12. Radon-Nikodym Theorem
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1 Tutorial 12: Radon-Nikodym Theorem Radon-Nikodym Theorem In the following, (Ω, F) is an arbitrary measurable space. Definition 96 Let μ and ν be two (possibly complex) measures on (Ω, F). We say that ν is absolutely continuous with respect to μ, and we write ν<<μ, if and only if, for all F: μ() =0 ν() =0 xercise 1. Let μ be a measure on (Ω, F) andν M 1 (Ω, F). Show that ν<<μis equivalent to ν << μ. xercise 2. Let μ be a measure on (Ω, F) andν M 1 (Ω, F). Let ɛ>0. Suppose there exists a sequence ( n ) n 1 in F such that: n 1, μ( n ) 1 2 n, ν( n) ɛ Define: = lim sup n = n 1 n 1 k n k
2 Tutorial 12: Radon-Nikodym Theorem 2 1. Show that: μ() = lim n + μ k n k =0 2. Show that: ν () = lim n + ν k n k ɛ 3. Let λ be a measure on (Ω, F). Can we conclude in general that: λ() = lim 4. Prove the following: n + λ k n k
3 Tutorial 12: Radon-Nikodym Theorem 3 Theorem 58 Let μ be a measure on (Ω, F) and ν be a complex measure on (Ω, F). The following are equivalent: (i) ν<<μ (ii) ν << μ (iii) ɛ >0, δ >0, F,μ() δ ν() <ɛ xercise 3. Let μ be a measure on (Ω, F) andν M 1 (Ω, F) such that ν<<μ.letν 1 = Re(ν) andν 2 = Im(ν). 1. Show that ν 1 << μ and ν 2 << μ. 2. Show that ν + 1,ν 1,ν+ 2,ν 2 are absolutely continuous w.r. to μ. xercise 4. Let μ be a finite measure on (Ω, F)andf L 1 C (Ω, F,μ). Let S be a closed proper subset of C. We assume that for all F
4 Tutorial 12: Radon-Nikodym Theorem 4 such that μ() > 0, we have: 1 fdμ S μ() 1. Show there is a sequence (D n ) n 1 of closed discs in C, with: S c = + D n n=1 Let α n C, r n > 0 be such that D n = {z C : z α n r n }. 2. Suppose μ( n ) > 0forsomen 1, where n = {f D n }. Show that: 1 fdμ α n μ( n ) 1 f α n dμ r n n μ( n ) n 3. Show that for all n 1, μ({f D n })=0. 4. Prove the following:
5 Tutorial 12: Radon-Nikodym Theorem 5 Theorem 59 Let μ be a finite measure on (Ω, F), f L 1 C (Ω, F,μ). Let S be a closed subset of C such that for all F with μ() > 0, we have: 1 fdμ S μ() Then, f Sμ-a.s. xercise 5. Let μ be a σ-finite measure on (Ω, F). Let ( n ) n 1 be a sequence in F such that n Ωandμ( n ) < + for all n 1. Define w :(Ω, F) (R, B(R)) as: w = + n= n 1+μ( n ) 1 n 1. Show that for all ω Ω, 0 <w(ω) Show that w L 1 R (Ω, F,μ).
