608 CHAPTER 20. REPRESENTATION THEOREMS

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).The following corollary is about representing a vector measure in terms of its total

variation. It is like representing a complex number in the form reiθ . The proof requires thefollowing lemma.

Lemma 20.2.7 Suppose (Ω,S ,µ) is a measure space and f is a function in L1(Ω,µ) withthe property that

|∫

Ef dµ| ≤ µ(E)

for all E ∈S . Then | f | ≤ 1 a.e.

Proof of the lemma: Consider the following picture.

1(0,0) .p

B(p,r)

where B(p,r)∩B(0,1) = /0. Let E = f−1(B(p,r)). In fact µ (E) = 0. If µ(E) ̸= 0 then∣∣∣∣ 1µ(E)

∫E

f dµ− p∣∣∣∣ =

∣∣∣∣ 1µ(E)

∫E( f − p)dµ

∣∣∣∣≤ 1

µ(E)

∫E| f − p|dµ < r

because on E, | f (x)− p|< r. Hence

| 1µ(E)

∫E

f dµ|> 1