206
IX – Multivariate Differential and Integral Calculus
Choosing g so that
g(−x)dm(x) = 1,
μ(f ) = μ(g)m(f )
for all f , qed.
Lemma b. For every g ∈ G, there is a number Δ(g) > 0 such that
f [g(x)] dm(x) = Δ(g)
f (x)dm(x)
(10.7)
for all f ∈ L(E).
Formula μ(f ) =
f [g(x)]dm(x) defines a Radon measure on E and μ is
clearly invariant, hence the formula. Δ(g) > 0 because if f is positive, then
so are the two integrals.
As can now be seen, if g is the diagonal matrix (t 1 , . . . , t n ), in which case
the left hand side of (7) is the integral of f (t 1 x
1 , . . . , t n x
n ), then the change
of x
i
→ t i x
i shows that Δ(g) = |t 1 . . . t n |
−1 = | det(g)|
−1 .
Lemma c. The map Δ is a continuous homomorphism from G to the multiplicative group R
∗
+ .
The relation Δ(gh) = Δ(g)Δ(h) can be obtained by apply lemma b twice.
To show that Δ is continuous, observe fist that, like g(x), f [g(x)] is a continuous function of the couple (g, x) ∈ G × E ; it is, moreover, zero outside
g
−1 (M ), where M ⊂ E is the compact support of f ; but when g varies
in a compact subset of G, the set g
−1 (M ) remains in the image K of the
compact set N × M under the map (g, x) → g
−1 (x). As g → g
−1 , and so
(g, x) → g
−1 (x) is also continuous, K is compact. Hence if g remains in a
fixed compact subset N of G, as can be seen, the integral of lemma b in
fact generalizes to a fixed compact subset of E. Continuity with respect to
the parameter g ∈ N is then clear (Chapter V, § 2, n
◦ 9, Theorem 9). Observing that G being an open subset of M n (R), all g ∈ G have a compact
neighbourhood N , leads to the conclusion.
The rest of the proof consists in determining all the continuous homomorphisms from G to R
∗
+ : they are all of the form g → | det(g)|
s , with s ∈ R.
Lemma d. For every matrix g ∈ G, there are orthogonal matrices u and v
and a diagonal matrix t = (t 1 , . . . , t n ), with t i > 0, such that g = utv.
Denote the usual scalar product on R
n by (x|y) and write g
for the
transpose of a matrix g. It is characterized by the identity
(g(x)|y) = (x|g
(y)) .
The orthogonal subgroupK = O n (R) of G is the set of matrices such that
g
g = 1, i.e. such that g(x) = x for all x ; it is obviously closed and
bounded in M n (R), and so compact. On the other hand, there are symmetric
matrices in G, and even in M n (R), i.e. such that h
= h ; for such matrices,
IX – Multivariate Differential and Integral Calculus
Choosing g so that
g(−x)dm(x) = 1,
μ(f ) = μ(g)m(f )
for all f , qed.
Lemma b. For every g ∈ G, there is a number Δ(g) > 0 such that
f [g(x)] dm(x) = Δ(g)
f (x)dm(x)
(10.7)
for all f ∈ L(E).
Formula μ(f ) =
f [g(x)]dm(x) defines a Radon measure on E and μ is
clearly invariant, hence the formula. Δ(g) > 0 because if f is positive, then
so are the two integrals.
As can now be seen, if g is the diagonal matrix (t 1 , . . . , t n ), in which case
the left hand side of (7) is the integral of f (t 1 x
1 , . . . , t n x
n ), then the change
of x
i
→ t i x
i shows that Δ(g) = |t 1 . . . t n |
−1 = | det(g)|
−1 .
Lemma c. The map Δ is a continuous homomorphism from G to the multiplicative group R
∗
+ .
The relation Δ(gh) = Δ(g)Δ(h) can be obtained by apply lemma b twice.
To show that Δ is continuous, observe fist that, like g(x), f [g(x)] is a continuous function of the couple (g, x) ∈ G × E ; it is, moreover, zero outside
g
−1 (M ), where M ⊂ E is the compact support of f ; but when g varies
in a compact subset of G, the set g
−1 (M ) remains in the image K of the
compact set N × M under the map (g, x) → g
−1 (x). As g → g
−1 , and so
(g, x) → g
−1 (x) is also continuous, K is compact. Hence if g remains in a
fixed compact subset N of G, as can be seen, the integral of lemma b in
fact generalizes to a fixed compact subset of E. Continuity with respect to
the parameter g ∈ N is then clear (Chapter V, § 2, n
◦ 9, Theorem 9). Observing that G being an open subset of M n (R), all g ∈ G have a compact
neighbourhood N , leads to the conclusion.
The rest of the proof consists in determining all the continuous homomorphisms from G to R
∗
+ : they are all of the form g → | det(g)|
s , with s ∈ R.
Lemma d. For every matrix g ∈ G, there are orthogonal matrices u and v
and a diagonal matrix t = (t 1 , . . . , t n ), with t i > 0, such that g = utv.
Denote the usual scalar product on R
n by (x|y) and write g
for the
transpose of a matrix g. It is characterized by the identity
(g(x)|y) = (x|g
(y)) .
The orthogonal subgroupK = O n (R) of G is the set of matrices such that
g
g = 1, i.e. such that g(x) = x for all x ; it is obviously closed and
bounded in M n (R), and so compact. On the other hand, there are symmetric
matrices in G, and even in M n (R), i.e. such that h
= h ; for such matrices,