---
tag: 7N0GFMW05R
kind: concept
name: Rectifiable set
msc: [28A75, 49Q15]
sources:
  - url: https://en.wikipedia.org/wiki/Rectifiable_set
same_as:
  wikidata: Q3038340
  wikipedia: Rectifiable set
---

## Summary

An $m$-rectifiable set is a subset of $\R^n$ covered by countably many Lipschitz images of $\R^m$.
The cover may miss a set of $m$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] zero.
Such sets are the measure-theoretic version of $m$-dimensional surfaces, and they may have corners, cusps and countably many pieces.
They are the basic objects of geometric measure theory, where they replace smooth submanifolds in problems such as the Plateau problem.
Every set of finite $m$-dimensional Hausdorff measure splits into a rectifiable part and a purely unrectifiable part, and the two parts behave very differently.

## Explanations

### L1 human

Think of a set that is built from countably many bent and crumpled copies of a flat $m$-dimensional piece.
It may look rough, but at almost every point it has an approximate tangent plane, much as a smooth surface does.
In the plane, a polygon, a circle, and a countable union of line segments are all $1$-rectifiable.
A filled square is not $1$-rectifiable, because countably many curves cover only a set of area zero.
Some fractal dust sets of dimension one, such as the four-corner Cantor set, are not rectifiable at all: they meet every curve in a set of length zero.