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.