Let $f$ be a nonnegative measurable function and $E\in \mathcal{M}$, where $\mathcal{M}$ is the sigma algebra of Lebesgue measurable sets of $\mathbb{R}$. Define $Y(E,f)=\{\int_E \phi dm: 0\le \phi \le f, \phi \; \text{is simple}\}$. Show that the set $Y(E,f)$ is always of the form $[0,x]$ or $[0,x)$, where the value $x=\infty$ is allowed.
In the text it is just said that "clearly" $Y(E,f)$ is always of the form $[0,x]$ or $[0,x)$, however, I cannot think of a way to show this apparently obvious fact. How can I show this? I would greatly appreciate any help.
Note that if $0\le\phi\le f$, then $0\le c\phi\le f$ for any $c\in[0,1]$. Since $\int_E c\phi\,dm=c\int_E\phi\,dm$ you obtain all values between $0$ and $\int_E\phi\,dm$ for each $\phi$, and thus all values between $0$ and $Y(E,f)$.