edit: I am using |*| as notation for measure, and all sets are assumed to be measurable.
Define $\liminf_{k \rightarrow \infty}E_k= \cup_{k=1} \cap_{j=k} E_j$ and $\limsup_{k \rightarrow \infty}E_k= \cap_{k=1} \cup_{j=k} E_j$.
I'm wondering if there is an example where $\liminf_{k \rightarrow \infty}E_j$ is a proper subset of $\limsup_{k \rightarrow \infty}E_j$.
Also, I'm trying to prove the inequality $|\liminf_{k \rightarrow \infty}E_k| \leq \liminf_{k \rightarrow \infty}|E_k|$
I think this is an interesting inequality, because the right hand is asking for the infimum of a sequence of positive numbers, where the left is the measure of a definition of infimum defined as a set operation.
Any advice is greatly appreciated! Thanks everyone!!
Best Answer
The inequality is trivial from the definition of $\liminf|E_j|$. Say $$I_k=\bigcap_{j\ge k}E_j.$$ Since $I_k\subset E_j$ for $j\ge k$ we have $$|I_k|\le\inf_{j\ge k}|E_j|.$$And now since $I_k\subset I_{k+1}$ it follows that $$|\bigcup_k I_k|=\lim_k|I_k|\le\lim_k\inf_{j\ge k}|E_j|=\liminf|E_j|.$$