It is known that a group action $\varphi:G\times X\to X$ on the compact metric space $X$ has a $\varphi$- invariant Borel probability measure on $X$ if and only if $G$ is amenable. Note that the group $G$ is amenable, if there is a sequence of finite sets $\{F_n\}_{n=0}^\infty\subseteq G$ such that
\begin{equation*} \lim _{n\to \infty}\frac{|gF_n\Delta F_n|}{|F_n|}=0, \forall g\in G \end{equation*}
In my research I need to work with a measure $\mu$ that is $\varphi$-invariant and $supp(\mu)=X$. What conditions imply it? Please help to know it.