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  "body": "\n## Definition\n\nLet $X$ be a [[set|0SJ3RX9TXA]].\nA *Dynkin system* on $X$ is a family $\\mathcal{D}$ of subsets of $X$ such that:\n\n1. $X \\in \\mathcal{D}$.\n2. If $A, B \\in \\mathcal{D}$ and $A \\subseteq B$, then $B \\setminus A \\in \\mathcal{D}$.\n3. If $A_1 \\subseteq A_2 \\subseteq \\cdots$ is an increasing sequence of members of $\\mathcal{D}$, then the [[union|07KVB4JSC7]] $\\bigcup_{n=1}^{\\infty} A_n$ is in $\\mathcal{D}$.\n\n## Remarks\n\nOn $X = \\set{1, 2, 3, 4}$, the family of subsets with an even number of elements is a Dynkin system.\nClause 1 holds since $X$ has $4$ elements.\nClause 2 holds since $B \\setminus A$ has $\\abs{B} - \\abs{A}$ elements when $A \\subseteq B$.\nClause 3 holds since an increasing sequence in a finite family is eventually constant.\nOn $X = \\set{1, 2}$, the family $\\set{\\varnothing, \\set{1}, X}$ is not a Dynkin system, since clause 2 fails: $X \\setminus \\set{1} = \\set{2}$ is missing.\nAn equivalent definition: $X \\in \\mathcal{D}$, $\\mathcal{D}$ is closed under [[complements|0B1FSS1VZW]], and $\\mathcal{D}$ is closed under countable unions of pairwise disjoint members.\nEvery [[σ-algebra|04WSW8T3SY]] is a Dynkin system.\nA Dynkin system that is also a [[π-system|4W9NM803BB]] is a σ-algebra.\nDynkin systems are also called λ-systems.\n",
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