Tricky measure subset problem

Question

Hi, I am unable to solve the first part, I have tried many things such as creating an infinite sequence of sets yet cannot seem to prove that they tend to infinity

Answer 1

Let the RHS of expression be $A'$. This case is trivial $\mu(A)< \infty$. If $\mu(A)=\infty$. Suppose $A'$ is finite. Prove the following:

1. Exists $B_n \subseteq A \in \mathcal{A}$ such that $\mu(B_n) \rightarrow A'$.
2. Modifying 1. Exists $B'\subseteq A \in \mathcal{A}$ such that $\mu(B') = A'$.
3. Then $\mu(A \setminus B') =?$ Does this yield a contradiction?