Varifold

Remarks

Example: 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 $x \in M$. Let $\mathcal{H}^m$ be the $m$-dimensional Hausdorff measure, which is surface area on $M$. Set $V(B) = \mathcal{H}^m(\set{x \in M : (x, T_x M) \in B})$ for Borel sets $B \subseteq U \times G(n, m)$. This 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 $M$. More generally, a rectifiable set with a positive density function gives a rectifiable varifold. Non-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.