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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a [[σ-finite|WEBBT18D4X]] measure space.\nLet $f \\colon X \\to [0, \\infty]$ be a measurable function with [[distribution function|HQVXKEMN5N]] $d_f(t) = \\mu(\\set{f \u003e t})$.\nThen\n$$\\int_X f \\, d\\mu = \\int_0^\\infty d_f(t) \\, dt,$$\nwhere the left side is the [[integral of a non-negative function|DG76G5Z6XW]] and the right side is taken with [[Lebesgue measure|2Z2QVWZ262]].\nMore generally, for $0 \u003c p \u003c \\infty$,\n$$\\int_X f^p \\, d\\mu = p \\int_0^\\infty t^{p-1} d_f(t) \\, dt.$$\n\n## Remarks\n\nThe name comes from the pointwise identity $f(x) = \\int_0^\\infty \\mathbf{1}_{\\set{f \u003e t}}(x) \\, dt$.\nIt writes $f$ as a stack of thin horizontal layers, one for each height $t$.\nIntegrating over $X$ and exchanging the two integrals by Tonelli's theorem gives the formula.\nThe second formula follows in the same way from $f(x)^p = \\int_0^{f(x)} p t^{p-1} \\, dt$.\n\nExample: for $f(x) = x$ on $[0, 1]$ with Lebesgue measure, $d_f(t) = 1 - t$ for $0 \\le t \\le 1$ and $0$ after.\nBoth sides equal $1/2$.\n\nFor a non-negative random variable $Y$ the first formula reads $\\mathbb{E}[Y] = \\int_0^\\infty P(Y \u003e t) \\, dt$.\n\n## Explanations\n\n### L1 human\n\nTo find the volume under a graph, one can slice it vertically or horizontally.\nThe layer cake formula slices horizontally.\nAt each height $t$ it measures the size of the region where the function is above $t$, and then adds these sizes over all heights.\n\n### L2 human\n\nThe formula turns integrals of $f$ into integrals of its distribution function.\nThis is why the $L^p$ norms depend only on $d_f$, so equimeasurable functions and decreasing rearrangements have the same norms.\nIt is the basic tool behind weak $L^p$ estimates, the Marcinkiewicz interpolation theorem and the bounds for the Hardy–Littlewood maximal function.\n",
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