A footnote about outer measure 
This is the theorem about in Royden's real analysis book. And in the book there is a footnote I am confusing:

Can anyone help me understanding it with examples~~~
 A: The outer measure of $E$ is defined of as the infimum of the following set 
$$\left \{\sum_{k=1}^{\infty} \mu(E_k) \colon \{E_k\}_{k=1}^\infty \text{with $E_k\in S$ such that  $E \subset \bigcup_{k=1}^\infty E_k$ } \right \}$$
Now nothing guarantees that for some set $E$ there is even one $\{E_k\}_{k=1}^\infty$ with $E_k\in S$ such that  $E \subset \bigcup_{k=1}^\infty E_k$. In this case the set above is empty, and it the outer measure of such an $E$ is then the infimum of the empty set. It is thus necessary to know what the infimum of the empty set would be, and this is what is stated in the footnote.
A: There is only one example: $\inf\emptyset=+\infty$. I can explain you the reason. The definition of $\inf$, i.e., greatest lower bound, which is equivalent to $\sup\{x\in\mathbb R:x<y\:\forall y\in\emptyset\}$.  Think about it: this means
$\inf\emptyset=\sup\mathbb R$ and thus $+\infty$.
A: Let $S := \{ \{x\} \mid x \in \mathbb{R}\}$. Then $\mathbb{R}$ cannot be covered by countably many elements of $S$.
