 Given an open subset formula_1 of Euclidean space formula_2, an "m"-dimensional varifold on formula_1 is defined as a Radon measure on the set

where formula_5 is the Grassmannian of all "m"-dimensional linear subspaces of an "n"-dimensional vector space.The Grassmannian is used to allow the construction of analogs to differential forms as duals to vector fields in the approximate tangent space of the set formula_1.The particular case of a rectifiable varifold is the data of a "m"-rectifiable set "M" (which is measurable with respect to the "m"-dimensional Hausdorff measure), and a density function defined on "M", which is a positive function θ measurable and locally integrable with respect to the "m"-dimensional Hausdorff measure.