13.6. CHANGE OF VARIABLES FOR C1 FUNCTIONS 357

It follows from Lemma 13.5.1 that h(U \∪iBi) also has measure zero. Then from Corollary13.6.2 ∫

VXE (y)dmn = ∑

i

∫h(Bi)

XE∩h(Bi) (y)dmn

= ∑i

∫Bi

XE (h(x)) |det(Dh(x))|dmn

=∫

UXE (h(x)) |det(Dh(x))|dmn.

This proves the corollary.With this corollary, the main theorem follows.

Theorem 13.6.4 Let U and V be open sets in Rn and let h,h−1 be C1 functions such thath(U) =V. Then if g is a nonnegative Lebesgue measurable function,∫

Vg(y)dy =

∫U

g(h(x)) |det(Dh(x))|dx. (13.6.16)

Proof: From Corollary 13.6.3, 13.6.16 holds for any nonnegative simple function inplace of g. In general, let {sk} be an increasing sequence of simple functions which con-verges to g pointwise. Then from the monotone convergence theorem∫

Vg(y)dy = lim

k→∞

∫V

skdy = limk→∞

∫U

sk (h(x)) |det(Dh(x))|dx

=∫

Ug(h(x)) |det(Dh(x))|dx.

This proves the theorem.This is a pretty good theorem but it isn’t too hard to generalize it. In particular, it is not

necessary to assume h−1 is C1.

Lemma 13.6.5 (Sard) Let U be an open set in Rn and let h : U → Rn be C1. Let

Z ≡ {x ∈U : detDh(x) = 0} .

Then mn (h(Z)) = 0.

Proof: Let Zk denote those points x of Z such that ||Dh(x)|| ≤ k and such that |x|< k.Let ε > 0 be given. For x ∈ Zk,

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

and so whenever r is small enough,

h(x+B(0,r)) = h(B(x,r))⊆ h(x)+Dh(x)B(0,r)+B(0,rε)

Note Dh(x)B(0,r) is contained in an n− 1 dimensional subspace of Rn due to the factDh(x) has rank less than n. Now let Q denote an orthogonal transformation preserving alldistances,

QQ∗ = Q∗Q = I,