6 Tutorial 12: Radon-Nikodym Theorem 6 xercise 6. Let μ be a σ-finite measure on (Ω, F) andν be a finite measure on (Ω, F), such that ν<<μ. Let w L 1 R (Ω, F,μ)besuch that 0 <w 1. We define μ = wdμ, i.e. F, μ() = wdμ 1. Show that μ is a finite measure on (Ω, F). 2. Show that φ = ν + μ is also a finite measure on (Ω, F). 3. Show that for all f L 1 C (Ω, F,φ), we have f L1 C (Ω, F,ν), fw L 1 C (Ω, F,μ), and: fdφ = fdν + fwdμ
7 Tutorial 12: Radon-Nikodym Theorem 7 4. Show that for all f L 2 C (Ω, F,φ), we have: ( ) 1 f dν f dφ f dφ (φ(ω)) 2 5. Show that L 2 C (Ω, F,φ) L1 C (Ω, F,ν), and for f L2 C (Ω, F,φ): fdν φ(ω). f 2 6. Show the existence of g L 2 C (Ω, F,φ) such that: f L 2 C (Ω, F,φ), fdν = fgdφ (1) 7. Show that for all F such that φ() > 0, we have: 1 gdφ [0, 1] φ()
8 Tutorial 12: Radon-Nikodym Theorem 8 8. Show the existence of g L 2 C (Ω, F,φ) such that g(ω) [0, 1] for all ω Ω, and (1) still holds. 9. Show that for all f L 2 C (Ω, F,φ), we have: f(1 g)dν = fgwdμ 10. Show that for all n 1and F, f =(1+g g n )1 L 2 C(Ω, F,φ) 11. Show that for all n 1and F, (1 g n+1 )dν = g(1 + g g n )wdμ 12. Define: ( + ) h = gw g n n=0
9 Tutorial 12: Radon-Nikodym Theorem 9 Show that if A = {0 g<1}, then for all F: ν( A) = hdμ 13. Show that {h =+ } = A c and conclude that μ(a c )= Show that for all F,wehaveν() = hdμ. 15. Show that if μ is σ-finite on (Ω, F), and ν is a finite measure on (Ω, F) such that ν<<μ,thereexistsh L 1 R (Ω, F,μ), such that h 0 and: F,ν() = hdμ 16. Prove the following:
10 Tutorial 12: Radon-Nikodym Theorem 10 Theorem 60 (Radon-Nikodym:1) Let μ be a σ-finite measure on (Ω, F). Let ν be a complex measure on (Ω, F) such that ν << μ. Then, there exists some h L 1 C (Ω, F,μ) such that: F, ν() = hdμ If ν is a signed measure on (Ω, F), we can assume h L 1 R (Ω, F,μ). If ν is a finite measure on (Ω, F), we can assume h 0. xercise 7. Let f = u + iv L 1 C (Ω, F,μ), such that: F, fdμ =0 where μ is a measure on (Ω, F). 1. Show that: u + dμ = udμ {u 0}
11 Tutorial 12: Radon-Nikodym Theorem Show that f =0μ-a.s. 3. State and prove some uniqueness property in theorem (60). xercise 8. Let μ and ν be two σ-finite measures on (Ω, F) such that ν<<μ.let( n ) n 1 be a sequence in F such that n Ωand ν( n ) < + for all n 1. We define: n 1, ν n = ν n = ν(n ) 1. Show that there exists h n L 1 R (Ω, F,μ)withh n 0 and: F, ν n () = h n dμ (2) for all n Show that for all F, h n dμ h n+1 dμ
12 Tutorial 12: Radon-Nikodym Theorem Show that for all n, p 1, μ({h n h n+1 > 1 p })=0 4. Show that h n h n+1 μ-a.s. 5. Show the existence of a sequence (h n ) n 1 in L 1 R (Ω, F,μ)such that 0 h n h n+1 for all n 1andwith(2) still holding. 6. Let h =sup n 1 h n. Show that: F,ν() = hdμ (3) 7. Show that for all n 1, n hdμ < Show that h<+ μ-a.s. 9. Show there exists h :(Ω, F) R + measurable, while (3) holds. 10. Show that for all n 1, h L 1 R (Ω, F,μn ).
13 Tutorial 12: Radon-Nikodym Theorem 13 Theorem 61 (Radon-Nikodym:2) Let μ and ν be two σ-finite measures on (Ω, F) such that ν << μ. There exists a measurable map h :(Ω, F) (R +, B(R + )) such that: F, ν() = hdμ xercise 9. Let h, h :(Ω, F) [0, + ] be two non-negative and measurable maps. Let μ be a σ-finite measure on (Ω, F). We assume: F, hdμ = h dμ Let ( n ) n 1 be a sequence in F with n Ωandμ( n ) < + for all n 1. We define F n = n {h n} for all n Show that for all n and F, hdμfn = h dμ Fn < Show that for all n, p 1, μ(f n {h>h +1/p}) =0.
