This post is just a collection of basic results I have compiled for referring to in desperate times Nothing too deep except Haar-von Neumann’s theorem.

A measure $\mu$ on a locally compact (Hausdorff, always) group $G$ is left-invariant if

$\boxed{ \int_G f(s^{-1}x) \text{d} \mu(x) = \int_G f(x) \text{d} \mu(x). }$

The most important theorem in this topic is the existence and uniqueness of the Haar measure (proven by Haar and von Neumann, respectively).

Theorem 1: On every locally compact group $G$ there is a unique (up to a positive constant of proportionality) left-invariant positive measure $\mu \neq 0$.

Proposition 2: $G, \mu$ as usual. For $G$ to have a finite measure, it is necessary and sufficient that $G$ be compact!

Proposition 3: There is a continuous group-homomorphism (called right modulus), $\triangle_r : G \to \mathbb R^*_+$ such that

$f(xs^{-1}) \text{d} \mu(x) = \triangle_r(s) \int f(x) \text{d} \mu(x).$

Proposition 4: Let $\mu$ (respectively, $\nu$) be a left-invariant Haar measure on locally compact groups $H$ (resp., $K$). Then the product integral on $G = H \times K$ is left-invariant and

$\boxed{\triangle^G_r(s,t) = \triangle^H_r(s) . \triangle^K_r(t).}$

Corollary 5: $G$ is unimodular (i.e., $\triangle_r =1$) precisely when $H$ and $K$ are so.

The most important example (for me) is the general linear group over number fields or $p$-adics and its subgroups. All semisimple (and more generally reductive) groups are unimodular. Compact groups are unimodular. Abelian groups are trivially so. However the measure on Borel (and parabolic) subgroups (i.e., upper triangular matrices) is not unimodular. The above proposition 4 allows one to transfer the Levi decomposition on the groups to their measures.