There is a theorem in our textbook that says:
Suppose that $E$ is measurable. "For each $\epsilon > 0$ , there is a closed set $F$ contained in $E$ for which $m*(E~F) \lt \epsilon$."
Since E is measurable, we know that,
$m^*(F) = m^*(F \cap E) + m^*(F \cap E^c) \rightarrow m^*(F \cap E^c) = m^*(F) - m^*(F \cap E)$ But since F is contained in E, $m^*(F) = m^*(F \cap E)$ So isn't $m^*(E \sim F)=m^*(F \cap E^c) = 0$? Then why do we need to consider epsilon?
Thanks in advance