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  "body": "\n## Summary\n\nThe distribution function of a measurable function $f$ records, for each level $t \\ge 0$, the measure of the set where $\\abs{f}$ is larger than $t$.\nIt forgets where the large values of $f$ sit and keeps only how much room they take.\nMany quantities, such as the $L^p$ norms, depend on $f$ only through its distribution function.\nIt is the starting point for the layer cake formula, rearrangements, weak $L^p$ spaces and interpolation.\n\n## Explanations\n\n### L1 human\n\nThink of the graph of $\\abs{f}$ as a mountain range and of $t$ as the height of a flood.\nThe distribution function at $t$ is the total length, area or volume of the land that stays above the water.\nAs the water rises, this amount can only shrink.\nFor example, for $f(x) = x$ on $[0, 1]$ with length as the measure, the set where $f \u003e t$ is $(t, 1]$, so the distribution function is $1 - t$ for $0 \\le t \\le 1$ and $0$ for $t \\ge 1$.\n",
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