The ergodic decomposition defined by actions of amenable groups

Let $G$ be a locally compact separable metric group, and assume that $G$ is amenable. Let $m_{_{\mathrm {L}}}$ be a left Haar measure on $G$, and let $\alpha =(F_n)_{n\in \mathbb {N}}$ be a tempered Følner sequence. Let $(X,d)$ be a locally compact s