What is the value of an infimum if it's taken over an empty set? I just got to this theorem in Royden where, a set function, $\mu$, over an arbitrary collection of subsets, S, of a set X can induce an outer measure.  But the outer measure is defined as an infimum of a sum with the infimum taken over countable collections that cover any set.  If $E\subseteq X$, then define

$\mu ^* E= \inf \sum_{k=1}^\infty \mu E_k$ over collections $\left\{E_k\right\}$ in S that cover E.

I'm just confused when there are no collections in S that cover E.  Or should I not bother thinking about this because useful set functions should at least be defined on X and evaluate $\mu X$.
The infimum is the greatest lower bound.  It is as "great" as it can get subject to the constraints imposed by the members of the set whose infimum you're taking.  For each such member, the constraint is merely that the infimum is not allowed to be greater than that number.  If there are no such constraints, then the infimum is $+\infty$.

PS (later edit):  Here's a practical (?) example.  Suppose the distance between two points on a manifold is defined as the infimum of the set of all lengths of paths connecting the two points.  Now suppose the two points are in different connected components of the space.  Then, since the greatest lower bound of the empty set is infinite, the distance between two points not connected by any path is infinite.  I seem to recall seeing some proposition somehere, in which this way of looking at it obviated the need for a hypothesis that the space is path-connected.
The null set is covered by any subset.  The infimum you see is not a void infimium.