According with the book Lages Lima - Análise real, a subset $X \subset \mathbb{R}$ has zero measure whenever for every $\epsilon>0$ there is a countable cover made up of open intervals, such that the sum of the lenght of all those intervals is less than $\epsilon$.
My question is : why do we require the cover to be countable?
Because the sum of uncountably many positive numbers is necessarily infinite. See:
Can we add an uncountable number of positive elements, and can this sum be finite?