Timeline for Finiteness of Hausdorff measure of balls
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 21, 2021 at 17:56 | comment | added | Martin Väth | An infinite-dimensional normed space is a famous example where no natural approach is known. In practice, all "usable" measures on such spaces have the property that most sets have infinite measure. If you want to have a ball with a finite measure in such a space, the measure becomes very unnatural (not translation invariant, extremely weighted, most sets have measure 0, etc.). | |
| Jun 21, 2021 at 5:29 | vote | accept | John D | ||
| Jun 21, 2021 at 0:37 | comment | added | Jacob Manaker | @JohnD: If your space is countable, then take counting measure. If it is uncountable and separable, then you can pull back Lebesgue measure via a Polish-space isomorphism to $\mathbb{R}$. If it is inseparable, then that sounds like a hard problem. (Do you consider a nontrivial measure on a separable strict subspace trivial?) | |
| Jun 20, 2021 at 11:05 | comment | added | John D | Thanks for the answer! Follow up question: are you aware of a general method to define a finite(nontrivial) measure on the Borel sigma algebra of an arbitrary metric space? | |
| Jun 20, 2021 at 7:25 | history | answered | Martin Väth | CC BY-SA 4.0 |