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  "body": "\n## Summary\n\nThe $s$-dimensional Hausdorff measure of a subset of a [[metric space|0VKD1TP7F3]] measures its $s$-dimensional size, for any real $s \\ge 0$.\nIt is built from covers by small sets, each counted by its [[diameter|0KW3PN547K]] to the power $s$.\nFor whole numbers $s = n$ it extends length, area and volume to rough sets, and in $\\R^n$ it agrees with [[Lebesgue measure|2Z2QVWZ262]] up to a constant.\nIt defines Hausdorff dimension and is the basic measure of fractal geometry.\n\n## Explanations\n\n### L1 human\n\nTo measure the length of a curve, cover it by many tiny pieces and add up their sizes.\nFor area one adds the squares of the sizes, and for volume the cubes.\nHausdorff measure allows any power $s$, also non-integer powers such as $0.63$.\nFor a given set, there is one critical power: below it the measure is infinite, above it the measure is zero.\nFor example, a line segment of length $1$ has $1$-dimensional measure $1$ and $2$-dimensional measure $0$.\n",
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