Compact and locally compact abelian groups are amenable

Let $K$ be compact. By [A1, F1], choose a left Haar measure $\nu$ and set $\mu :=\nu \mathrm{/}\nu (K)$; [F1] gives $0<\nu (K)<\mathrm{\infty }$, and positive scalar rescaling preserves left invariance and regularity, so $\mu$ is a normalized Haar probability. For $\phi \in L^{\mathrm{\infty }}(K,\mu ;\mathbb{C})$ and every $\eta >0$, the definition of essential supremum gives $\mathrm{\mid }\phi \mathrm{\mid }\le \mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}+\eta$ almost everywhere. Since $\mu (K)=1$, monotonicity of the nonnegative integral gives ${\int }_K\mathrm{\mid }\phi \mathrm{\mid }d\mu \le \mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}+\eta <\mathrm{\infty }$, so $\phi$ is integrable by [F2]. Define $m(\phi ):={\int }_K\phi d\mu$. Its definition is independent of the representative by [F2]; linearity and positivity follow from [F3], while the simple-function integral gives $m(1_K)=\mu (K)=1$. The triangle inequality gives $\mathrm{\mid }m(\phi )\mathrm{\mid }\le \mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}+\eta$ for every $\eta >0$, hence $\mathrm{\mid }m(\phi )\mathrm{\mid }\le \mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$. Thus $m$ is a bounded positive unital functional, hence a mean by [F5].

Let $G$ be locally compact Hausdorff and abelian. By [F7], every continuous affine action of $G$ on a nonempty compact convex subset of a Hausdorff locally convex space has a fixed point. Thus $G$ has the fixed-point property required by [F8]. Applying [F8] proves that $G$ is amenable.

For $g\in K$, the left translation $T_g(x)=g^{-1}x$ is Borel and measure-preserving by [F4]. It therefore preserves null sets, so composition defines the same $L^{\mathrm{\infty }}$ class independently of the representative. The integral invariance in [F4] gives $m(L_g\phi )={\int }_K\phi \circ T_{g}d\mu ={\int }_K\phi d\mu =m(\phi )$ for every $\phi \in L^{\mathrm{\infty }}(K)$. Thus the mean from step 1.1 is left-invariant, and $K$ is amenable by [F5, F6].

Step 2.1 proves part (1), and step 1.2 proves part (2), with no countability or unimodularity assumption.