Statement I is Halmos's definition of completeness; statement II omits $\mu(E_n)<\infty$

Consider the statements: I. A measure $\mu$ on a ring $\mathbf R$ is complete if $E\in\mathbf R$, $F\subseteq E$ and $\mu(E)=0$ then $F\in\mathbf R$. II. A measure $\mu$ on a ring $\mathbf R$ is $\sigma$-finite if, for every set $E\in\mathbf R$, $E=\bigcup_{n=1}^\infty E_n$ for some sequence $\{E_n\}$ such that $E_n\in\mathbf R$.