This section is a generalization of earlier material in which a measure was differentiated with
respect to Lebesgue measure. Here an arbitrary Radon measure will be differentiated with
respect to another arbitrary Radon measure. In this section, B
(x,r)
will denote a ball with
center x and radius r. Also, let λ and μ be Radon measures and as above, Z will denote a μ
measure zero set off of which μ
(B (x,r))
> 0 for all r > 0.
Definition 18.4.1Forx
∈∕
Z, define the upper and lower symmetric derivativesas
in the case when both the upper and lower derivatives are equal.
Lemma 18.4.2Let λ and μ be Radon measures. If A is a bounded subset of
{ -- }
x∈∕Z :D μλ(x) ≥ a
, then
-- --
λ(A) ≥ aμ(A)
and if A is a bounded subset of
{x ∕∈ Z : Dμλ (x) ≤ a}
, then
--
λ(A) ≤ aμ(A)
Proof: Suppose first that A is a bounded subset of
{ -- }
x∈∕Z :D μλ(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 balls
constitutes a Vitali cover for A. By Corollary 18.3.4 there is a disjoint sequence of these balls