212
IX – Multivariate Differential and Integral Calculus
The measure of K(r) being proportional to that of r
n , (20) shows that
(1 − q
)
n ≤ m [ϕ(K)] /|J ϕ (a)|m(K) ≤ (1 + q
)
n .
(10.21)
However, condition (17) has been imposed on q
; it necessarily follows that
1 − q ≤ m [ϕ(K)] /|J ϕ (a)|m(K) ≤ 1 + q ,
and hence (16) holds, qed.
Lemma 4. Let U be an open subset and ϕ a C
1 map defined on U . Then
ϕ(M ) has measure zero for any compact set
43 of measure zero M ⊂ U .
Let d > 0 be the distance from the compact subspace M to the border
of U and M
the set of x ∈ U such that d(x, M ) ≤ d/2. It is also a compact
set contained in U . Let k be an integer > 0 such that 1/2
k < d/2 and let us
take a grid of R
n by hyperplanes defined by a single equation x
i = p/2
k , with
p ∈ Z and i ∈ {1, . . . , n}. R
n can thereby be decomposed into cubes of the
form K(a, 1/2
k+1 ) whose pairwise intersections are at most faces of dimension
≤ n − 1. Finitely many of these cubes intersect M non-trivially since M is
bounded; they are all contained in M
since their diameter d k is < d/2 ; and
finally, they cover M . Let M k be their union. ϕ(M ) ⊂ ϕ(M k ) =
ϕ(K),
where K varies in the set of cubes K(a, 1/2
k+1 ) comprising M k .
To find an upper bound for the measures of these ϕ(K), observe that,
these cubes being convex, by the (FT),
ϕ(x) − ϕ(y) =
1
0
ϕ
[tx + (1 − t)y] (x − y)dt
and so
|ϕ(x) − ϕ(y)| ≤ |x − y|.ϕ
K ≤ |x − y|.ϕ
M ,
for all x, y ∈ K. In conclusion, since all cubes K considered have the same
diameter d k , ϕ(K) is contained in a cube of diameter ≤ cd k where c =
ϕ
M , a uniform norm on M
. The measure of a cube of diameter d being
proportional to d
n , it follows that
m [ϕ(K)] ≤ cm(K) .
However, m(M k ) =
m(K), where summation is over the cubes comprising
M k , because their pairwise intersections have measure zero since they are
contained in the hyperplanes of R
n . Also,
43 There is a far stronger result: if U is an open subset of R
n , any C
1 map ϕ :
U −→ R
n transforms sets contained in U into sets of measure zero, without any
compactness assumption. See Dieudonn´ e, El´ ements d’analyse, XVI.22, exercises 1
and 2. More complete results can be found in Rudin, Real and Complex Analysis,
chap. 8.
IX – Multivariate Differential and Integral Calculus
The measure of K(r) being proportional to that of r
n , (20) shows that
(1 − q
)
n ≤ m [ϕ(K)] /|J ϕ (a)|m(K) ≤ (1 + q
)
n .
(10.21)
However, condition (17) has been imposed on q
; it necessarily follows that
1 − q ≤ m [ϕ(K)] /|J ϕ (a)|m(K) ≤ 1 + q ,
and hence (16) holds, qed.
Lemma 4. Let U be an open subset and ϕ a C
1 map defined on U . Then
ϕ(M ) has measure zero for any compact set
43 of measure zero M ⊂ U .
Let d > 0 be the distance from the compact subspace M to the border
of U and M
the set of x ∈ U such that d(x, M ) ≤ d/2. It is also a compact
set contained in U . Let k be an integer > 0 such that 1/2
k < d/2 and let us
take a grid of R
n by hyperplanes defined by a single equation x
i = p/2
k , with
p ∈ Z and i ∈ {1, . . . , n}. R
n can thereby be decomposed into cubes of the
form K(a, 1/2
k+1 ) whose pairwise intersections are at most faces of dimension
≤ n − 1. Finitely many of these cubes intersect M non-trivially since M is
bounded; they are all contained in M
since their diameter d k is < d/2 ; and
finally, they cover M . Let M k be their union. ϕ(M ) ⊂ ϕ(M k ) =
ϕ(K),
where K varies in the set of cubes K(a, 1/2
k+1 ) comprising M k .
To find an upper bound for the measures of these ϕ(K), observe that,
these cubes being convex, by the (FT),
ϕ(x) − ϕ(y) =
1
0
ϕ
[tx + (1 − t)y] (x − y)dt
and so
|ϕ(x) − ϕ(y)| ≤ |x − y|.ϕ
K ≤ |x − y|.ϕ
M ,
for all x, y ∈ K. In conclusion, since all cubes K considered have the same
diameter d k , ϕ(K) is contained in a cube of diameter ≤ cd k where c =
ϕ
M , a uniform norm on M
. The measure of a cube of diameter d being
proportional to d
n , it follows that
m [ϕ(K)] ≤ cm(K) .
However, m(M k ) =
m(K), where summation is over the cubes comprising
M k , because their pairwise intersections have measure zero since they are
contained in the hyperplanes of R
n . Also,
43 There is a far stronger result: if U is an open subset of R
n , any C
1 map ϕ :
U −→ R
n transforms sets contained in U into sets of measure zero, without any
compactness assumption. See Dieudonn´ e, El´ ements d’analyse, XVI.22, exercises 1
and 2. More complete results can be found in Rudin, Real and Complex Analysis,
chap. 8.