Subject: Cantor's triadic set
Date: Fri, 13 Mar 1998 19:31:40 +0100
From: Abdelilah Elanbari
First I hope my question will be understandable since I don't know the english translation of what is known in french as 'Ensemble triadique de Cantor'.
The problem is about measures. It's known and easy to demonstrate, that within R, a set a infinite but countable points has a null measure. But the inverse is not true. In that a set of infinite uncountable points my have a null measure.
That's what Cantor proved by its 'triadic set'. This set is defined as below:
- take the range [0,1] (whose mesure is one) and divide it in three equal parts.
- Remove the central part: ]1/3,2/3[. The set is now [1,1/3] U [2/3,1] and its measure is 2/3
- Repeat the steps above for each rangeand so on...
At step n the measure is (2/3)n. As n grows the measure tends to 0
So the mesaure of the limit set is 0. What I forgot and I cannot remmember is how to demonstrates that this limit set is uncountable... Any help?
Thanks a lot. Abdelilah