I met with a multiple choice question,
Q. Let $E\subset M \subset \Bbb R^n$, where $M$ is a measurable (Lebesgue) set with $m(M)<\infty, $ then $E$ is measurable if
($m^*$ denotes the Lebesgue outer measure)
$m(M)=m^*(E)+m^*(M \triangle E)$, $\triangle$ denotes the symmetric difference.
Caratheodory criterion says '$E$ is measurable if and only if $m^*(A)=m^*(A \cap E)+m^*(A \cap E^c)$ for all $A \subseteq \Bbb R^n$'. Thus, here I can see the measurability of $E$ implies the statements of First, Third and Fourth options (which are exactly the same). But how can we make a necessary and sufficient statement using the condition 'being a subset of a measurable set $M$'?