Question

Construct a set K similar to the Cantor set where in the n-th step we remove the middle interval of size 1/5n of all the intervals in the previous step. We have K₀​=[0,1],K₁​=[0,0.4]∪[0.6,1],K2​=[0,0.23]∪[0.27,0.4]∪[0.6,0.73]∪[0.77,1], and so on. Convince yourself that it has the same properties as the Cantor set we saw in class, except for the measure. Compute the measure of this set K.

Answer 1

The set K is similar to the Cantor set, with the same properties except for the measure, which is zero.

Step-by-step explanation:

The set K you have described is similar to the Cantor set. It is constructed by removing the middle interval of size 1/5n from each interval in the previous step. The Cantor set has several properties like being closed, perfect, totally disconnected, and uncountable. However, its measure is zero. Similarly, the measure of set K is also zero because, at each step, the removed intervals become smaller and the total measure decreases.