Cantor Set

From Optimal Transport Wiki
Jump to navigation Jump to search

Cantor Ternary Set

A Cantor ternary set of base-3 can be constructed from through the iterative process of removing the open middle third from each closed interval. Specifically, starting from a closed interval , one can first remove the open middle third, , to get the remaining union of closed intervals . Next, one repeat the process of removing open middle thirds from each closed interval, ie . Each is then constructed iteratively by removing the middle one third from each closed intervals of . The Cantor set is then obtained when one repeats the process infinitely many times, or equivalently:[1][2]

Properties of Cantor Sets

A Cantor set has the following properties.[1]

  • is closed, compact, nowhere dense, and totally disconnected. Moreover, has no isolated points.
  • is Lebesgue measurable, with Lebesgue measure .
  • Cantor set is in bijection with , giving us a counterexample of a noncountable set having zero measure.

Cantor Function

The Cantor set can be used to define Cantor Function, an increasing function which is continuous but has zero derivative almost everywhere.


  1. 1.0 1.1 Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §1.5
  2. Craig, Katy. MATH 201A HW 5. UC Santa Barbara, Fall 2020.