# 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 $C+C$?

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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 .

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George S: Excellent article man! Thanks +1 –  anonymous Aug 14 '10 at 9:16
@Chandru: No problem. You are welcome. :) –  user1119 Aug 14 '10 at 9:20