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  "body": "\n## Definition\n\nLet $(X, \\mathcal{A}, \\mu)$ be a [[measure space|P15VQS5DJ9]] and let $G$ be a [[countable|01TQB88H4C]] group that [[acts|0WCY42EV9V]] on $X$ by [[measure-preserving|556WBDAHNP]] measurable maps $x \\mapsto gx$.\nA measurable set $F \\subseteq X$ is a *fundamental domain* for this action if:\n\n1. the translates cover almost all of $X$: $\\mu\\big(X \\setminus \\bigcup_{g \\in G} gF\\big) = 0$;\n2. distinct translates are almost disjoint: $\\mu(gF \\cap hF) = 0$ for all $g, h \\in G$ with $g \\ne h$.\n\n## Remarks\n\nExample: $G = \\Z$ acting on $X = \\R$ with Lebesgue measure by $n \\cdot x = x + n$, and $F = [0, 1)$.\nThe translates $F + n = [n, n+1)$ cover $\\R$ and are pairwise disjoint, so both clauses hold.\n\nNon-example: for the same action, $F = [0, 2)$ covers $\\R$ but fails clause 2, since $F \\cap (F + 1) = [1, 2)$ has measure $1$.\n\nWithout a measure, a *strict fundamental domain* is a set that meets each [[orbit|0YQPZB5Q1G]] in exactly one point.\nIn geometry one often asks instead for a closed region $D$ whose translates cover $X$ and whose interiors are pairwise disjoint.\nAny two measurable fundamental domains of the same action have the same measure.\n",
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