Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How can one show that the Lebesgue measure is not complete on $(\mathcal{R}, \mathbf{B}(\mathcal{R}))$?

share|cite|improve this question

$\mathcal B(\mathbb R)$ has the cardinality of the continuum. Cantor set too and moreover it has measure $0$. Hence there's at least a subset of Cantor set which is not measurable.

share|cite|improve this answer

How: You would have to give an example of a subset of measure zero with a subset which is not measurable.

share|cite|improve this answer

Construct a subset $C$ of measure zero, and $|C|=|R|$, notice that the set of Borel sets has the same cardinality as $R$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.