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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]] and let $f \\colon X \\to [0, \\infty]$ be a [[measurable function|YYNCJSR0PJ]].\nThe *measure with density* $f$ is the map $f\\mu \\colon \\Sigma \\to [0, \\infty]$ given by\n$$(f\\mu)(A) = \\int_A f \\, d\\mu = \\int_X f \\, \\mathbf{1}_A \\, d\\mu,$$\nwhere the integral is the [[integral of a non-negative function|DG76G5Z6XW]].\n\n## Remarks\n\nThe map $f\\mu$ is a [[measure|PR11BZQQPG]]: $(f\\mu)(\\emptyset) = 0$, and countable additivity follows from the monotone convergence theorem applied to $f \\mathbf{1}_{A_1 \\cup \\dots \\cup A_n}$.\nIt is absolutely continuous with respect to $\\mu$, since $\\mu(A) = 0$ implies $\\int_A f \\, d\\mu = 0$.\n\nExample: on $\\R$ with Lebesgue measure and $f = \\mathbf{1}_{[0,1]}$, the measure $f\\mu$ is Lebesgue measure restricted to $[0, 1]$.\nCheck: $(f\\mu)(A) = \\int \\mathbf{1}_{A \\cap [0,1]} \\, dx$, the length of $A \\cap [0, 1]$.\n\nNon-example: $f(x) = x$ on $\\R$ takes negative values, so it is not an admissible density.\nThe formula $A \\mapsto \\int_A x \\, dx$ is not even defined for $A = \\R$.\n\nFor measurable $g \\ge 0$, $\\int g \\, d(f\\mu) = \\int g f \\, d\\mu$.\nNotation: $f\\mu$, $f \\cdot \\mu$, or $d\\nu = f \\, d\\mu$.\n",
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