13.5. CHANGE OF VARIABLES FOR LINEAR MAPS 351

Proof: Let F be the collection of closures of balls in F . Then F covers E in the senseof Vitali and so from Corollary 13.4.4 there exists a sequence of disjoint closed balls fromF satisfying mn

(E \∪∞

i=1Bi)= 0. Now boundaries of the balls, Bi have measure zero and

so {Bi} is a sequence of disjoint open balls satisfying mn (E \∪∞i=1Bi) = 0. The reason for

this is that

(E \∪∞i=1Bi)\

(E \∪∞

i=1Bi)⊆ ∪∞

i=1Bi \∪∞i=1Bi ⊆ ∪∞

i=1Bi \Bi,

a set of measure zero. Therefore,

E \∪∞i=1Bi ⊆

(E \∪∞

i=1Bi)∪(∪∞

i=1Bi \Bi)

and so

mn (E \∪∞i=1Bi) ≤ mn

(E \∪∞

i=1Bi)+mn

(∪∞

i=1Bi \Bi)

= mn(E \∪∞

i=1Bi)= 0.

This implies you can fill up an open set with balls which cover the open set in the senseof Vitali.

Corollary 13.4.6 Let U ⊆ Rn be an open set and let F be a collection of closed or evenopen balls of bounded radii contained in U such that F covers U in the sense of Vitali.Then there exists a countable collection of disjoint balls from F , {B j}∞

j=1, such that mn(U \∪∞

j=1B j) = 0.

13.5 Change of Variables for Linear MapsTo begin with certain kinds of functions map measurable sets to measurable sets. It will beassumed that U is an open set in Rn and that h : U → Rn satisfies

Dh(x) exists for all x ∈U, (13.5.11)

Lemma 13.5.1 Let h satisfy 13.5.11. If T ⊆U and mn (T ) = 0, then mn (h(T )) = 0.

Proof: LetTk ≡ {x ∈ T : ||Dh(x)||< k}

and let ε > 0 be given. Now by outer regularity, there exists an open set, V , containing Tkwhich is contained in U such that mn (V )< ε . Let x ∈ Tk. Then by differentiability,

h(x+v) = h(x)+Dh(x)v+o(v)

and so there exist arbitrarily small rx < 1 such that B(x,5rx) ⊆ V and whenever |v| ≤rx, |o(v)|< k |v| . Thus

h(B(x,rx))⊆ B(h(x) ,2krx) .

From the Vitali covering theorem there exists a countable disjoint sequence of thesesets, {B(xi,ri)}∞

i=1 such that {B(xi,5ri)}∞

i=1 ={

B̂i

}∞

i=1covers Tk Then letting mn denote

the outer measure determined by mn,

mn (h(Tk))≤ mn

(h(∪∞

i=1B̂i

))