The Lebesgue Measure of the Cantor Set Why is it that m(C+C) = 2, where C is the Cantor set, but C is itself measure zero?  What is happening that makes this change occur?  Is it true generally that m(A+B) =/= m(A) + m(B)?  Are there cases in which this necessarily holds?  I should note that this is all within the usual topology on R.  Thanks!

Yes, in general $m(A+B) \ne m(A)+m(B)$.  Equality does hold when $m(A+B)=0$, but for trivial reasons.
