Thread: lebesgue outermeasure

Hi,
I have this problem that i don't know how to start with:
If E is a set on the real line with lebesgue outermeasure zero then its complement is dense in R
Thank you

Try the contrapositive: if the complement of E is NOT dense in R, then what can you say about E?

if it's not dense then its closure is a subset of R and from there we prove that E has a measure different than zero?

If the complement of E is not dense, then E contains an open set.

ok so if i go roughly like this would it be true:
Suppose E^c is not dense then there exist an interval I in R such that I is contained in E^c.
the lebesque outermeasure is denoterd by ð*. so ð*(E)=ð*(EI)+ð*(EI^c)= 0 + ð*(I^c) which is differenty than zero since I is an interval.