26.7. DIFFERENTIATION OF MEASURES 963

Now there exists a subsequence such that uε → u pointwise a.e. and also each uε,i → u,ipointwise a.e. as well as in Lp. Therefore, for x,y not in a set of measure zero,

|u(x)−u(y)| ≤C(∫

B(x,2|x−y|)|∇u(z)|p dz

)1/p(|x−y|1−n/p

)which shows the claim about u being equal to a continuous function off a set of measurezero. Thus h is also continuous off a set of measure zero.

As before, letting g(y) ≡ uε (y)− uε (x)−∇uε (x) · (y−x) and writing Morrey’s in-equality,

|uε (y)−uε (x)−∇uε (x) · (y−x)|

≤ C(∫

B(x,2|x−y|)|∇uε (z)−∇uε (x)|p dz

)1/p(|x−y|1−n/p

)Then taking a suitable subsequence and passing to the limit while also letting v = y−x, itfollows

|u(x+v)−u(x)−∇u(x) ·v|

≤ C(∫

B(x,2|v|)|∇u(z)−∇u(x)|p dz

)1/p(|v|1−n/p

)= C

(1|v|n

∫B(x,2|v|)

|∇u(z)−∇u(x)|p dz)1/p

|v|

= C′(

1B(x,2 |v|)

∫B(x,2|v|)

|∇u(z)−∇u(x)|p dz)1/p

|v|

for all x,x+v /∈ N, a set of measure zero. Defining u,∇u at the points of N so that theinequality continues to hold, Du(x) exists at every Lebesgue point of ∇u. Also Dh existsa.e. because this is true of the component functions. This proves the theorem.

26.7 Differentiation Of MeasuresRecall the Vitali covering theorem in Corollary 13.4.5 on Page 350.

Corollary 26.7.1 Let E ⊆ Rn and let F , be a collection of open balls of bounded radiisuch that F covers E in the sense of Vitali. Then there exists a countable collection ofdisjoint balls from F , {B j}∞

j=1, such that m(E \∪∞j=1B j) = 0.

Definition 26.7.2 Let µ be a Radon measure defined on Rn. Then

dµ

dm(x)≡ lim

r→0

µ (B(x,r))m(B(x,r))

whenever this limit exists.

It turns out this limit exists for m a.e. x. To verify this here is another definition.