Measure of the Cantor set plus the Cantor set
Here is the full question from Halmos's Problems for mathematicians, young and old:
Here's the hint:
And here's the solution:
If you are asking the case where $C$ is the Cantor ternary set, then you can show that $C+ C$ is actually $[0,2]$.
For more general Cantor sets, you can find a description in the paper: On the topological structure of the arithmetic sum of two Cantor sets, P Mendes and F Oliveira, available at http://iopscience.iop.org/0951-7715/7/2/002 .