Let $\{E_n\}$ be a collection of measurable sets. Define $$\lim_{n\rightarrow \infty} \inf E_n=\bigcup_{n=1}^{\infty} \left(\bigcap_{k=n}^{\infty} E_k\right).$$ How does one show that $$ \mu\left(\lim_{n\rightarrow \infty} \inf E_n\right) \leq \lim_{n\rightarrow \infty} \inf ~\mu\left(E_n\right).$$
This is my attempt.
Since $\bigcap_{k=n}^{\infty} E_k\subseteq E_n~~n=1,2,3,\ldots~~,$ by monotonicity, we have $$ \mu\left(\bigcap_{k=n}^{\infty} E_k\right)\leq \mu\left(E_n\right).$$ This is all I have for now. I'll add more when I make progress.