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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a measure space and let $f$ be a real or complex measurable function on $X$.\nLet $d_f(s) = \\mu(\\set{\\abs{f} \u003e s})$ be the [[distribution function|HQVXKEMN5N]] of $f$.\nThe *decreasing rearrangement* of $f$ is the function $f^* \\colon [0, \\infty) \\to [0, \\infty]$ given by the [[infimum|04KFTA34C5]]\n$$f^*(t) = \\inf \\set{s \\ge 0 : d_f(s) \\le t},$$\nwith the convention $\\inf \\varnothing = \\infty$.\n\n## Remarks\n\nExample: let $X = [0, 1]$ with Lebesgue measure and $f(x) = x$.\nThen $d_f(s) = 1 - s$ for $0 \\le s \\le 1$ and $d_f(s) = 0$ for $s \\ge 1$.\nFor $0 \\le t \u003c 1$ the condition $d_f(s) \\le t$ means $s \\ge 1 - t$, so $f^*(t) = 1 - t$.\nFor $t \\ge 1$ every $s \\ge 0$ works, so $f^*(t) = 0$.\n\nNon-example: the function $f(x) = x$ on $[0, 1]$ is not its own decreasing rearrangement, since it is increasing.\nA function $g \\ge 0$ on $[0, \\infty)$ is its own rearrangement exactly when it is non-increasing and right-continuous.\n\nThe function $f^*$ is non-increasing and right-continuous.\nIt has the same distribution function as $\\abs{f}$ for Lebesgue measure on $[0, \\infty)$.\nHence $\\int_0^\\infty (f^*)^p \\, dt = \\int_X \\abs{f}^p \\, d\\mu$ for $0 \u003c p \u003c \\infty$.\nThe *symmetric decreasing rearrangement* of a function on $\\R^n$ is a related notion: it is radial, decreasing in $\\abs{x}$, and equimeasurable with $\\abs{f}$.\n",
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