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  "body": "\n## Definition\n\nLet $\\Omega$ be a [[set|0SJ3RX9TXA]].\nA σ-algebra on $\\Omega$ is a subset $\\mathcal{F}$ of the [[power set|0BC7KX51Z8]] of $\\Omega$ with these three properties.\nThe set $\\Omega$ is in $\\mathcal{F}$.\nIf $A \\in \\mathcal{F}$, then its [[complement|0B1FSS1VZW]] $\\Omega \\setminus A$ is in $\\mathcal{F}$.\nIf $A_1, A_2, A_3, \\dots$ is a sequence of elements of $\\mathcal{F}$, then the [[union|07KVB4JSC7]] $\\bigcup_{n=1}^{\\infty} A_n$ is in $\\mathcal{F}$.\nThe pair $(\\Omega, \\mathcal{F})$ is a measurable space, and the elements of $\\mathcal{F}$ are the measurable sets.\n\n## Remarks\n\nA σ-algebra contains $\\varnothing$ and is closed under countable intersections, by De Morgan's laws.\nThe power set of $\\Omega$ and the family $\\{\\varnothing, \\Omega\\}$ are σ-algebras on $\\Omega$. They are the largest and the smallest σ-algebras on $\\Omega$.\nFor each family $\\mathcal{C}$ of subsets of $\\Omega$, the intersection of all σ-algebras that contain $\\mathcal{C}$ is a σ-algebra. It is the σ-algebra $\\sigma(\\mathcal{C})$ generated by $\\mathcal{C}$.\nThe [[Borel σ-algebra|0BVHZXWKCD]] on $\\mathbb{R}$ is the σ-algebra generated by the open intervals.\n\n## Explanations\n\n### L1 human\n\nA σ-algebra is the list of the sets that we are allowed to measure.\nIf we can measure a set, then we can measure its complement.\nIf we can measure each set of a sequence, then we can measure their union.\nFor a finite sample space we can usually measure every subset, so the σ-algebra is the power set.\n",
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