I need to find a set $X$, a $\sigma$-algebra $\mathcal{S}$ on $X$ and a measure $\mu$ on $(X, \mathcal{S})$ such that
$$ \{\mu(E): E \in \mathcal{S}\} = \{\infty\}\cup \bigcup_{n=0}^{\infty} \left[3n, 3n+1\right]. $$
I have no idea how to start this. Any tips are appreciated.
Let $X=[0,\infty)$ and let $\mathcal S$ denote the collection of Borel subsets of $[0,\infty)$.
Define $\rho=3\sum_{n=1}^{\infty}\delta_n$ where $\delta_n(B)=\mathsf1_B(n)$ and let $\nu$ be prescribed by $B\mapsto\lambda([0,1]\cap B)$ where $\lambda$ denotes the Lebesgue measure.
Then $\mu:=\rho+\nu$ does the job.
We have $\mu(X)=\infty$ and if $B\in\mathcal S$ with $\mu(B)<\infty$ then $\rho(B)$ is an nonnegative integer with $3\mid \rho(B)$.
Next to that it is evident that $\{\nu(B)\mid B\in\mathcal S\}=[0,1]$.