If $\sum_{k=1}^\infty |E_k| < \infty$, then
$\displaystyle |\limsup_{k\rightarrow \infty} E_k|=0$.
Where $|*|$ denotes the Lebesgue measure, all sets are assumed to be measurable, and $\limsup_{k\rightarrow \infty} E_k = \bigcap_{k=1} \bigcup_{j=k} E_k$
Been struggling with this one, any help is appreciated! Thanks!
Best Answer
For $N$ large, we use that $\limsup E_n$ is an intersection, and subadditivity of the measure: $$|\limsup E_n|\leq|\bigcup_{n=N}^\infty E_n|\leq\sum_{n=N}^\infty|E_n|$$ and the right-hand side is a tail of a convergent series, so it goes to $0$