$X:= \Bbb Q$ and $\mathcal S:=\mathcal I^1 | \Bbb Q := \{]a,b] \cap \Bbb Q | a,b \in \Bbb R , a\le b\} $ and define the content $$\mu : \mathcal S \to [0,\infty[, \:\:\: \mu(]a,b] \cap \Bbb Q ) := b-a \:\:\:\:\:\:\:(a \le b)$$
I've already shown that $\mathcal S$ is a semiring and that $\mu$ is a finite content, but now I have to show that $\mu$ is NOT a premeasure, i.e. i have to find a sequence of disjoint sets $A_i \in \mathcal S, \: \: \:\bigcup_{i=1}^\infty A_i \in \mathcal S$ for which the following is true: $$\mu \Big(\bigcup_{i=1}^\infty A_i \Big ) \not = \sum_{i=1}^\infty \mu(A_i)$$
Normally contetns aren't premeasures because of some set that would deliver infinity somewhere, but I can't really see where I can show this for $\mu$.
Any ideas or tipps? Thanks in advance!