11. Complex Measures - Probability

1 Tutorial 11: Complex Measures111. Complex MeasuresIn the following, ( ,F) denotes an arbitrary measurable 90 Let(an)n 1be a sequence of Complex numbers. Wesay that(an)n 1has thepermutation propertyif and only if, forall bijections :N N ,theseries + k=1a (k)converges inC1 Exercise (an)n 1be a sequence of Complex Show that if (an)n 1has the permutation property, then thesame is true of (Re(an))n 1and (Im(an))n Supposean Rfor alln 1. Show that if + k=1akconverges:+ k=1|ak|=+ + k=1a+k=+ k=1a k=+ 1which excludes as 11: Complex Measures2 Exercise (an)n 1be a sequence inR, such that the series + k=1akconverges, and + k=1|ak|=+.

2 LetA>0. We define:N+ ={k 1:ak 0},N ={k 1:ak<0}1. Show thatN+andN are Let +:N N+and :N N be two bijections. Showthe existence ofk1 1 such that:k1 k=1a +(k) A3. Show the existence of an increasing sequence (kp)p 1such that:kp k=kp 1+1a +(k) 11: Complex Measures3for allp 1, wherek0= Consider the permutation :N N defined informally by:( (1), +(1),.., +(k1) , (2), +(k1+1),.., +(k2) ,..)representing ( (1), (2),..). More specifically, definek 0=0andk p=kp+pfor allp 1. For alln N andp 1with:2k p 1<n k p(1)we define: (n)={ (p)ifn=k p 1+1 +(n p)ifn>k p 1+1(2)Show that :N N is indeed a 1, there exists a uniquep 1 such that (1) 11: Complex Measures45.}

3 Show that if + k=1a (k)converges, there isN 1, such that:n N,p 1 n+p k=n+1a (k) <A6. Explain why (an)n 1cannot have the permutation Prove the following theorem:Theorem 56 Let(an)n 1be a sequence of Complex numbers suchthat for all bijections :N N ,theseries + k=1a (k) , the series + k=1akconverges absolutely, + k=1|ak|<+ 11: Complex Measures5 Definition 91 Let( ,F)be a measurable space andE lmeasurable partitionofE, any sequence(En)n 1of pairwisedisjoint elements ofF, such thatE= n 92We callcomplex measureon a measurable space( ,F)any map :F C, such that for allE Fand(En)n 1measurable partition ofE,theseries + n=1 (En)converges to (E).

4 The set of all Complex Measures on( ,F)is denotedM1( ,F).Definition 93We callsigned measureon a measurable space( ,F), any Complex measure on( ,F)with values Show that a measure on ( ,F) may not be a Complex Show that for all M1( ,F), ( )= these tutorials, signed measure may not have values in{ ,+ }. 11: Complex Measures63. Show that a finite measure on ( ,F) is a Complex measure withvalues inR+, and Let M1( ,F). LetE Fand (En)n 1be a measurablepartition ofE. Show that:+ n=1| (En)|<+ 5. Let be a measure on ( ,F)andf L1C( ,F, ). Define: E F, (E) = Efd Show that is a Complex measure on ( ,F).

5 11: Complex Measures7 Definition 94 Let be a Complex measure on a measurable space( ,F).Wecalltotal variationof ,themap| |:F [0,+ ],defined by: E F,| |(E) =sup+ n=1| (En)|where the sup is taken over all measurable partitions(En)n be a Complex measure on ( ,F).1. Show that for allE F,| (E)| | |(E).2. Show that| |( )= be a Complex measure on ( ,F). LetE Fand(En)n 1be a measurable partition Show that there exists (tn)n 1inR,withtn<| |(En) for 11: Complex Measures82. Show that for alln 1, there exists a measurable partition(Epn)p 1ofEnsuch that:tn<+ p=1| (Epn)|3. Show that (Epn)n,p 1is a measurable partition Show that for allN 1, we have Nn=1tn | |(E).

