$(X,\Sigma,\mu)$ with $\{\mu(A):A\in\Sigma\}=[0,1]\cup[2,3]$

Question

Are there any measure space $(X,\Sigma,\mu)$ with $\{\mu(A):A\in\Sigma\}=[0,1]\cup[2,3]$?

Answer 1

Yes, consider $[0,1]\cup\{2\}$ where we put Lebesgue measure on $[0,1]$ and measure $2$ on $\{2\}$.

Answer 2

Yes.

Let $X=[0,1]$ and let $\Sigma$ denote the collections of Borel-subsets of $X$.

Let $\mu=\lambda+2\delta_0$ where $\lambda$ is the Lebesgue measure and $\delta_0$ is prescribed by $B\mapsto\mathsf1_B(0)$.

Then $\{\mu(A)\mid A\in\Sigma\wedge 0\notin A\}=[0,1]$ and $\{\mu(A)\mid A\in\Sigma\wedge 0\in A\}=[2,3]$.