I am trying to prove that if U is a non-empty subset of the real line, then the measure of U denoted m(U) > 0.

If this said "open set" then I could prove it because every open set contains at least one open interval and open intervals have a positive measure. I'm not sure how to proceed with this one since is just say "non-empty subset" of the real line.

  • 2
    $\begingroup$ The (Lebesgue) measure of a one-element set is $0$. And there are many non-empty subsets of the line with measure $0$, some very "large" (the same cardinality as the full real line). $\endgroup$ – André Nicolas Feb 18 '16 at 3:02
  • 1
    $\begingroup$ To justify Andre's statement: Take a point. Its measure has to be less than that of any open interval containing it, so its measure must be zero. $\endgroup$ – peter a g Feb 18 '16 at 3:03

What if U is a countable set of points? Then it's a nonempty subset of the real line that has measure 0.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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