The question is basically in the title. We have $\mu^{\star}:2^{\mathbb{R}}\to\mathbb{R}^{+}$ whose definition can be taken as $\mu^{\star}(A)=\inf_{\bigcup_{j=1}^{\infty}Q_{j}\supset A}\sum\limits_{j=1}^{\infty}|Q_{j}|$ where the $\{Q_{j}\}_{j=1}^{\infty}$ are countable coverings of the set $A$ by simple sets (e.g. cubes, rectangles, balls, etc.).
Clearly $\mu^{\star}$ is well defined, in the sense that if the infimum exists for a set $A$, it is unique. But whoever said it exists to begin with? None of the books I have read on the subject mention this; they seem to all take for granted that the infimum always exists, no matter what $A$ is.
The same question applies to outer and inner Jordan measures too.
And I'm not talking about Jordan or Lebesgue measurable sets; I mean for any set $A$, why do these outer (and inner for Jordan) measures exist?