The following is Sard’s lemma. In the proof, it does not matter which norm you use in
defining balls but it may be easiest to consider the norm

||x ||

≡ max

{|x|,i = 1,⋅⋅⋅,p}
i

.

Lemma 9.8.1(Sard) Let U be an open set in ℝ^{p}and let h : U → ℝ^{p}be differentiable.Let

Z ≡ {x ∈ U : detDh (x) = 0}.

Then m_{p}

(h(Z ))

= 0.

Proof: For convenience, assume the balls in the following argument come from

||⋅||

_{∞}.
First note that Z is a Borel set because h is continuous and so the component functions of the
Jacobian matrix are each Borel measurable. Hence the determinant is also Borel
measurable.

Suppose that U is a bounded open set. Let ε > 0 be given. Also let V ⊇ Z with V ⊆ U
open, and

mp (Z )+ ε > mp (V) .

Now let x ∈ Z. Then since h is differentiable at x, there exists δ_{x}> 0 such that if r < δ_{x},
then B

(x,r)

⊆ V and also,

h (B (x,r)) ⊆ h(x)+ Dh (x)(B (0,r))+ B (0,rη), η < 1.

Regard Dh

(x)

as an n × n matrix, the matrix of the linear transformation Dh

(x)

with
respect to the usual coordinates. Since x ∈ Z, it follows that there exists an invertible matrix
A such that ADh

(x)

is in row reduced echelon form with a row of zeros on the bottom.
Therefore,

mp (A (h (B(x,r)))) ≤ mp (ADh (x)(B (0,r))+ AB (0,rη)) (9.15)

(9.15)

The diameter of ADh

(x)

(B (0,r))

is no larger than

||A||

||Dh (x)||

2r and it lies in
ℝ^{p−1}×

{0}

. The diameter of AB

(0,rη)

is no more than

||A ||

(2rη)

.Therefore, the measure of
the right side in 9.15 is no more than

p−1
[(||A||||Dh (x)||2r+ ||A||(2η))r] (rη)
≤ C (||A||,||Dh (x)||)(2r)pη

Hence from the change of variables formula for linear maps,

∑ ∑
mp (h (Z)) ≤ mp (h(Bi)) ≤ ε mp (Bi)
i i
≤ ε(mp (V )) ≤ ε(mp (Z)+ ε).

Since ε is arbitrary, this shows m_{p}

(h(Z ))

= 0. What if U is not bounded? Then consider
Z_{n} = Z ∩ B

(0,n)

. From what was just shown, h

(Zn)

has measure 0 and so it
follows that h

(Z)

also does, being the countable union of sets of measure zero.
■

With this important lemma, here is a generalization of Theorem 9.7.5.

Theorem 9.8.2Let U be an open set and let h be a 1 − 1, C^{1}(U) function withvalues in ℝ^{p}. Then if g is a nonnegative Lebesgue measurable function,

∫ ∫
g (y )dmp = g (h (x))|det(Dh (x))|dmp. (9.16)
h(U) U

(9.16)

Proof:Let Z =

{x : det (Dh (x)) = 0}

, a closed set. Then by the inverse function
theorem, h^{−1} is C^{1} on h