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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.

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    For the future, you might consider editing your title to be more informative. That way, you're more likely to get more views and more answers.2010-12-06

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