Reiter's condition

Remarks

The functions $f$ are almost invariant probability densities. From such densities, an invariant mean on $L^\infty(G)$ is obtained as a weak-$*$ cluster point of the functionals $\varphi \mapsto \int \varphi f$. Conversely, an invariant mean gives densities that are almost invariant in the weak topology of $L^1(G)$. A convexity argument, based on the Hahn–Banach theorem, upgrades them to the norm condition.

Example: for $G = \Z$ and $K = \set{-k, \ldots, k}$, take $f = \frac{1}{n} \mathbf{1}_{\set{0, \ldots, n-1}}$. Then $\norm{L_j f - f}_1 = 2\abs{j}/n \le 2k/n$ for $j \in K$, which is less than $\varepsilon$ when $n$ is large.

For a discrete group the condition says that there are finitely supported probability vectors that are almost invariant in $\ell^1(G)$. A Følner sequence gives such vectors by normalizing indicator functions.

Going deeper

For a student

Reiter's condition is the bridge between invariant means and the Følner condition. The $L^2$ version, with $\norm{\cdot}_2$ and unit vectors, says that the left regular representation has almost invariant vectors, which is Hulanicki's theorem. It is also the form of amenability used for the group algebra, as in Johnson's theorem that $L^1(G)$ is an amenable Banach algebra exactly when $G$ is amenable.