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  "body": "\n## Definition\n\nLet $X$ be a [[set|0SJ3RX9TXA]].\nA *π-system* on $X$ is a nonempty family $\\mathcal{P}$ of subsets of $X$ such that $A \\cap B \\in \\mathcal{P}$ for all $A, B \\in \\mathcal{P}$, where $A \\cap B$ is the [[intersection|0HBRW4E6G1]].\n\n## Remarks\n\nThe family $\\set{(-\\infty, a] : a \\in \\R}$ is a π-system on $\\R$, since $(-\\infty, a] \\cap (-\\infty, b] = (-\\infty, \\min(a, b)]$.\nThe family $\\set{[0, 1], [1, 2]}$ is not a π-system, since $[0, 1] \\cap [1, 2] = \\set{1}$ is not in it.\nEvery [[σ-algebra|04WSW8T3SY]] is a π-system.\nBy induction, a π-system is closed under all finite intersections.\nThe name is also written $\\pi$-system.\n",
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