Kenneth Kuttler

610 CHAPTER 20. REPRESENTATION THEOREMS

Proof: By Corollary 20.2.8 and Theorem 20.2.5 which says that |λ | is finite, thereexists a unique f such that | f |= 1 |λ | a.e. and

λ (E) =∫

Ef d |λ | .

Now |λ | ≪ µ and so it follows from Corollary 20.1.3 there exists a unique nonnegativemeasurable function h such that for all E measurable,

|λ |(E) =∫

Ehdµ

where since |λ | is finite, h ∈ L1 (Ω,µ) . It follows from approximating f with simple func-tions and using the above formula that

λ (E) =∫

Ef hdµ.

Then let g = L1 (Ω,µ) . This proves the corollary.

Corollary 20.2.10 Suppose (Ω,S ) is a measure space and µ is a finite nonnegative mea-sure on S . Then for h ∈ L1 (µ) , define a complex measure, λ by

λ (E)≡∫

Ehdµ.

Then|λ |(E) =

∫E|h|dµ.

Furthermore, |h|= gh where gd |λ | is the polar decomposition of λ ,

λ (E) =∫

Egd |λ |

Proof: From Corollary 20.2.8 there exists g such that |g|= 1, |λ | a.e. and for all E ∈S

λ (E) =∫

Egd |λ |=

∫E

hdµ.

Let sn be a sequence of simple functions converging pointwise to g. Then from the above,∫E

gsnd |λ |=∫

Esnhdµ.

Passing to the limit using the dominated convergence theorem,∫E

d |λ |=∫

Eghdµ.

It follows gh ≥ 0 a.e. and |g| = 1. Therefore, |h| = |gh| = gh. It follows from the above,that

|λ |(E) =∫

Ed |λ |=

∫E

ghdµ =∫

Ed |λ |=

∫E|h|dµ

and this proves the corollary.

610 CHAPTER 20. REPRESENTATION THEOREMSProof: By Corollary 20.2.8 and Theorem 20.2.5 which says that |A| is finite, thereexists a unique f such that |f| = 1 |A| a.e. andA(E) = | fala.Now |A| < yu and so it follows from Corollary 20.1.3 there exists a unique nonnegativemeasurable function h such that for all E measurable,A\(E) = f hdwhere since |A| is finite, h € L! (Q, 1). It follows from approximating f with simple func-tions and using the above formula thatA(E) = [ fhay.Then let g = L' (Q,w) . This proves the corollary.Corollary 20.2.10 Suppose (Q,./) is a measure space and [ is a finite nonnegative mea-sure on SY. Then for h € L' (uw), define a complex measure, A byA(E)= / hdEThenAl(E) = f Ihldy.Furthermore, |h| = Zh where gd |A| is the polar decomposition of A,A(E) = | aaEProof: From Corollary 20.2.8 there exists g such that |g| = 1,|A| a.e. and for all E € 7A(E) = | gd\a\= ff ha.Let s, be a sequence of simple functions converging pointwise to g. Then from the above,[gs \al= | suhdp.E EPassing to the limit using the dominated convergence theorem,[aiai= | ehdy.E EIt follows gh > 0 a.e. and |g| = 1. Therefore, |h| = |gh| = Zh. It follows from the above,thata\(E)= [alal= f gnaw = f aial= | inlayE E E Eand this proves the corollary.