How is this $\mu_0$ not a premeasure?

Solution 1:

Let $(\varepsilon_n)_{n\in\mathbb{N}}$ be a sequence of rational numbers with $\sum_{n=1}^\infty\epsilon_n<1$, and let $(q_n)_{n\in\mathbb{N}}$ be an enumeration of $(0,1]\cap\mathbb{Q}$.

Define intervals $I_n$ for $n=1,2,\ldots$ as follows: If $q_n\in\bigcup_{k=1}^{n-1}I_n$, lelt $I_n=\emptyset$. Otherwise, let $I_n=(p_n,q_n]$ where $p_n$ is the largest of the numbers $0$, $q_n-\varepsilon_n$, $q_1$, … $q_{n-1}$ less than $q_n$.

Now $(0,1]\cap\mathbb{Q}$ is the disjoint union of the rational intervals $I_n\cap\mathbb{Q}$, and the sum of the lengths of these intervals is less than $1$.