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