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!

When $m(A)=m(B)=0$ and $A$ is countable then $m(A+B)=0$. This is so because $A+B=\bigcup _{a\in A}a+B$ is a countable union, and $m$ is translation invariant so $m(a+B)=m(B)=0$, so countable (sub)additivity gives that $m(A+B)=0$. 
However, when both $A$ and $B$ are uncountable, this argument fails, as then $A+B=\bigcup _{a\in A}a+B$ is still true but this is not a countable union. The Cantor set illustrates that the measure of $A+B$ can then indeed increase.