14 Tutorial 12: Radon-Nikodym Theorem Show that for all n 1, μ({f n {h h })=0. 4. Show that μ({h h } {h<+ }) =0. 5. Show that h = h μ-a.s. 6. State and prove some uniqueness property in theorem (61). xercise 10. Take Ω = { } and F = P(Ω) = {, { }}. Let μ be the measure on (Ω, F) defined by μ( ) =0andμ({ }) =+. Let h, h :(Ω, F) [0, + ] be defined by h( ) =1 2=h ( ). Show that we have: F, hdμ = h dμ xplain why this does not contradict the previous exercise. xercise 11. Let μ be a complex measure on (Ω, F). 1. Show that μ<< μ.
15 Tutorial 12: Radon-Nikodym Theorem Show the existence of some h L 1 C (Ω, F, μ ) such that: F,μ() = hd μ 3. If μ is a signed measure, can we assume h L 1 R (Ω, F, μ )? xercise 12. Further to ex. (11), define A r = { h <r} for all r>0. 1. Show that for all measurable partition ( n ) n 1 of A r : + n=1 μ( n ) r μ (A r ) 2. Show that μ (A r ) = 0 for all 0 <r<1. 3. Show that h 1 μ -a.s.
16 Tutorial 12: Radon-Nikodym Theorem Suppose that F is such that μ () > 0. Show that: 1 hd μ μ () 1 5. Show that h 1 μ -a.s. 6. Prove the following: Theorem 62 For all complex measure μ on (Ω, F), thereexistsh belonging to L 1 C (Ω, F, μ ) such that h =1and: F, μ() = hd μ If μ is a signed measure on (Ω, F), we can assume h L 1 R (Ω, F, μ ). xercise 13. Let A F,and(A n ) n 1 be a sequence in F.
17 Tutorial 12: Radon-Nikodym Theorem Show that if A n A then 1 An 1 A. 2. Show that if A n A then 1 An 1 A. 3. Show that if 1 An 1 A, then for all μ M 1 (Ω, F): μ(a) = lim μ(a n) n + xercise 14. Let μ be a measure on (Ω, F) andf L 1 C (Ω, F,μ). 1. Show that ν = fdμ M 1 (Ω, F). 2. Let h L 1 C (Ω, F, ν ) be such that h =1andν = hd ν. Show that for all,f F: f1 F dμ = h1 F d ν
18 Tutorial 12: Radon-Nikodym Theorem Show that if g :(Ω, F) (C, B(C)) is bounded and measurable: F, fgdμ = hgd ν 4. Show that: F, ν () = f hdμ 5. Show that for all F, Re(f h)dμ 0, Im(f h)dμ =0 6. Show that f h R + μ-a.s. 7. Show that f h = f μ-a.s. 8. Prove the following:
19 Tutorial 12: Radon-Nikodym Theorem 19 Theorem 63 Let μ be a measure on (Ω, F) and f L 1 C (Ω, F,μ). Then, ν = fdμ defined by: F, ν() = fdμ is a complex measure on (Ω, F) with total variation: F, ν () = f dμ xercise 15. Let μ M 1 (Ω, F) be a signed measure. Suppose that h L 1 R (Ω, F, μ ) is such that h = 1 and μ = hd μ. Define A = {h =1} and B = {h = 1}. 1. Show that for all F, μ + () = 1 2 (1 + h)d μ. 2. Show that for all F, μ () = 1 2 (1 h)d μ. 3. Show that μ + = μ A = μ(a ).
