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  "body": "\n## Summary\n\nGiven a [[measure space|P15VQS5DJ9]] $(X, \\Sigma, \\mu)$ and a measurable function $f \\ge 0$, the measure with density $f$ gives each set $A$ the mass $\\int_A f \\, d\\mu$.\nIt is written $f \\mu$ or $f \\, d\\mu$.\nIt is the basic way to build new measures from old ones.\nThe Radon–Nikodym theorem says that, for σ-finite measures, every measure absolutely continuous with respect to $\\mu$ is of this form.\n\n## Explanations\n\n### L1 human\n\nStart with length on the real line and a weight function $f$.\nThe new measure of an interval is the area under the graph of $f$ over that interval.\nWhere $f$ is large, the new measure packs more mass.\nWith $f(x) = e^{-x^2/2} / \\sqrt{2\\pi}$ and Lebesgue measure, one gets the standard normal distribution.\n",
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