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  "body": "\n## Statement\n\nLet $X$ be a set.\nA *monotone class* on $X$ is a family $\\mathcal{M}$ of subsets of $X$ with these two properties.\nIf $A_1 \\subseteq A_2 \\subseteq \\cdots$ are in $\\mathcal{M}$, then $\\bigcup_{n=1}^{\\infty} A_n \\in \\mathcal{M}$.\nIf $A_1 \\supseteq A_2 \\supseteq \\cdots$ are in $\\mathcal{M}$, then $\\bigcap_{n=1}^{\\infty} A_n \\in \\mathcal{M}$.\n\nLet $\\mathcal{A}$ be an [[algebra of sets|PFMGZ7988V]] on $X$.\nThen the smallest monotone class on $X$ that contains $\\mathcal{A}$ is equal to the [[σ-algebra|0SV8K0TDGY]] $\\sigma(\\mathcal{A})$ generated by $\\mathcal{A}$.\n\n## Remarks\n\nEvery σ-algebra is a monotone class, so the smallest monotone class $m(\\mathcal{A})$ is contained in $\\sigma(\\mathcal{A})$.\nFor the other inclusion, one shows that $m(\\mathcal{A})$ is an algebra.\nFor fixed $A$, the sets $B$ with $A \\cup B$, $A \\setminus B$ and $B \\setminus A$ in $m(\\mathcal{A})$ form a monotone class; this gives closure first for $A \\in \\mathcal{A}$, then for $A \\in m(\\mathcal{A})$.\nAn algebra that is a monotone class is a σ-algebra, since a countable union is the increasing union of finite unions.\n\nAs an application, let $\\mu$ and $\\nu$ be finite measures on $(X, \\sigma(\\mathcal{A}))$ with $\\mu(X) = \\nu(X)$ that agree on $\\mathcal{A}$.\nThe sets on which they agree form a monotone class, by continuity of measures, so $\\mu = \\nu$ on $\\sigma(\\mathcal{A})$.\n\nThe hypothesis that $\\mathcal{A}$ is an algebra is needed.\nOn $X = \\set{1, 2}$, the family $\\set{\\varnothing, \\set{1}}$ is already a monotone class, but the σ-algebra it generates also contains $\\set{2}$.\n\n## Explanations\n\n### L1 human\n\nA monotone class is a family of sets that is stable under limits of growing sequences and of shrinking sequences.\nThe theorem says: start from an algebra of sets, and close it under such limits only.\nThen you already get every set of the generated σ-algebra.\nSo a property that survives monotone limits and holds on the algebra holds on the whole σ-algebra.\n\n### L2 human\n\nThe theorem is a tool to extend statements from simple sets to all measurable sets.\nIt is used to prove uniqueness of extended measures and to set up product measures and the Fubini–Tonelli theorem.\nDynkin's π-λ theorem is a variant in which the algebra is replaced by a π-system and the monotone class by a Dynkin system.\nThere is also a version for classes of functions, used in probability theory.\n",
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