Let $(X,M,\mu)$ be a measure space. Why are the following statements equivalent?
i) There exists a sequence of pairwise disjoint measurable sets $\{A_{n}\}$ in $M$ , each of finite measure, such that $\mu(A) = \sum_{n=1}^{\infty} \mu(A \cap A_{n})$ for every measurable set $A \in M$.
ii) $\mu$ is a countable sum of pairwise mutually singular finite measures.