TITLE: Measure of the Cantor set plus the Cantor set
QUESTION [14 upvotes]: 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 $C+C$?

REPLY [31 votes]: Here is the full question from Halmos's Problems for mathematicians, young and old:

Here's the hint:

And here's the solution: