Timeline for Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(a,b]$ and infinitesimals to singletons
Current License: CC BY-SA 3.0
8 events
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| Sep 27, 2012 at 20:27 | comment | added | David Milovich | @Gerald Edgar: Indeed, we can do much much more with the same technique, which boils down to replacing a single ultraproduct with an iterated ultraproduct to get more control. (In this case, a two-step iteration was sufficient.) To my mind, the most salient barriers are finite obstructions like the Banach-Tarski obstruction to congruency invariance in three dimensions. I don't see a finite obstruction to what you propose regarding Hausdorff measure. | |
| Sep 27, 2012 at 17:57 | comment | added | Gerald Edgar | But probably we want more than the OP asks for. In addition to assigning lengths to intervals and $\delta$ to points, maybe we want to pick a certain infinitesimal $\delta_s$ for each $0<s<1$ and try to get $\delta_s$ times the $s$-dimensional Hausdorff measure when we have a set of dimension $s$ (up to smaller-size infinitesimals). And, for that matter, when we do Hausdorff measures there is no reason to use only constant gauge functions. Maybe we would want to do this with come other nonarchimedean extension of the reals, rather than NSA. | |
| Sep 27, 2012 at 16:58 | comment | added | David Milovich | I just minorly corrected the answer: the parenthetical paragraph now correctly handles cases such as $[0,1/3)\cup[2/3,1)\in A$ but $[0,1/3)\not\in A$. | |
| Sep 27, 2012 at 16:56 | history | edited | David Milovich | CC BY-SA 3.0 |
Minor correction to handle minimal finite unions of intervals
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| Sep 27, 2012 at 16:51 | vote | accept | Alexander Pruss | ||
| Sep 27, 2012 at 15:22 | history | edited | David Milovich | CC BY-SA 3.0 |
minor typos
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| Sep 27, 2012 at 15:05 | history | edited | David Milovich | CC BY-SA 3.0 |
minor correction to $\mu_A$ construction; added 42 characters in body
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| Sep 27, 2012 at 14:57 | history | answered | David Milovich | CC BY-SA 3.0 |