Compact, discrete and abelian groups are unimodular

Compact case. By [F2] and [A1] there is a left Haar probability $\lambda$ on $G$ that is right invariant, so ${\int }_{G}f(xg)d\lambda (x)={\int }_{G}fd\lambda (x)$ for every $f\in C_c(G)$ and every $g$; by [F4] there is $t>0$ with $\lambda =t\mu$ on Borel sets. Hence ${\int }_{G}f(xg)d\mu (x)=t^{-1}{\int }_{G}f(xg)d\lambda (x)=t^{-1}{\int }_{G}fd\lambda (x)={\int }_{G}fd\mu (x)$ for every $f\in C_c(G)$, so the scalar attached to $\mu$ by [F1] satisfies $c(g)=1$ by its uniqueness, hence ${\mathrm{\Delta }}_G(g)=c(g^{-1})=1$ for every $g$ and $G$ is unimodular by [F1].

Abelian case. Fix $g\in G$ and $f\in C_c(G)$. Since $xg=gx$ and $\mu$ is left invariant, ${\int }_{G}f(xg)d\mu (x)={\int }_{G}f(gx)d\mu (x)={\int }_{G}f(x)d\mu (x)$, so $c(g)=1$ by the uniqueness in [F1]; as $g$ was arbitrary, ${\mathrm{\Delta }}_G\equiv 1$ and $G$ is unimodular.

Discrete case. Let $G$ be discrete and let $\nu$ be counting measure. By [F5] the measure $\nu$ is a left Haar measure and a right Haar measure on $G$, and the fixed left Haar measure $\mu$ equals $c\nu$ for some $c>0$; since $\nu$ is right invariant, so is $\mu$, hence $c(g)=1$ for every $g$ by the uniqueness in [F1] and ${\mathrm{\Delta }}_G\equiv 1$.

Every compact, discrete or abelian LCH group therefore falls under one of steps 1.1–1.3 and is unimodular. ∎