How do I show this? I am having a hard time grasping this. I have read that this is the 1-dimensional case of Brunn-Minkowski Inequality but I have no idea how to prove this:
m(A + B) $\geq$ m(A $\cup$ B)
where A and B are disjoint subsets of $\mathbb{R}$.
I'm thinking I could use sub additivity and/or the fact that m(x+A) = m(A), x $\in$ $\mathbb{R}$ but I do not know how to to start.
m(A) is the outer measure of A: inf{$\sum$I: I is an open interval and A $\subseteq \cup I$}