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  "body": "\n## Statement\n\nLet $X$ be a set, let $\\mathcal{P}$ be a [[π-system|4W9NM803BB]] on $X$, and let $\\mathcal{D}$ be a [[Dynkin system|V5M2CKB591]] on $X$ with $\\mathcal{P} \\subseteq \\mathcal{D}$.\nLet $\\sigma(\\mathcal{P})$ be the [[σ-algebra|0SV8K0TDGY]] generated by $\\mathcal{P}$.\nThen $\\sigma(\\mathcal{P}) \\subseteq \\mathcal{D}$.\n\n## Remarks\n\nLet $d(\\mathcal{P})$ be the smallest Dynkin system that contains $\\mathcal{P}$.\nFor a fixed $A \\in \\mathcal{P}$, the sets $B$ with $A \\cap B \\in d(\\mathcal{P})$ form a Dynkin system that contains $\\mathcal{P}$, so it contains $d(\\mathcal{P})$.\nRepeating the argument with $A \\in d(\\mathcal{P})$ shows that $d(\\mathcal{P})$ is a π-system.\nA Dynkin system that is a π-system is a σ-algebra, so $\\sigma(\\mathcal{P}) = d(\\mathcal{P}) \\subseteq \\mathcal{D}$.\n\nThe main use is uniqueness of measures.\nLet $\\mu$ and $\\nu$ be [[probability measures|059TJN0NZ1]] on $(X, \\sigma(\\mathcal{P}))$ that agree on $\\mathcal{P}$.\nThe sets on which $\\mu$ and $\\nu$ agree form a Dynkin system that contains $\\mathcal{P}$, so $\\mu = \\nu$ on $\\sigma(\\mathcal{P})$.\nFor example, a probability measure on the Borel sets of $\\R$ is fixed by its values on the rays $(-\\infty, a]$.\n\nThe hypothesis that $\\mathcal{P}$ is a π-system is needed.\nOn $X = \\set{1, 2, 3, 4}$, let $\\mathcal{P} = \\set{\\set{1, 2}, \\set{2, 3}}$ and let $\\mathcal{D}$ be the family of subsets with an even number of elements.\nThen $\\mathcal{D}$ is a Dynkin system that contains $\\mathcal{P}$, but $\\set{2} = \\set{1, 2} \\cap \\set{2, 3}$ lies in $\\sigma(\\mathcal{P})$ and not in $\\mathcal{D}$.\n\n## Explanations\n\n### L1 human\n\nTo show that a property holds for every set of a σ-algebra, it is often enough to check it on a simple generating family.\nThe theorem says this works when the simple family is closed under intersections and the good sets form a Dynkin system.\nDynkin systems are easy to check, since they only ask for complements and disjoint unions.\n\n### L2 human\n\nThe π-λ theorem is the workhorse behind uniqueness statements in measure theory and probability.\nIt shows that a distribution function determines a probability measure on $\\R$, and that a measure on a product is fixed by its values on rectangles.\nIt also gives the independence of σ-algebras generated by independent π-systems.\nThe monotone class theorem plays the same role with algebras of sets in place of π-systems.\n",
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