964 CHAPTER 26. INTEGRALS AND DERIVATIVES

Definition 26.7.3 Let f (r) be a function having values in [−∞,∞] . Then

lim supr→0+

f (r) ≡ limr→0

(sup{ f (t) : t ∈ [0,r]})

lim infr→0+

f (r) ≡ limr→0

(inf{ f (t) : t ∈ [0,r]})

This is well defined because the function r→ inf{ f (t) : t ∈ [0,r]} is increasing and r→sup{ f (t) : t ∈ [0,r]} is decreasing. Also note that limr→0+ f (r) exists if and only if

lim supr→0+

f (r) = lim infr→0+

f (r)

and if this happens

limr→0+

f (r) = lim infr→0+

f (r) = lim supr→0+

f (r) .

The claims made in the above definition follow immediately from the definition of whatis meant by a limit in [−∞,∞] and are left for the reader.

Theorem 26.7.4 Let µ be a Borel measure on Rn then dµ

dm (x) exists in [−∞,∞] m a.e.

Proof:Let p < q and let p,q be rational numbers. Define

Npq (M) ≡{

x ∈ Rn such that lim supr→0+

µ (B(x,r))m(B(x,r))

> q

> p > lim infr→0+

µ (B(x,r))m(B(x,r))

}∩B(0,M) ,

Npq ≡{

x ∈ Rn such that lim supr→0+

µ (B(x,r))m(B(x,r))

> q

> p > lim infr→0+

µ (B(x,r))m(B(x,r))

},

N ≡{

x ∈ Rn such that lim supr→0+

µ (B(x,r))m(B(x,r))

>

lim infr→0+

µ (B(x,r))m(B(x,r))

}.

I will show m(Npq (M)) = 0. Use outer regularity to obtain an open set, V containingNpq (M) such that

m(Npq (M))+ ε > m(V ) .

From the definition of Npq (M) , it follows that for each x ∈ Npq (M) there exist arbitrar-ily small r > 0 such that

µ (B(x,r))m(B(x,r))

< p.