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  "body": "\n## Definition\n\nLet $n \\ge 1$ and $0 \\le m \\le n$.\nFor $E \\subseteq \\R^n$, let $\\mathcal{H}^m(E)$ be its $m$-dimensional [[Hausdorff measure|Q1TGZK8RN6]].\nA set $E \\subseteq \\R^n$ is *countably $m$-rectifiable* if there are [[Lipschitz maps|0QNVQ4VD9T]] $f_j \\colon \\R^m \\to \\R^n$, $j = 1, 2, \\dots$, with $\\mathcal{H}^m\\left( E \\setminus \\bigcup_j f_j(\\R^m) \\right) = 0$.\n*Geometric measure theory* is the branch of analysis that studies:\n\n1. the measure-theoretic structure of subsets of $\\R^n$ and of Riemannian manifolds, measured by Hausdorff measures, in particular the [[rectifiable sets|7N0GFMW05R]] and their approximate tangent planes;\n2. generalised surfaces built from them, such as [[currents|360H7MD9RD]] and [[varifolds|N3QA80M44M]];\n3. variational problems for these objects, such as the minimisation of area, with the existence and regularity of their solutions.\n\n## Remarks\n\nThe rectifiable sets are the sets on which the tools of calculus, such as tangent planes and the area formula, still work almost everywhere.\n\nExample: the graph $E = \\set{(x, \\abs{x}) : x \\in [-1, 1]}$ in $\\R^2$ is $1$-rectifiable, since it is the image of a Lipschitz map of an interval, and $\\mathcal{H}^1(E) = 2\\sqrt{2}$.\nIts tangent line exists at every point except the corner $(0, 0)$.\n\nNon-example: the four-corner Cantor set in $\\R^2$, obtained by keeping repeatedly the four corner squares of side $1/4$ of each square, has $0 \u003c \\mathcal{H}^1 \u003c \\infty$.\nIt is purely $1$-unrectifiable: it meets every Lipschitz image of $\\R$ in a set of $\\mathcal{H}^1$-measure $0$.\n\n## Explanations\n\n### L1 human\n\nGeometric measure theory measures the size of rough sets and builds generalised surfaces from them, so that problems such as finding a least-area surface always have solutions.\n",
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