Rectifiable set

In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set. As such, it has many of the desirable properties of smooth manifolds, including tangent spaces that are defined almost everywhere. Rectifiable sets are the underlying object of study in geometric measure theory.A subset E {displaystyle E}   of Euclidean space R n {displaystyle mathbb {R} ^{n}}   is said to be m {displaystyle m}  -rectifiable set if there exist a countable collection { f i } {displaystyle {f_{i}}}   of continuously differentiable maps