§ 3. Integration of Differential Forms
205
invertible matrices and, contrary to some shorter ones, requiring little integral
calculations.
Set E = R
n and let L(E) be the set of continuous functions with compact
support in E. We will not differentiate between a matrix g ∈ G and the linear
map x → g(x) corresponding to it on E and for which g is the matrix with
respect to the canonical basis (e i ) : the formulas
g (e i ) = g
j
i e j , g(x)
i = g
i
j x
j ,
(10.5)
where the g(x)
i are the canonical coordinates of the vector g(x), in conformity
with tensor conventions, then follow. The Jacobian of y = g(x) with respect
to x is then equal to det(g), so that it suffices to prove that for all f ∈ L(E),
f [g(x)] dx = | det(g)|
−1
f (x)dx ,
(10.6)
where integration is over all of E.
Lemma a. Let μ be a Radon measure on R
n . Assume μ to be invariant
under translations. Then μ is proportional to the Lebesgue measure m.
Note first that, for any f, g ∈ L(E),the function (x, y) → f (x)g(y − x)
has compact support in E × E, since if f and g are zero outside the compact
sets M and N , then it can be = 0 only if (x, y) belongs to the compact
set M × (M + N ). The most elementary Lebesgue-Fubini theorem can then
be applied to this functions, and in the following, formal calculations are
possible. Having said this,
m(f )μ(g) =
f (x)g(y)dm(x)dμ(y) =
f (x)g(y − x)dm(x)dμ(y)
by the change of variable y
→ y − x in the integration with respect to μ ; the
change of variable x → x + y in the integration with respect to m then gives
m(f )μ(g) =
f (x + y)g(−x)dm(x)dμ(y) =
=
g(−x)dm(x)
f (x + y)dμ(y) =
= μ(f )
g(−x)dm(x) .
linear form, this category contains “ exceptional ” groups whose construction is
far less obvious.
For a more elementary but far less instructive proof, see for example Rudin,
Real and Complex Analysis, end of Chapter 8, where a proof of the complete result analogous to ours can also be found, as well as more subtle results from the
theory of integration. Dieudonn´ e, El´ ements d’analyse (vol. 3, XVI.22) uses the
fact that, locally, all diffeomorphisms decompose into simpler maps (changing
one variable at a time) for which the formula is more or less obvious, thereby
avoiding all approximation calculations presented in parts (ii) and (iii) below.
205
invertible matrices and, contrary to some shorter ones, requiring little integral
calculations.
Set E = R
n and let L(E) be the set of continuous functions with compact
support in E. We will not differentiate between a matrix g ∈ G and the linear
map x → g(x) corresponding to it on E and for which g is the matrix with
respect to the canonical basis (e i ) : the formulas
g (e i ) = g
j
i e j , g(x)
i = g
i
j x
j ,
(10.5)
where the g(x)
i are the canonical coordinates of the vector g(x), in conformity
with tensor conventions, then follow. The Jacobian of y = g(x) with respect
to x is then equal to det(g), so that it suffices to prove that for all f ∈ L(E),
f [g(x)] dx = | det(g)|
−1
f (x)dx ,
(10.6)
where integration is over all of E.
Lemma a. Let μ be a Radon measure on R
n . Assume μ to be invariant
under translations. Then μ is proportional to the Lebesgue measure m.
Note first that, for any f, g ∈ L(E),the function (x, y) → f (x)g(y − x)
has compact support in E × E, since if f and g are zero outside the compact
sets M and N , then it can be = 0 only if (x, y) belongs to the compact
set M × (M + N ). The most elementary Lebesgue-Fubini theorem can then
be applied to this functions, and in the following, formal calculations are
possible. Having said this,
m(f )μ(g) =
f (x)g(y)dm(x)dμ(y) =
f (x)g(y − x)dm(x)dμ(y)
by the change of variable y
→ y − x in the integration with respect to μ ; the
change of variable x → x + y in the integration with respect to m then gives
m(f )μ(g) =
f (x + y)g(−x)dm(x)dμ(y) =
=
g(−x)dm(x)
f (x + y)dμ(y) =
= μ(f )
g(−x)dm(x) .
linear form, this category contains “ exceptional ” groups whose construction is
far less obvious.
For a more elementary but far less instructive proof, see for example Rudin,
Real and Complex Analysis, end of Chapter 8, where a proof of the complete result analogous to ours can also be found, as well as more subtle results from the
theory of integration. Dieudonn´ e, El´ ements d’analyse (vol. 3, XVI.22) uses the
fact that, locally, all diffeomorphisms decompose into simpler maps (changing
one variable at a time) for which the formula is more or less obvious, thereby
avoiding all approximation calculations presented in parts (ii) and (iii) below.