How to construct a set with cardinality of continuum and a Lebesgue measure of zero?

For instance, a set of all rational numbers within (0,1) is a countable set with Lebesgue measure of zero. What about continuum?

  • 4
    $\begingroup$ Are you familiar with the Cantor set? $\endgroup$ Commented Apr 7, 2017 at 18:16
  • $\begingroup$ @Noah Schweber - that answers my question. Thanks. $\endgroup$
    – Stepan
    Commented Apr 7, 2017 at 18:17

1 Answer 1


Cantor set enter image description hereis indeed a simple example.

It has cardinality of continuum, because it contains all numbers that don't have "1" in ternary numeral system. If we replace "2" with "1" and switch to binary representation, we cover (0,1) except for a countable number of (cut) points. Hence, it has cardinality of continuum - countable = continuum.

Lebesgue measure $λ(E) = \lim_{n \to \infty} (\frac{2}{3})^n = 0$. Thus, Lebesgue measure of Cantor set is zero.


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