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"?


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.

  • $\begingroup$ 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

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