Measure of the Cantor set plus the Cantor set

The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets:

If $C$ is the cantor set, then what is the measure of $C+C$?

Solution 1:

Here is the full question from Halmos's Problems for mathematicians, young and old:

Here's the hint:

And here's the solution:

Solution 2:

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 .