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 on a locally compact (Hausdorff, always) group is **left-invariant** if

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 there is a unique (up to a positive constant of proportionality) left-invariant positive measure .

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

**Proposition 3: **There is a continuous group-homomorphism (called **right modulus**), such that

**Proposition 4:** Let (respectively, ) be a left-invariant Haar measure on locally compact groups (resp., ). Then the product integral on is left-invariant and

**Corollary 5:** is unimodular (i.e., ) precisely when and are so.

The most important example (for me) is the general linear group over number fields or -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.

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