966 CHAPTER 26. INTEGRALS AND DERIVATIVES

These are measures thanks to Theorem 20.2.3 on Page 603 and µ+−µ− = µ . These mea-sures have values in [0,∞). They are called the positive and negative parts of µ respectively.For µ a complex measure, define Re µ and Im µ by

Re µ (E) ≡ 12

(µ (E)+µ (E)

)Im µ (E) ≡ 1

2i

(µ (E)−µ (E)

)Then Re µ and Im µ are both real measures. Thus for µ a complex measure,

µ = Re µ+−Re µ

−+ i(Im µ

+− Im µ−)

= ν1−ν1 + i(ν3−ν4)

where each ν i is a real measure having values in [0,∞).

Then there is an obvious corollary to Theorem 26.7.4.

Corollary 26.7.6 Let µ be a complex Borel measure on Rn. Then dµ

dm (x) exists a.e.

Proof: Letting ν i be defined in Definition 26.7.5. By Theorem 26.7.4, for m a.e. x,dν idm (x) exists. This proves the corollary because µ is just a finite sum of these ν i.

Theorem 20.1.2 on Page 597, the Radon Nikodym theorem, implies that if you have twofinite measures, µ and λ , you can write λ as the sum of a measure absolutely continuouswith respect to µ and one which is singular to µ in a unique way. The next topic is relatedto this. It has to do with the differentiation of a measure which is singular with respect toLebesgue measure.

Theorem 26.7.7 Let µ be a Radon measure onRn and suppose there exists a µ measurableset, N such that for all Borel sets, E, µ (E) = µ (E ∩N) where m(N) = 0. Then

dµ

dm(x) = 0 m a.e.

Proof: For k ∈ N, let

Bk (M) ≡{

x ∈ NC : lim supr→0+

µ (B(x,r))m(B(x,r))

>1k

}∩B(0,M) ,

Bk ≡{

x ∈ NC : lim supr→0+

µ (B(x,r))m(B(x,r))

>1k

},

B ≡{

x ∈ NC : lim supr→0+

µ (B(x,r))m(B(x,r))

> 0}.

Let ε > 0. Since µ is regular, there exists H, a compact set such that H ⊆ N ∩B(0,M)and

µ (N∩B(0,M)\H)< ε.