5
$\begingroup$

Sry if my question is stupid, but I just wondered if is there is like a corresponding counterpart to the Lebesgue measure on $\mathbb{R}^n$ for (some?) metric spaces $(E,d)$? Since the natural way to measure distances in $(E,d)$ is given by $d$, shouldn't there be a "natural" way to measure sets as well? Sure I'm aware that the structure of $\mathbb{R}^n$ is very special but is there a theory out there about how to measure sets in metric spaces "naturally"?

$\endgroup$
3
$\begingroup$

Well, the real difficult here is what is the equivalent of invariance under translations.

For instance in a normed space of infinite dimension, how many balls of radius $r/4$ can you put in one of radius $r$? Answer: many.

$\endgroup$
  • $\begingroup$ thank's for that input. Maybe if one only considers finite dimensionals metric spaces? $\endgroup$ – Mr. Barrrington Oct 29 '14 at 12:51
  • $\begingroup$ Yes, this is back to $\Bbb R^d$ via an isomorphism. $\endgroup$ – mookid Oct 29 '14 at 12:59
  • $\begingroup$ So basically whenever the space infinite dimensional, i.e. not isomorphic to some $\mathbb{R}^n$ this is not possible. Thx! $\endgroup$ – Mr. Barrrington Oct 29 '14 at 13:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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