1090 CHAPTER 31. DIFFERENTIATION, RADON MEASURES

31.3 Differentiation of Radon MeasuresThis section is a generalization of earlier ideas in which differentiation was with respectto Lebesgue measure. Here an arbitrary Radon measure, not necessarily an integral withrespect to Lebesuge measure, will be differentiated with respect to another arbitrary Radonmeasure. This requires a more sophisticated covering theorem. In this section, B(x,r) willdenote a closed ball with center x and radius r. Also, let λ and µ be Radon measures andas above, Z will denote a µ measure zero set off of which µ (B(x,r))> 0 for all r > 0.

Definition 31.3.1 For x /∈Z, define the upper and lower symmetric derivatives as

Dµ λ (x)≡ limsupr→0

λ (B(x,r))µ (B(x,r))

, Dµ λ (x)≡ lim infr→0

λ (B(x,r))µ (B(x,r))

.

respectively. Also defineDµ λ (x)≡ Dµ λ (x) = Dµ λ (x)

in the case when both the upper and lower derivatives are equal.

Lemma 31.3.2 Let λ and µ be Radon measures. If A is a bounded subset of{x /∈ Z : Dµ λ (x)≥ a

},

thenλ (A)≥ aµ (A)

and if A is a bounded subset of{

x /∈ Z : Dµ λ (x)≤ a}, then

λ (A)≤ aµ (A)

The same conclusion holds even if A is not necessarily bounded.

Proof: Suppose first that A is a bounded subset of{

x /∈ Z : Dµ λ (x)≥ a}

, let ε > 0,and let V be a bounded open set with V ⊇ A and λ (V )− ε < λ (A) ,µ (V )− ε < µ (A) .Then if x ∈ A,

λ (B(x,r))µ (B(x,r))

> a− ε, B(x,r)⊆V,

for infinitely many values of r which are arbitrarily small. Thus the collection of such ballsconstitutes a Vitali cover for A. By Corollary 13.14.3 there is a disjoint sequence of theseclosed balls {Bi} such that

µ (A\∪∞i=1Bi) = 0. (31.3.11)

Therefore,

(a− ε)∞

∑i=1

µ (Bi)<∞

∑i=1

λ (Bi)≤ λ (V )< ε +λ (A)

and so

a∞

∑i=1

µ (Bi) ≤ ε + εµ (V )+λ (A)

≤ ε + ε (µ (A)+ ε)+λ (A) (31.3.12)