Cardinality of the continuum and Lebesgue measure of zero

Question

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?

Are you familiar with the Cantor set? – Noah Schweber Apr 07 '17 at 18:16 — Noah Schweber, Apr 07 '17 at 18:16
@Noah Schweber - that answers my question. Thanks. – Stepan Apr 07 '17 at 18:17 — Stepan, Apr 07 '17 at 18:17

Stepan · Accepted Answer · 2017-04-08T17:06:32.110

Cantor set is 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.

Cardinality of the continuum and Lebesgue measure of zero

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