I am supposed to prove the following theorem
Let $(X,\mathcal{F}, \mu)$ be a probability space and $E_n \in \mathcal{F}$ for $n \in N$. Show that
$$\mu(\lim_{n\rightarrow \infty} \inf E_n) \leq \lim_{n\rightarrow \infty} \inf \mu(E_n)\leq \lim_{n\rightarrow \infty} \sup \mu(E_n) \leq \mu(\lim_{n\rightarrow \infty} \sup E_n)
$$
I tried to proceed by breaking the proof into three parts
$$ 1) \ \mu(\lim_{n\rightarrow \infty} \inf E_n) \leq \lim_{n\rightarrow \infty} \inf \mu(E_n) $$
$$ 2) \ \lim_{n\rightarrow \infty} \inf \mu(E_n)\leq \lim_{n\rightarrow \infty} \sup \mu(E_n) $$
$$ 3) \ \lim_{n\rightarrow \infty} \sup \mu(E_n) \leq \mu(\lim_{n\rightarrow \infty} \sup E_n)$$
I think the 2nd part can be easily proved since $\mu(E_n)$ is a bounded sequence because $\mu(E_n)\leq \mu(X)=1 \ \forall \ n \in N$ since $(X,\mathcal{F}, \mu)$ is a probability space and for a bounded sequence $a_n ,\ \lim_{n\rightarrow \infty} \inf a_n\leq \lim_{n\rightarrow \infty} \sup a_n $. Correct me if there is any mistake
But for the first and third part I only know this much. I know that $\mu(\lim_{n \rightarrow \infty}\inf E_n) = \mu\left(\bigcup\limits_{N=1}^{\infty} \left(\bigcap\limits_{n=N}^{\infty} E_n\right)\right)$ and that
$\mu(\lim_{n \rightarrow \infty}\sup E_n) = \mu\left(\bigcap\limits_{N=1}^{\infty} \left(\bigcup\limits_{n=N}^{\infty} E_n\right)\right)$ and also that I somehow need to show that
$\mu(\lim_{n \rightarrow \infty}\inf E_n) $ is less than $inf \{\mu(E_n)|\ m \leq n < \infty \} $ for some $m \in N$ and that $\mu(\lim_{n \rightarrow \infty}\sup E_n) $ is greater than $sup \{\mu(E_n)|\ m \leq n < \infty \} $ for some $m \in N$. How should I proceed now ? I also welcome any other method of solution.
Best Answer
Fatou's lemma applied to $\chi_{E_n}$ gives $\mu(\liminf E_n) \leq \liminf \mu(E_n)$. Reverse Fatou lemma applied to $\chi_{E_n}$ with dominating function $1 \in L^1(X, \mu)$ gives $\mu(\limsup E_n) \geq \limsup(\mu(E_n))$. The middle inequality is trivial.