I have to prove that if $E$ is a set such that $\mu (E)=1$, then, there is a subset $E_t\subset E$ such that $\mu(E_t)=t$ for every $0\leq t\leq 1$
I think, that given a measurable set $E$ there is a Borel Set $F$ such that $\mu(F)=\mu(E)$ and then $F$ is union of disjoint intervals...so that would be enough, but I am not sure about the last affirmation. Is it true that every Borel set a union of disjoint intervals?
(Here $\mu$ is the Lebesgue measure)
Let $f(x) = \int_{(-\infty, x]} 1_E$. The function $f$ is continuous, $\lim_{x \to -\infty} = 0$ and $\lim_{x \to +\infty} = 1$. The intermediate value theorem does the rest.