Let $(E_n)_{n \in \mathbb{N}}$ be some measurable sets. Define $\lim \inf E_n$ to be
$$\bigcup_{n \in \mathbb{N}} \bigcap_{m = n}^\infty E_m$$
I want to show that $\mu(\lim \inf E_n) \leq \lim \inf E_n$. I thought we might take an increasing sequence $(F_n)$ where $F_n = E_1 \cup … \cup E_n$ and use that $\mu(\lim \inf F_n) = \lim_{n \to \infty} \mu(\bigcap_{m = n}^\infty F_m)$, but I am stuck at relation $\bigcap_{m = n}^\infty F_m$ to the $E_n$ sequence and I don't see how the $\lim \inf$ of a sequence factors in.
Best Answer
$X_n = \bigcap_{m=n}^{\infty} E_m$ is an increasing sequence of sets, hence $\mu(\bigcup_{n=1}^\infty X_n) = \lim \mu(X_n)$ by sigma additivity. But $X_n\subset E_n$, hence $\mu(X_n)\le \mu(E_n)$. Hence \begin{equation} \mu(\liminf E_n) = \lim \mu(X_n) = \liminf\mu(X_n)\le\liminf\mu(E_n) \end{equation}