Corollary

If G is discrete, or compact, or commutative, then G is unimodular.

This is moreover trivial when G is commutative. Note also that if G is discrete, then the measure on G for which each point has mass 1 is obviously a left and right Haar measure on G, called the normalized Haar measure on G. If G is compact, there exists one and only one Haar measure on G such that $\mu(G) = 1$; it is called the normalized Haar measure of G. The preceding two conventions are not in accord when G is both discrete and compact, that is, finite; when we are in this case we shall always explicitly specify what is meant by normalized Haar measure.

Subgroups and quotient groups of a unimodular group are not always unimodular (§ 2, Exer. 5). See, however, Prop. 10 of § 2, No. 7.

We shall see later that semi-simple or nilpotent connected Lie groups are unimodular.