From nowhere dense perfect set to zero measure set.

Question

I know that Cantor set is nowhere dense and perfect. But if I have a nowhere dense perfect set, can I call it a Cantor set?

Also, I already proved that a certain subset of the real line is a nowhere dense perfect set. Knowing this, what else should I consider if I want to prove that its measure is zero?

Answer 1

Every nonempty totally disconnected perfect compact metric space is homeomorphic to the Cantor set (Wikipedia). On the real line, having empty interior implies being totally disconnected: indeed, every connected set is an interval, hence has nonempty interior.

For the second question: Cantor-type sets can have positive measure. To prove that the measure is zero, you need to find a way to cover it by intervals of total length $<\epsilon$, for any $\epsilon>0$.