For a presentation, I am learning about the Cantor Set and how it is homeomorphic to the p-adic numbers. I was reading section two of this paper. In it it states that the Cantor Set has a vanishing Lebesgue measure.
Wikipedia says: Given a subset ${\displaystyle E\subseteq \mathbb {R} } $, with the length of interval ${\displaystyle I=[a,b]({\text{or }}I=(a,b))} $ given by ${\displaystyle \ell (I)=b-a} $, the Lebesgue outer measure ${\displaystyle \lambda ^{*}(E)} $ is defined as
$${\displaystyle \lambda ^{*}(E)=\operatorname {inf} \left\{\sum _{k=1}^{\infty }\ell (I_{k}):{(I_{k})_{k\in \mathbb {N} }}{\text{ is a sequence of open intervals with }}E\subseteq \bigcup _{k=1}^{\infty }I_{k}\right\}} $$.
Why would this be zero for the Cantor Set?
Show that the complement of the Cantor set in $[0,1]$ has measure 1. You will also have to use that the Cantor set is measurable, hence $\lambda^*([0,1]) = \lambda^*(E\cap[0,1]) + \lambda^*(E^c\cap[0,1])$, in which $E$ is the Cantor set.
What is your background on Measure Theory?