# Why does the Lebesgue outer measure on $\mathbb{R}^{d}$ always exist for any set $A$?

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?

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Every bounded from below non-empty set of real numbers has an infimum. The set you are asking about is bounded by 0 from below, and is not empty (if the set is empty then the measure of $A$ is infinite).
How dare I forget about the least upper bound property of $\mathbb{R}$...You get so caught up with the domain of $\mu^{\star}$ (e.g. sets) that you forget its range is just $[0,\infty]$. –  Taylor Martin Oct 12 '12 at 21:49