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  "body": "\n## Definition\n\nLet $n$ and $m$ be integers with $1 \\le m \\le n$, and let $\\mathcal{H}^m$ be the $m$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] on $\\R^n$.\nA set $E \\subseteq \\R^n$ is *$m$-rectifiable* if there are [[Lipschitz maps|0TMZ21WKCN]] $f_i \\colon \\R^m \\to \\R^n$, for $i = 1, 2, \\dots$, such that\n$$\\mathcal{H}^m\\Big(E \\setminus \\bigcup_{i=1}^{\\infty} f_i(\\R^m)\\Big) = 0.$$\nA set $E \\subseteq \\R^n$ is *purely $m$-unrectifiable* if $\\mathcal{H}^m(E \\cap F) = 0$ for every $m$-rectifiable set $F \\subseteq \\R^n$.\n\n## Remarks\n\nThis is the countable rectifiability of Federer and Mattila.\nSome authors also require $\\mathcal{H}^m(E) \u003c \\infty$, or use $C^1$ maps instead of Lipschitz maps; the second change gives the same class of sets.\nExample: the unit circle $S \\subseteq \\R^2$ is $1$-rectifiable, since the Lipschitz map $f(t) = (\\cos t, \\sin t)$ satisfies $f(\\R) = S$.\nNon-example: the unit square $Q = [0,1]^2$ is not $1$-rectifiable.\nEach set $f_i(\\R)$ has $\\sigma$-finite $\\mathcal{H}^1$ measure, so it has Lebesgue measure zero in $\\R^2$.\nHence $Q \\setminus \\bigcup_i f_i(\\R)$ has positive area, and so its $\\mathcal{H}^1$ measure is infinite, not zero.\nThe image of a [[rectifiable curve|0HATW2VWAZ]] is $1$-rectifiable.\n",
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