$\{µ(E) : E ∈ S\} = [0, 1] ∪ [3, c]$. Prove that $c ≥ 4$. Can you give an example of $(X, S, µ)$ if $c = 4$?

Question

Suppose that there exists a measure space $(X, S, µ)$ with $\{µ(E) : E ∈ S\} = [0, 1] ∪ [3, c]$. Prove that $c ≥ 4$. Can you give an example of $(X, S, µ)$ if $c = 4$?

My Try: Clearly $\mu(X)=c$ Now suppose we have $A,B\in S$ s.t $\mu(A)=3$ and if $\mu(B)=1$. If $\mu(A \cap B)=0$. We are done. Otherwise we have $\mu(A \cap B)\neq0$ then $0<\mu(A \cap B) \leq \mu(B)=1$ then $2\leq \mu(A\setminus (A\cap B))=\mu(A)-\mu(A \cap B)<3$.

What about the example?

Answer 1

Your proof is correct. Example: let $P$ be a probability measure on $(-\infty ,0)$ with a density and $\mu =P+3\delta_1$. Note that $\{\int_A f(x)\, dx:A \subset (0,\infty)$ Borel $\}$ is $[0,1]$ by IVP. Hence the range of $\mu$ is $[0,1] \cup [3,4]$. [Split $\mu (A)$ as $\mu (A\cap (-\infty,0))+\mu (A\cap (0,\infty))$. The scond term is either $0$ or $3$].