How can $\bigcup\limits_{i=1}^{\infty}E_i\in\mathscr{R}$?

Question

Theorem: Suppose $\mu:\mathscr{R}\to\mathbb{R}^+$ is non-negative and additive on a ring $\mathscr{R}$. Then

(i) if $E\in\mathscr{R}$, and $\{E_i\}$ is a sequence of disjoint sets of $\mathscr{R}$ such that $E\supset\bigcup\limits_{i=1}^{\infty}E_i$

$\mu(E)\geqslant\sum_\limits{i=1}^{\infty}\mu(E_i)$;

(ii) if $E\in\mathscr{R}$, and $\{E_i\}$ is a sequence of disjoint sets of $\mathscr{R}$ such that $E\subset\bigcup\limits_{i=1}^{\infty}E_i$

$\mu(E)\leqslant\sum_\limits{i=1}^{\infty}\mu(E_i)$;

Question:

On this theorem it is stated that $E\in\mathscr{R}$, and $E=\bigcup_\limits{i=1}^{\infty}$. However by the definition the ring is only closed for finite unions not for infinite unions. How can $E=\bigcup_\limits{i=1}^{\infty}\in\mathscr{R}$?

Is the author implying the Monotone class generated by the ring?

Thanks in advance!