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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]] and let $0 \\le m \\le n$.\nLet $G(n, m)$ be the Grassmannian of $m$-dimensional linear subspaces of $\\R^n$, with its usual compact metric topology.\nAn *$m$-dimensional varifold* in $U$ is a [[Radon measure|HZNEQQBCW4]] $V$ on the locally compact space $U \\times G(n, m)$.\nThe *weight* of $V$ is the Radon measure $\\norm{V}$ on $U$ given by $\\norm{V}(A) = V(A \\times G(n, m))$.\n\n## Remarks\n\nExample: let $M \\subseteq U$ be an $m$-dimensional $C^1$ submanifold that is a closed subset of $U$, with tangent plane $T_x M$ at $x \\in M$.\nLet $\\mathcal{H}^m$ be the $m$-dimensional [[Hausdorff measure|Q1TGZK8RN6]], which is surface area on $M$.\nSet $V(B) = \\mathcal{H}^m(\\set{x \\in M : (x, T_x M) \\in B})$ for Borel sets $B \\subseteq U \\times G(n, m)$.\nThis is a Borel measure that is finite on compact sets, since $M$ has locally finite area, so $V$ is a varifold; its weight is $\\mathcal{H}^m$ restricted to $M$.\nMore generally, a [[rectifiable set|7N0GFMW05R]] with a positive density function gives a *rectifiable varifold*.\nNon-example: counting measure on a countable dense subset of $U \\times G(n, m)$ is not finite on compact sets with interior, so it is not a Radon measure and not a varifold.\n",
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