The question I have is the following:
Let $X: [0, \infty]\rightarrow [0, \infty]$ be a continuous function such that $X(0) = 0$. We define the "$X$-outer measure" $\mu^*$ to be $\mu^*(A) =$ inf$\{\displaystyle\sum\limits_{n=1}^\infty X(|I_n|): E\subset \cup I_n\}$ for any set $A\subset \mathbb{R}$, where the infimum is over coverings of A by countably many intervals $(I_n)_{n=1}^\infty$. Prove that $\mu^*(E) = 0$ whenever $E$ is countable.
How would I go about proving this? Any help would be greatly appreciated!