Let $G$ be a locally compact and $\sigma$-compact Hausdorff group. Is true that $G$ is amenable? My attempt is to consider a sequence $(K_n)$ of compact subsets of $G$ such that $G=\bigcup_{n\in\mathbb{N}} K_n$ and for each $n$ define $m_n:L^\infty(G)\to\mathbb{R}$ as $$ m_n(f) = \frac{1}{\mu(K_n)}\int_{K_n}f\textrm{ d}\mu,$$
where $\mu$ is the Haar measure, and prove that for all ultrafilter $\mathcal{U}$ on $\mathbb{N}$, the limit
$$m_{\mathcal{U}}=\underset{n\in\mathbb{N}}{\mathcal{U}-\lim}\ m_n$$
in the weak*-topology of $L^\infty(G)^*$ defines a left-invarian mean on $G$. Is this correct?
No, any countable group is $\sigma$-compact but not all such groups are amenable, e.g. the free group on two generators.