Given an amenable group, it is a standard trick to turn a left-invariant mean ( i.e. a continuous positive normalised linear functional $m:\ell_\infty(G) \to \mathbb{R}$ such that $\forall g \in G, m \circ \lambda_g = m$ where $\lambda_g: \ell_\infty(G) \to \ell_\infty(G)$ is the left-regular action of $G$) into a bi-invariant mean (also invariant under pre-composition with the right-regular action of $G$).

From this bi-invariant mean one gets a sequence (or a net, when $G$ is uncountable) of almost invariant probability measures (i.e. $\xi_n \in \ell_1G$ with $\| \lambda_g \xi_n - \xi_n\|_1 \to 0$ and $\| \rho_g \xi_n - \xi_n\|_1 \to 0$).

$\mathbf{Question}$: Does there exists a bi-invariant Folner sequence? (i.e. a sequence of finite set $F_n$ such that $\xi_n = \chi_{F_n} /|F_n|$ is a sequence of almost invariant probability measure)

In other words, is it obvious that the `bi-invariant property'' follow through the same proof as`

left-'' and ``right-'' do, or, better, does there exist a simpler argument to show such a sequence exists?