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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]], and for a subset $F \\subseteq G$ and $g \\in G$ let $gF = \\set{gx : x \\in F}$.\nThe group $G$ satisfies the *Følner condition* if the following holds.\nFor every finite set $S \\subseteq G$ and every $\\varepsilon \u003e 0$, there is a finite nonempty set $F \\subseteq G$ with\n$$\\abs{sF \\,\\triangle\\, F} \u003c \\varepsilon \\abs{F} \\quad \\text{for all } s \\in S,$$\nwhere $\\triangle$ is the symmetric difference.\nFor a countable group, a *Følner sequence* is a sequence $(F_n)$ of finite nonempty subsets with $\\abs{gF_n \\triangle F_n} / \\abs{F_n} \\to 0$ for every $g \\in G$.\n\n## Remarks\n\nExample: $G = \\Z$ with $F_n = \\set{0, 1, \\ldots, n-1}$.\nFor $k \\in \\Z$ with $\\abs{k} \\le n$, the sets $k + F_n$ and $F_n$ differ in $2\\abs{k}$ points, so $\\abs{(k + F_n) \\triangle F_n} / n = 2\\abs{k}/n \\to 0$.\nSo $(F_n)$ is a Følner sequence.\n\nNon-example: the free group $F_2$ on two generators fails the condition.\nEvery finite nonempty $F \\subseteq F_2$ has some generator $s$ with $\\abs{sF \\triangle F}$ at least a fixed positive fraction of $\\abs{F}$.\nThis reflects that $F_2$ is not [[amenable|CEAT9BESKK]].\n\nFølner's theorem says that a discrete group satisfies the Følner condition exactly when it is amenable.\nFor a [[locally compact group|AHV05ZBCYW]], one uses compact sets $F$ of positive [[Haar measure|VY0HXD3GC4]] and compact sets $S$ instead.\n",
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