# Measure of a non-empty subset is greater than 0

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.

• 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). – André Nicolas Feb 18 '16 at 3:02
• 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. – peter a g Feb 18 '16 at 3:03