§ 3. Integration of Differential Forms
213
m [ϕ(M )] ≤ m [ϕ (M k )] ≤
m [ϕ(K)] ≤
cm(K) = cm (M k ) .
(10.22)
The Lemma will, therefore, follow once we have shown that lim m(M k ) = 0.
But let us compare the sets M k and M k+1 . To obtain the first one, take a
grid of R
n by the hyperplanes x
i = p/2
k , whereas the second ones is obtained
by using the hyperplanes x
i = q/2
k+1 . Clearly, any cube K of the second grid
is contained in at least one cube K
of the first ; if K meets M , the same holds
for K
. Hence M k+1 ⊂ M k , which gives a decreasing sequence of compact sets
contained in M . By definition, any point of M k belongs to a cube of radius
1/2
k meeting M , and so is at a distance ≤ 1/2
k from M . Any point common
to all the M k is, therefore, at distance zero from M , in other words belongs
to M since M is closed.
However, when there is a decreasing sequence of closed sets
44 M k with intersection M , we know that m(M ) = lim m(M k ) ; this was shown in Chapter
V, § 2, end of n
◦ 11, for an increasing sequence of open sets, but, as mentioned
then, the result and the proof remain the same for a decreasing sequence of
closed sets. Since here m(M ) = 0, relation (22) shows that m[ϕ(M )] = 0,
qed.
(iii) Change of variable formula. The general formula
ϕ(A)
f (x)dx =
A
f [ϕ(t)] .|J ϕ (t)|dt
can now be proved by replacing A = ¯
U with simpler sets, unions of cubes,
and then passing to the limit.
Choose a number r > 0. For any integer k > 0, let us once again take
a grid of R
n by the hyperplanes x
i = p/2
k . Denote the finitely many cubes
in this grid contained in U by K 1 , . . . , K N and let A k ⊂ U be their union.
The pairwise intersections of these cubes being compact and having measure
zero, the same holds (lemma 4) for their images; hence
A k
f [ϕ(t)] .|J ϕ (t)|dt =
Ki
f [ϕ(t)] .|J ϕ (t)|dm(t) .
(10.23)
The function integrated in (23) being uniformly continuous on the compact
set A, for any r > 0, k may be assumed to be sufficiently large for it to be
constant, up to r, in each cube K i . Setting a i to be the centre of K i and
b i = ϕ(a i ), the general term of the right hand side of (23) is seen to be equal
to f (b i ).|J ϕ (a i )|m(K i ), up to m(K i )r.
Now, setting D i = ϕ(K i ), B k = ϕ(A k ) =
D i ⊂ B = ϕ(A), lemma
4 shows that the pairwise intersections of the ϕ(A k ) have measure zero. So
once again,
B k
f (x)dx =
Di
f (x)dx .
(10.23’)
44 more generally of “ measurable ” sets as Lebesgue’s complete theory will show.
213
m [ϕ(M )] ≤ m [ϕ (M k )] ≤
m [ϕ(K)] ≤
cm(K) = cm (M k ) .
(10.22)
The Lemma will, therefore, follow once we have shown that lim m(M k ) = 0.
But let us compare the sets M k and M k+1 . To obtain the first one, take a
grid of R
n by the hyperplanes x
i = p/2
k , whereas the second ones is obtained
by using the hyperplanes x
i = q/2
k+1 . Clearly, any cube K of the second grid
is contained in at least one cube K
of the first ; if K meets M , the same holds
for K
. Hence M k+1 ⊂ M k , which gives a decreasing sequence of compact sets
contained in M . By definition, any point of M k belongs to a cube of radius
1/2
k meeting M , and so is at a distance ≤ 1/2
k from M . Any point common
to all the M k is, therefore, at distance zero from M , in other words belongs
to M since M is closed.
However, when there is a decreasing sequence of closed sets
44 M k with intersection M , we know that m(M ) = lim m(M k ) ; this was shown in Chapter
V, § 2, end of n
◦ 11, for an increasing sequence of open sets, but, as mentioned
then, the result and the proof remain the same for a decreasing sequence of
closed sets. Since here m(M ) = 0, relation (22) shows that m[ϕ(M )] = 0,
qed.
(iii) Change of variable formula. The general formula
ϕ(A)
f (x)dx =
A
f [ϕ(t)] .|J ϕ (t)|dt
can now be proved by replacing A = ¯
U with simpler sets, unions of cubes,
and then passing to the limit.
Choose a number r > 0. For any integer k > 0, let us once again take
a grid of R
n by the hyperplanes x
i = p/2
k . Denote the finitely many cubes
in this grid contained in U by K 1 , . . . , K N and let A k ⊂ U be their union.
The pairwise intersections of these cubes being compact and having measure
zero, the same holds (lemma 4) for their images; hence
A k
f [ϕ(t)] .|J ϕ (t)|dt =
Ki
f [ϕ(t)] .|J ϕ (t)|dm(t) .
(10.23)
The function integrated in (23) being uniformly continuous on the compact
set A, for any r > 0, k may be assumed to be sufficiently large for it to be
constant, up to r, in each cube K i . Setting a i to be the centre of K i and
b i = ϕ(a i ), the general term of the right hand side of (23) is seen to be equal
to f (b i ).|J ϕ (a i )|m(K i ), up to m(K i )r.
Now, setting D i = ϕ(K i ), B k = ϕ(A k ) =
D i ⊂ B = ϕ(A), lemma
4 shows that the pairwise intersections of the ϕ(A k ) have measure zero. So
once again,
B k
f (x)dx =
Di
f (x)dx .
(10.23’)
44 more generally of “ measurable ” sets as Lebesgue’s complete theory will show.