h hf d hf d, (*) It follows that for all M, ( ) 0 50

51 for ay f C ( ) 0 Se C( ) s dese L,, (see Theorem23, Covex Futos, L p Spaes, Spae of Cotuous Futos, Lus s Theorem), whe C( ) s edowed wth the L, orm ad se h L,, we a take a sequee of futos f C( ) suh that hf d 0 as It follows the from (*) that ( ) 0 L, so that f h Note that gve a bouded lear futoal o C ( ), f 0 0, we may ormalse t by takg so that ts orm s uty If 0, we a just take the trval Borel measure So ow we assume that 0ad ormalse t by osderg We shall thus assume wthout lost of geeralty that The key to the proof s to use the postve measure verso of the Resz Represetato Theorem (Theorem, Postve Borel Measure ad Resz Represetato Theorem) Assume that we a ostrut a postve omplex lear futoal o C( ) suh that ( f) f f () u The we a apply the postve measure verso of Resz Represetato Theorem (Theorem, Postve Borel Measure ad Resz Represetato Theorem) to to gve a postve omplete Borel measure,, whh s outer regular ad er regular wth respet to ope set ad sets of fte measure, suh that ( f) f d, for all f C ( ) 5