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  "body": "\n## Summary\n\nA group satisfies the Følner condition when it has finite subsets that are almost invariant under translation.\nMoving such a set by a given element changes only a small fraction of it.\nA sequence of such sets is a Følner sequence.\nBy Følner's theorem, a discrete group satisfies the condition exactly when it is [[amenable|CEAT9BESKK]].\nFølner sequences are the natural sets over which to take averages in ergodic theorems for group actions.\n\n## Explanations\n\n### L1 human\n\nIn the integers, the block $\\set{0, 1, \\ldots, n-1}$ is almost unchanged when it is shifted by $1$: only one point leaves and one point enters.\nAs $n$ grows, the fraction of points that change goes to $0$.\nGroups with such almost invariant finite sets are said to satisfy the Følner condition.\nIn the grid $\\Z^2$, large squares work, since their boundary is small compared with their area.\nIn a free group, every finite set has a boundary comparable to its size, so the condition fails.\n",
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