Can anyone help me with the following question? Thank you very much in advance :)
Let $u^*$ be a Borel measure on ${\mathbb R}^{n}.$ Show that $u^{*}(A \cup B) = u^{*}(A) + u^{*}(B)$ whenever $A$, $B$ are subsets of ${\mathbb R}^{n}$ with dist$(A,B)>0.$
I am new to this site, measure theory and real analysis, so please dont bash me if you think this question is trivial.I am only willing to learn and that once I get the hang of all this I shall use proper fonts to ask questions/or hopefully one day answer some of your queries :)
Thank you once again!!! cheers!
Hint: If $\operatorname{dist}(A,B) > 0$ then there are disjoint open sets $U$ and $V$ such that $A \subset U$ and $B \subset V$, which further implies that for any open sets $E$ and $F$ with $A \subset E$ and $B \subset F$ you have that $A \subset U \cap E$, $B \subset V \cap F$, and $\mu^*((U \cap E) \cup (V \cap F)) = \mu^*(U \cap E) + \mu^*(V \cap F)$. In particular, this holds when $E$ and $F$ are closely approximating $A$ and $B$ from above.