Suppose $\mathfrak{R}$ is a $\sigma$-ring. Suppose $E_n \in \mathfrak{R}$. How would we show that $\bigcap_{n=1}^{\infty} E_n \in \mathfrak{R}$?
We know that $\bigcup_{n=1}^{\infty} E_n = \bigcap_{n=1}^{\infty} E_{n}^{c} \in \mathfrak{R}$. Likewise, how would we show that $\overline{\lim} \ E_n \in \mathfrak{R}$ and $\underline{\lim} \ E_n \in \mathfrak{R}$? The former is the set of points that are in infinitely many $E_n$. The latter is the set of points that are in all but a finite number of $E_n$.
2. 1)For difference, do not use the whole space X, use $\bigcup \limits_{n =1}^\infty E_n$ instead.
2)Another equivalent definition of $\overline{\lim} \ E_n$ is $\bigcap \limits_{k = 1}^\infty \bigcup \limits_{n = k}^\infty E_n$, and $\underline{\lim} \ E_n$ is $\bigcup \limits_{k = 1}^\infty \bigcap \limits_{n = k}^\infty E_n$,