Amenability is equivalent to Reiter's condition (P1)

Suppose $G$ is amenable. Let $\nu$ be a left-invariant mean on $L^{\mathrm{\infty }}(G)$ and define $m(\psi ):=\nu ([\psi ])$ for $\psi \in \mathrm{U}\mathrm{C}\mathrm{B}(G)$. By [F1], the class map is well-defined and injective; positivity, complex linearity, normalization, and left invariance pass from $\nu$ to $m$. Hence [F2] gives Reiter's condition (P1).

Suppose $G$ satisfies (P1). By [F3], choose a Reiter net $(f_i)\subseteq \mathcal{P}$. By [A1] and [F5], the ultrafilter lemma holds. The cluster-point result [F6] gives a left-invariant mean on $L^{\mathrm{\infty }}(G)$, so $G$ is amenable by [F4].

Let ${\mu }^{\mathrm{\prime }}=c\mu$ for $c>0$. The measures have the same null sets, and $f\mapsto f^{\mathrm{\prime }}:=c^{-1}f$ maps ${\mathcal{P}}_{\mu }$ bijectively to ${\mathcal{P}}_{{\mu }^{\mathrm{\prime }}}$. For every compact $Q$, $L_x{f}^{\mathrm{\prime }}=c^{-1}{L}_{x}f$ on the common almost-everywhere classes, so ${\mathrm{\Delta }}_{Q,{\mu }^{\mathrm{\prime }}}(f^{\mathrm{\prime }})={\mathrm{\Delta }}_{Q,\mu }(f)$. Thus (P1) is independent of this rescaling. Amenability is likewise normalization-independent by [F4], with ${\mu }^{\mathrm{\prime }}$ still a left Haar measure by [F7].

Steps 1.1 and 1.2 prove the two implications for the fixed left Haar measure, and step 1.3 proves normalization independence. Therefore the stated equivalence holds.