20 Tutorial 12: Radon-Nikodym Theorem Show that μ = μ B = μ(b ). Theorem 64 (Hahn Decomposition) Let μ beasignedmeasure on (Ω, F). There exist A, B F, such that A B =, Ω=A B and for all F, μ + () =μ(a ) and μ () = μ(b ). Definition 97 Let μ be a complex measure on (Ω, F). We define: L 1 C(Ω, F,μ) = L 1 C(Ω, F, μ ) and for all f L 1 C (Ω, F,μ), thelebesgue integral of f with respect to μ, is defined as: fdμ = fhd μ where h L 1 C (Ω, F, μ ) is such that h =1and μ = hd μ. xercise 16. Let μ be a complex measure on (Ω, F).
21 Tutorial 12: Radon-Nikodym Theorem Show that for all f :(Ω, F) (C, B(C)) measurable: f L 1 C(Ω, F,μ) f d μ < + 2. Show that for f L 1 C (Ω, F,μ), fdμ is unambiguously defined. 3. Show that for all F,1 L 1 C (Ω, F,μ)and 1 dμ = μ(). 4. Show that if μ is a finite measure, then μ = μ. 5. Show that if μ is a finite measure, definition (97) ofintegral and space L 1 C (Ω, F,μ) is consistent with that already known for measures. 6. Show that L 1 C (Ω, F,μ)isaC-vector space and that: (f + αg)dμ = fdμ+ α gdμ for all f,g L 1 C (Ω, F,μ)andα C.
22 Tutorial 12: Radon-Nikodym Theorem Show that for all f L 1 C (Ω, F,μ), we have: fdμ f d μ xercise 17. Let μ, ν M 1 (Ω, F), let α C. 1. Show that αν = α. ν 2. Show that μ + ν μ + ν 3. Show that L 1 C (Ω, F,μ) L1 C (Ω, F,ν) L1 C (Ω, F,μ+ αν) 4. Show that for all F: 1 d(μ + αν) = 1 dμ + α 1 dν 5. Show that for all f L 1 C (Ω, F,μ) L1 C (Ω, F,ν): fd(μ + αν) = fdμ+ α fdν
23 Tutorial 12: Radon-Nikodym Theorem 23 xercise 18. Let f :(Ω, F) [0, + ] be non-negative and measurable. Let μ and ν be measures on (Ω, F), and α [0, + ]: 1. Show that μ + αν is a measure on (Ω, F) and: fd(μ + αν) = fdμ+ α fdν 2. Show that if μ ν, then: fdμ fdν xercise 19. Let μ M 1 (Ω, F), μ 1 = Re(μ) andμ 2 = Im(μ). 1. Show that μ 1 μ and μ 2 μ. 2. Show that μ μ 1 + μ 2.
24 Tutorial 12: Radon-Nikodym Theorem Show that L 1 C (Ω, F,μ)=L1 C (Ω, F,μ 1) L 1 C (Ω, F,μ 2). 4. Show that: L 1 C (Ω, F,μ 1) = L 1 C (Ω, F,μ+ 1 ) L1 C (Ω, F,μ 1 ) L 1 C(Ω, F,μ 2 ) = L 1 C(Ω, F,μ + 2 ) L1 C(Ω, F,μ 2 ) 5. Show that for all f L 1 C (Ω, F,μ): ( fdμ = fdμ + 1 fdμ 1 + i fdμ + 2 fdμ 2 ) xercise 20. Let μ M 1 (Ω, F). Let A F.Leth L 1 C (Ω, F, μ ) be such that h =1andμ = hd μ. Recall that μ A = μ(a )and μ A = μ (F A ) where F A = {A, F} F. 1. Show that we also have F A = { : F, A}. 2. Show that μ A M 1 (Ω, F) andμ A M 1 (A, F A ).
