I am working on the following task:
Let $\phi:\mathcal{P}(X)\rightarrow[0,\infty]$ be an outer measure on $X$. Let $A,B\subseteq X$ and $A$ or $B$ $\phi$ measureable. Show the following statements:
(i) $\phi(A\cup B)+\phi(A\cap B)=\phi(A)+\phi(B)$
(ii) If $A$ and $B$ are disjoint and $C\subseteq A$ as well as $D\subseteq B$, then $\phi(C\cup D)=\phi(C)+\phi(D)$
But I really do not know how to prove the statements here. Can somebody help me with this task?
Let us assume that $A \in \mathcal{P}(X)$ is measurable, i.e. $\phi(M) = \phi(M \cap A) + \phi(M \cap A^C)$ for all $M \in \mathcal{P}(X).$ Then $$\phi(A) + \phi(B) = \phi(A) + \phi(A \cap B) + \phi(B \cap A^C) = \\ \phi(A \cap B) + \phi((A \cup B) \cap A) + \phi((A \cup B) \cap A^C) = \phi(A\cap B) + \phi(A \cup B).$$ For the second point, since $A$ and $B$ are disjoint and $A$ is $\phi-$measurable, we infer that $$\phi(C \cup D) = \phi((C \cup D) \cap A) + \phi((C \cup D) \cap A^C) = \phi(C) + \phi(D).$$ I hope this helps. :)