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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]] and let $A \\in \\Sigma$.\nThe *restriction* of $\\mu$ to $A$ is the map $\\mu|_A \\colon \\Sigma \\to [0, \\infty]$ given by\n$$\\mu|_A(B) = \\mu(A \\cap B) \\quad \\text{for } B \\in \\Sigma.$$\n\n## Remarks\n\nThe map $\\mu|_A$ is a [[measure|PR11BZQQPG]] on $(X, \\Sigma)$, because intersecting a disjoint family with $A$ gives a disjoint family with the same union intersected with $A$.\nExample: if $\\lambda$ is Lebesgue measure on $\\R$ and $A = [0, 1]$, then $\\lambda|_A([0, 5]) = \\lambda([0, 1]) = 1$.\nFor a measurable $f \\colon X \\to [0, \\infty]$ one has $\\int_X f \\, d(\\mu|_A) = \\int_A f \\, d\\mu$.\nNon-example: for a set $A \\notin \\Sigma$, the sets $A \\cap B$ need not lie in $\\Sigma$, so the formula need not define a measure on $\\Sigma$.\nA related view keeps only the points of $A$: the trace $\\sigma$-algebra $\\Sigma_A = \\set{A \\cap B : B \\in \\Sigma}$ on $A$, with the measure $\\mu$ restricted to $\\Sigma_A$.\nOther notations are $\\mu \\llcorner A$ and $\\mu(\\,\\cdot \\cap A)$.\n",
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