Okay. So I know that a set $E$ is called measurable if for any set $A$, we have $$ m^\ast(A)=m^\ast(A\cap E)+m^\ast(A\cap E^c).$$ Recently, I came across a Lemma which says that
a set $A\subset E$ is called measurable if and only if $$m^\ast(A) + m^\ast(E\setminus A)=b-a$$ where $a$ and $b$ are the end points of the interval $E$.
I would like to know how to proof the above lemma. I have been trying for some time now with no results.
Thanks.
Added: Following Sid's answer, I make some additions.
since $A\subset E$, $A\cap E=A$. so $m^\ast(A\cap E)=m^\ast(A)$. Also, $E\backslash A=E\cap A^c$. Thus $m^\ast(A) + m^\ast(E\setminus A)=b-a=m^\ast(E)$ implies that $$m^\ast(A\cap E) + m^\ast(E\cap A^c)=m^\ast(E).$$ Hence $A$ is measurable.
Conversely, suppose $A$ is measurable, then we have $$m^\ast(A\cap E) + m^\ast(E\cap A^c)=m^\ast(E).$$
since $A\subset E$, $A\cap E=A$ and $E\backslash A=E\cap A^c$, the result follows.