Hello I hope you are doing great.
I would like to know your opinion regarding this reasoning.
Assume you that you put a point in the $n$-th iteration of the ternary Cantor set. This set is formed by the union of disjoint intervals. Lets give to each interval the same probability of the point to be in it. So the probability of the point to be in any of the intervals is $\frac{1}{2^n}$. Clearly the probability of the point to be in the $n$-th iteration is $1$. We continue with this procedure, forcing that the point is indeed in the corresponding iteration of the Cantor set. If we consider this to the limit set, the Cantor set. The probability of the ball to be in a set (which is a point) is zero, but since we required that the point should be in the Cantor set, the probability of being in it is $1$. Reaching a contradiction.
I know my reasoning is false, but I'm not completely sure why.