1.$A,\ B\subseteq R$ and $m^*(A)<\infty$, show that $ m^*(B\cap A^c)\geq m^*(B)-m^*(A)$; here $m^*$ is the outer measure.
My work- we can write,
$m^*(B)\leq m^*((B\cap A^c)\cup A)\leq m^*(B\cap A^c)+m^*(A)$, by sigma additivity of outer measure.
then, $m^*(B)\leq m^*(B\cap A^c)+m^*(A)$
since, $m^*(A)$ is finite, we can substract it from both side. hence, $m^*(B)-m^*(A)\leq m^*(B\cup A^c)$
Is this correct proof?
2.In the second part, I'm asked to prove $m^*((a,b)\cup(c,d))=m^*(a,b)+m^*(c,d)$ where (a,b) and (c,d) are disjoint intervals. Should I use the above results for the proof?
Since outer measure is not additive , I don't think we can use additivity here.