So, this should be a little obvious, but not to me at the moment.
Why is $m(B = E - A) = m(E) - m(A)$?
Of course, all given sets are measurable.
Thanks.
So, this should be a little obvious, but not to me at the moment.
Why is $m(B = E - A) = m(E) - m(A)$?
Of course, all given sets are measurable.
Thanks.
This is false. For example, let $m$ be counting measure on (the power set of) $\{1,2\}$, let $B=E=\{1\}$ and let $A=\{2\}$.
You can add extra hypotheses which make the conclusion true. For example, if $A\subseteq E$ and $m(A)<\infty$ then $m(E-A)=m(E)-m(A)$. Indeed, since $A\subseteq E$ the (measurable) sets $E-A$ and $A$ are disjoint and have union $E$, so $m(E-A)+m(A)=m(E)$, and since $m(A)<\infty$ we can subtract it to get $m(E-A)=m(E)-m(A)$.
In general $m(B-A) \neq m(B)-m(A)$. If $B$ and $A$ are disjoint and $m(B)=m(A)>0$, then $m(B)=m(B-A)>0$ yet $m(B)-m(A)=0$.