Proposition 55

Suppose that K is locally compact. Let μ be a Haar measure on the additive group of K. Let G be a Lie group of finite dimension n.

(i) Let ω be a non-zero left invariant differential form of degree n on G. Then the measure $\mathrm{mod}(\omega)_\mu$ (Differentiable and Analytic Manifolds, R, 10.1.6) is a left Haar measure on G. If $K = \mathbf{R}$ and G has the orientation defined by ω, the measure defined by ω (Differentiable and Analytic Manifolds, R, 10.4.3) is a left Haar measure on G.

(ii) Let φ be an étale endomorphism of G. Then $\mathrm{mod}\, \phi = \mathrm{mod}\, \det L(\phi)$.

(i) is obvious. Let V, $V'$ be open neighbourhoods of e such that $\phi(V) = V'$ and $\phi|V$ is a local isomorphism of G into G. Then

whence $\mathrm{mod}\, \phi = \mathrm{mod}\, \det L(\phi)$ by definition of $\mathrm{mod}\, \phi$.