6 5. Show that for allN 1, we have:N n=1| |(En) | |(E)6. Suppose that (Ap)p 1is another arbitrary measurable 11: Complex Measures9ofE. Show that for allp 1:| (Ap)| + n=1| (Ap En)|7. Show that for alln 1:+ p=1| (Ap En)| | |(En)8. Show that:+ p=1| (Ap)| + n=1| |(En)9. Show that| |:F [0,+ ]isameasureon( ,F). 11: Complex Measures10 Exercise , b R,a< C1([a, b];R), and define: x [a, b],H(x) = xaF (t)dt1. Show thatH C1([a, b];R)andH =F .2. Show that:F(b) F(a)= baF (t)dt3. Show that:12 + /2 /2cos d =1 4. Letu Rnand u:Rn Rnbe the translation u(x)=x+ that the Lebesgue measuredxon (Rn,B(Rn)) is invariantby translation u, ({ u B})=dx(B) for allB B(Rn).

7 11: Complex Measures115. Show that for allf L1C(Rn,B(Rn),dx), andu Rn: Rnf(x+u)dx= Rnf(x)dx6. Show that for all R,wehave: + cos+( )d = + cos+ d 7. Let Randk Zsuch thatk /2 <k+ 1. Show: 2k < 2k + 8. Show that: 2k cos+ d = 2k + cos+ d 11: Complex Measures129. Show that: + cos+ d = 2k + 2k cos+ d = + cos+ d 10. Show that for all R:12 + cos+( )d =1 Exercise ,..,zNbeNcomplex numbers. Let k Rbesuch thatzk=|zk|ei k, for allk=1,..,N. For all [ ,+ ], wedefineS( )={k=1,..,N:cos( k )>0}.1. Show that for all [ ,+ ], we have: k S( )zk = k S( )zke i k S( )|zk|cos( k ) 11: Complex Measures132.

8 Define :[ ,+ ] Rby ( )= Nk=1|zk|cos+( k ).Show the existence of 0 [ ,+ ] such that: ( 0)= sup [ ,+ ] ( )3. Show that:12 + ( )d =1 N k=1|zk|4. Conclude that:1 N k=1|zk| k S( 0)zk Exercise M1( ,F). Suppose that| |(E)=+ forsomeE F. Definet= (1 +| (E)|) R+. 11: Complex Measures141. Show that there is a measurable partition (En)n 1ofE,with:t<+ n=1| (En)|2. Show the existence ofN 1 such that:t<N n=1| (En)|3. Show the existence ofS {1,..,N}such that:N n=1| (En)| n S (En) 4. Show that| (A)|>t/ ,whereA= n LetB=E\A. Show that| (B)| | (A)| | (E)|. 11: Complex Measures156.

9 Show thatE=A Bwith| (A)|>1and| (B)|> Show that| |(A)=+ or| |(B)=+ .Exercise M1( ,F). Suppose that| |( ) = + .1. Show the existence ofA1,B1 F, such that =A1 B1,| (A1)|>1and| |(B1)=+ .2. Show the existence of a sequence (An)n 1of pairwise disjointelements ofF, such that| (An)|>1 for alln Show that the series + n=1 (An) does not converge to (A)whereA= + n= Conclude that| |( )<+ . 11: Complex Measures16 Theorem 57 Let be a Complex measure on a measurable space( ,F). Then, its total variation| |is a finite measure on( ,F).Exercise thatM1( ,F)isaC-vector space, with:( + )(E) = (E)+ (E)( )(E) =.

10 (E)where , M1( ,F), C,andE 95 LetHbe aK-vector space, whereK= lnormonH,anymapN:H R+, with the following properties:(i) x H,(N(x)=0 x=0)(ii) x H, K,N( x)=| |N(x)(iii) x, y H,N(x+y) N(x)+N(y) 11: Complex Measures17 Exercise Explain why . pmay not be a norm onLpK( ,F, ).2. Show that = , is a norm, when , is an Show that =| |( ) defines a norm onM1( ,F).Exercise M1( ,F) be a signed measure . Show that: + =12(| |+ ) =12(| | )are finite Measures such that: = + ,| |= ++ Exercise M1( ,F)andl:R2 Rbe a linear 11: Complex Measures181. Show thatlis Show thatl is a signed measure on ( ,F).