Let $(S,\Sigma ,\mu)$ be $\sigma $ finite measure space and $\mu$ does not concentrate on any finite sets.(For example, Lebesgue measure space.)
I want to prove the existence of $(A_n)_{n\in \mathbb{N}}\subset \Sigma$ that is disjoint sequence and $\mu(A_n)>0$ for all $n$.
How do I prove this?
