35.6. CHANGE OF VARIABLES FORMULA LIPSCHITZ MAPS 1245

You see that if x were an interior point of E, then this limit will equal 1. However, itis sometimes the case that the limit equals 1 even when x is not an interior point. In fact,these points of density make sense even for sets that have empty interior.

Lemma 35.6.3 Let E be a Lebesgue measurable set. Then there exists a set of measurezero, N, such that if x ∈ E \N, then x is a point of density of E.

Proof: Consider the function, f (x) = XE (x). This function is in L1loc (Rn). Let NC

denote the Lebesgue points of f . Then for x ∈ E \N,

1 = XE (x) = limr→0

1mn (B(x,r))

∫B(x,r)

XE (y)dmn

= limr→0

mn (B(x,r)∩E)mn (B(x,r))

.

In this section, Ω will be a Lebesgue measurable set in Rn and h : Ω→ Rn will beLipschitz. Recall the following definition and theorems. See Page 13.4.2 for the proofs andmore discussion.

Definition 35.6.4 Let F be a collection of balls that cover a set, E, which have the prop-erty that if x ∈ E and ε > 0, then there exists B ∈F , diameter of B < ε and x ∈ B. Such acollection covers E in the sense of Vitali.

Theorem 35.6.5 Let E ⊆Rn and suppose mn(E)<∞ where mn is the outer measure deter-mined by mn, n dimensional Lebesgue measure, and let F , be a collection of closed ballsof bounded radii such that F covers E in the sense of Vitali. Then there exists a countablecollection of disjoint balls from F , {B j}∞

j=1, such that mn(E \∪∞j=1B j) = 0.

Now this theorem implies a simple lemma which is what will be used.

Lemma 35.6.6 Let V be an open set in Rr,mr (V ) < ∞. Then there exists a sequence ofdisjoint open balls {Bi} having radii less than δ and a set of measure 0, T , such that

V = (∪∞i=1Bi)∪T.

As in the proof of the change of variables theorem given earlier, the first step is toshow that h maps Lebesgue measurable sets to Lebesgue measurable sets. In showing thisthe key result is the next lemma which states that h maps sets of measure zero to sets ofmeasure zero.

Lemma 35.6.7 If mn (T ) = 0 then mn (h(T )) = 0.

Proof: Let V be an open set containing T whose measure is less than ε . Now using theVitali covering theorem, there exists a sequence of disjoint balls {Bi}, Bi = B(xi,ri) whichare contained in V such that the sequence of enlarged balls,

{B̂i

}, having the same center

but 5 times the radius, covers T . Then

mn (h(T ))≤ mn

(h(∪∞

i=1B̂i

))