I want to prove that $[0,1]$ does not have measure zero but the book says $“$Explain why the following observation is not a solution to the problem: Every open interval that contains $[a,b]$ has length $> b-a",$ which was exactly identical to my initial reasoning.
Why is this bad reasoning?
Best Answer
To see why this reasoning doesn't work, this following statement is true but does not imply that the set is not of measure 0.
Every open interval that contains all the rationals in $[0,1]$, or $\mathbb{Q} \cap [0,1]$, is of length $> 1 - \epsilon$, but the rationals are of measure 0.