Let $(X, M, \mu)$ be a measure space and $\{A_n\}$ be a sequence of measurable sets.
I want to show that if $\mu$ is a finite measure and $\mu(A_n) > \epsilon>0$ for each n, then $\mu ($lim sup $A_N)≥\epsilon $.
Since lim sup $A_n$ = $\bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k$, we have $$ \mu(\text{lim sup } A_n) ≤ \mu(\bigcup_{k=n}^\infty A_k)$$ for all $n$, but I'm not sure where to go from here to show the desired result. I would appreciate any help on how to proceed. Thanks in advance!