Given a set $E$ which is bounded, measurable and $m^{\ast}(E)=x >0$, then for each $y \in (0,x)$ we can find a measurable set $A \subset E$, such that $m^{\ast}(A)=y$. To see this one considers measure as a continuous function and applies the Intermediate Value property.
Now we extend the question in the following manner: Suppose $E$ is a set which has finite measure, then for each positive $x < m^{\ast}(E)$ prove that there exists a perfect set $A \subset E$, such that $m^{\ast}(A)=x$.