$ \text{Given: } K \subset \mathbb{R}^n \text{ is compact, } E \subset \mathbb{R}^n, \ K \cap E = \emptyset $
$ \text{Show: } |K \cup E|_e = |K| + |E|_e $
$ \text{Proof: K compact implies K measurable. By Carath} \acute{\text{e}} \text{odory, } $
$ |K \cup E|_e = |K \cap (K \cup E)|_e + |(K \cup E) - K|_e = |K|_e + |E|_e = |K| + |E|_e $