25 Tutorial 12: Radon-Nikodym Theorem Let F and ( n ) n 1 be a measurable partition of. Show: + n=1 4. Show that we have μ A μ A. μ A ( n ) μ A () 5. Let Fand ( n ) n 1 be a measurable partition of A. Show that: + μ( n ) μ A (A ) n=1 6. Show that μ A (A c )=0. 7. Show that μ A = μ A. 8. Let F A and ( n ) n 1 be an F A -measurable partition of.
26 Tutorial 12: Radon-Nikodym Theorem 26 Show that: + n=1 9. Show that μ A μ A. μ A ( n ) μ A () 10. Let F A F and ( n ) n 1 be a measurable partition of. Show that ( n ) n 1 is also an F A -measurable partition of, and conclude: + μ( n ) μ A () n=1 11. Show that μ A = μ A. 12. Show that μ A = hd μ A. 13. Show that h A L 1 C (A, F A, μ A )andμ A = h A d μ A.
27 Tutorial 12: Radon-Nikodym Theorem Show that for all f L 1 C (Ω, F,μ), we have: f1 A L 1 C(Ω, F,μ), f L 1 C(Ω, F,μ A ),f A L 1 C(A, F A,μ A ) and: f1 A dμ = fdμ A = f A dμ A Definition 98 Let f L 1 C (Ω, F,μ),whereμ is a complex measure on (Ω, F). leta F.Wecallpartial Lebesgue integral of f with respect to μ over A, the integral denoted A fdμ, defined as: fdμ = (f1 A )dμ = fdμ A = (f A )dμ A A where μ A is the complex measure on (Ω, F), μ A = μ(a ), f A is the restriction of f to A and μ A is the restriction of μ to F A,the trace of F on A. xercise 21. Prove the following:
28 Tutorial 12: Radon-Nikodym Theorem 28 Theorem 65 Let f L 1 C (Ω, F,μ), whereμ is a complex measure on (Ω, F). Then,ν = fdμ defined as: F, ν() = fdμ is a complex measure on (Ω, F), with total variation: F, ν () = f d μ Moreover, for all measurable map g :(Ω, F) (C, B(C)), we have: g L 1 C(Ω, F,ν) gf L 1 C(Ω, F,μ) and when such condition is satisfied: gdν = gfdμ xercise 22. Let (Ω 1, F 1 ),...,(Ω n, F n ) be measurable spaces, where n 2. Let μ 1 M 1 (Ω 1, F 1 ),..., μ n M 1 (Ω n, F n ). For all
29 Tutorial 12: Radon-Nikodym Theorem 29 i N n,leth i belonging to L 1 C (Ω i, F i, μ i ) be such that h i =1 and μ i = h i d μ i. For all F 1... F n, we define: μ() = h 1...h n d μ 1... μ n 1. Show that μ M 1 (Ω 1... Ω n, F 1... F n ) 2. Show that for all measurable rectangle A 1... A n : μ(a 1... A n )=μ 1 (A 1 )...μ n (A n ) 3. Prove the following: Theorem 66 Let μ 1,...,μ n be n complex measures on measurable spaces (Ω 1, F 1 ),...,(Ω n, F n ) respectively, where n 2. There exists a unique complex measure μ 1... μ n on (Ω 1... Ω n, F 1... F n ) such that for all measurable rectangle A 1... A n, we have: μ 1... μ n (A 1... A n )=μ 1 (A 1 )...μ n (A n )
30 Tutorial 12: Radon-Nikodym Theorem 30 xercise 23. Further to theorem (66) and exercise (22): 1. Show that μ 1... μ n = μ 1... μ n. 2. Show that μ 1... μ n = μ 1... μ n. 3. Show that for all F 1... F n : μ 1... μ n () = h 1...h n d μ 1... μ n 4. Let f L 1 C (Ω 1... Ω n, F 1... F n,μ 1... μ n ). Show: fdμ 1... μ n = fh 1...h n d μ 1... μ n 5. let σ be a permutation of {1,...,n}. Show that: fdμ 1... μ n =... fdμ σ(1)...dμ σ(n) Ω σ(n) Ω σ(1)
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