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"?
2026-04-07 06:49:13.1775544553
"Lebesgue measure" on metric spaces?
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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.