Consider a closed and bounded set $ F $ in the open interval $ (-n,n) $ of $ \mathbb{R} $. Then in the usual topology with Lebesgue measure $ \mu(F) = 2n - \mu((-n,n) \setminus F) $. The Cantor set satisfies this property so it can be our $ F $. That is, $ C $ is in $ [0,1] $ and can also be said to be in $ [-1,1] $. $ C $ is bounded because it has a supremum and infimum, in this case $ 1 $ and $ -1 $.
Write $ (-1,1) \setminus C = (-1,0) \cup [0,1] \setminus C $.
So $ \mu((-1,1) \setminus C) = 1 + \mu([0,1] \setminus C) $, and this apparently implies $ C $ has measure zero.
This proof was given to me and I am not sure where the errors are if there are any. Would anyone be able to explain or correct this proof?