Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

up vote 2 down vote accepted

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).

share|improve this answer
1  
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 Oct 12 '12 at 21:49
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.