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  "body": "\n## Definition\n\nLet $X$ be a [[set|0SJ3RX9TXA]].\nA *measurable space* is a pair $(X, \\Sigma)$, where $\\Sigma$ is a [[σ-algebra|04WSW8T3SY]] on $X$.\nThe elements of $\\Sigma$ are the *measurable sets* of $(X, \\Sigma)$.\n\n## Remarks\n\nThe real line with its [[Borel σ-algebra|0BVHZXWKCD]] $\\mathcal{B}(\\R)$ is a measurable space, because $\\mathcal{B}(\\R)$ is a σ-algebra by its definition.\nAny set $X$ with its power set is a measurable space, and so is $X$ with $\\set{\\varnothing, X}$.\nFor $X = \\set{1, 2}$, the pair $(X, \\set{\\varnothing, \\set{1}, X})$ is not a measurable space.\nThe family is not closed under complements, since $X \\setminus \\set{1} = \\set{2}$ is missing.\nA map $f \\colon X \\to Y$ between measurable spaces $(X, \\Sigma)$ and $(Y, T)$ is called measurable when $f^{-1}(B) \\in \\Sigma$ for every $B \\in T$.\nSome authors call a measurable space a *Borel space*.\n",
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