rectifiable

Rectifiable is an adjective used in geometry and geometric measure theory to describe objects that have finite length or, more generally, finite m-dimensional measure in a controlled, decomposable way.

Rectifiable curves. A curve γ: [a,b] → R^n is rectifiable if its length L(γ) is finite, where L(γ) = sup { ∑ |γ(t_i) − γ(t_{i−1})| } over all partitions a=t_0<...<t_m=b. If γ is rectifiable, it has a well-defined length equal to L(γ); it is of bounded variation and, on many curves, admits an arc-length parametrization with |dγ/ds| = 1 almost everywhere. If γ is C^1, then L(γ) = ∫_a^b |γ′(t)| dt.

Rectifiable sets. In geometric measure theory, a subset E ⊂ R^n is m-rectifiable if there exist Lipschitz maps f_k from subsets A_k ⊂ R^m into R^n such that E is covered, up to a set of m-dimensional Hausdorff measure zero, by the union ∪_k f_k(A_k). Often one also requires H^m(E) < ∞. Equivalently, E is contained in a countable union of Lipschitz images of subsets of R^m up to H^m-null sets. In particular, 1-rectifiable sets are (up to null sets) unions of Lipschitz images of curves; smooth submanifolds are rectifiable, while certain pathological sets can be purely unrectifiable.

Notes. Rectifiability generalizes the notion of finite length for curves to higher-dimensional measure theory and provides a framework for distinguishing “well-behaved” geometric sets from fractal or highly irregular ones.