13.2. VECTOR MEASURES 323

Proof: It is clear that Reλ and Imλ are real-valued vector measures on S . Since|λ |(Ω)< ∞, it follows easily that |Reλ |(Ω) and | Imλ |(Ω)< ∞. This is clear because

|λ (E)| ≥ |Reλ (E)| , |Imλ (E)| .

Therefore, each of

|Reλ |+Reλ

2,|Reλ |−Re(λ )

2,| Imλ |+ Imλ

2, and

| Imλ |− Im(λ )

2

are finite measures on S . It is also clear that each of these finite measures are abso-lutely continuous with respect to µ and so there exist unique nonnegative functions inL1(Ω), f1, f2, g1, g2 such that for all E ∈S ,

12(|Reλ |+Reλ )(E) =

∫E

f1dµ,

12(|Reλ |−Reλ )(E) =

∫E

f2dµ,

12(| Imλ |+ Imλ )(E) =

∫E

g1dµ,

12(| Imλ |− Imλ )(E) =

∫E

g2dµ.

Now let f = f1− f2 + i(g1−g2). ■

Theorem 13.2.8 The following hold where λ will be a complex measure so |λ | isfinite and µ will be a finite measure, both defined on S .

1. If µ is a finite nonnegative measure on S , and λ (E) ≡∫

E hdµ for h ∈ L1 (Ω,µ),then |λ |(E) =

∫E |h|dµ.

2. If |∫

E f dµ| ≤ µ(E) for all E ∈ S , then | f | ≤ 1 a.e. If |∫

E f dµ| = µ(E) for allE ∈S , then | f |= 1 µ a.e.

3. Letting g be such that λ (E) =∫

E gd |λ | , it follows that |g| = 1 for |λ | a.e. If alsoλ (E) =

∫E hdµ for h ∈ L1 (Ω,µ) , then |h|= gh µ a.e.

Proof: 1.) Letting π (E) = {F1, ...,Fn} ,n

∑k=1|λ (Fk)|=

n

∑k=1

∣∣∣∣∫Fk

hdµ

∣∣∣∣≤ n

∑k=1

∫Fk

|h|dµ =∫

E|h|dµ

and so, taking the sup for all such partitions, |λ |(E) ≤∫

E |h|dµ. Let simple functionssn→ sgn(h) where |sgn(h)|= 1 and sgn(h)h = |h|. We can assume also that |sn| ≤ 1. Saysn = ∑

mni=1 cn

i XFni

where the Fni are disjoint, {Fn

i }mni=1 a partition of E. Then |λ |(E)≤

∫E|h|dµ =

∫E

sgn(h)hdµ = limn→∞

∫E

snhdµ = limn→∞

mn

∑i=1

∫Fn

i

cni hdµ

≤ limn→∞

∣∣∣∣∣mn

∑i=1

∫Fn

i

cni hdµ

∣∣∣∣∣≤ lim infn→∞

mn

∑i=1

∣∣∣∣∫Fni

hdµ

∣∣∣∣≤ supn

mn

∑i=1|λ (Fn

i )| ≤ |λ |(E)