Timeline for Extension of a function from almost everywhere to everywhere
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| May 18, 2016 at 9:25 | comment | added | Bogdan Grechuk | Bill Johnson - thank you, I knew about notions of null sets, but not about this book. YCor - yes, this is a disadvantage of the proposed approach. Piero D'Ancona - yes, I thought about this, simple method, works for all functions for which liminf is finite. However, I have strong intuition that, for example, sign(0) should be defined as 0, not -1. Dirk - thank you, finally I have a name for the concept I am talking about - this is exactly "precise representative"! Are their any reasonable definition of precise representative in infinite dimension? | |
| May 17, 2016 at 18:56 | comment | added | Dirk | Maybe you are also interested in the notion of "precise representative" which exists for Sobolev (BV) functions that are regular enough. You can find these in Evans and Gariepy's "Measure theory and fine properties of functions". | |
| May 17, 2016 at 17:01 | comment | added | Piero D'Ancona | You might define $f(x)=\liminf_{y\to x} f(y)$ | |
| May 17, 2016 at 13:28 | history | edited | Bogdan Grechuk | CC BY-SA 3.0 |
added 742 characters in body
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| May 16, 2016 at 15:59 | vote | accept | Bogdan Grechuk | ||
| May 13, 2016 at 20:08 | comment | added | YCor | This definition of $f(x)$ using centered balls is sensitive to the choice of norm on $\mathbf{R}^n$. | |
| May 13, 2016 at 20:00 | answer | added | Willie Wong | timeline score: 4 | |
| May 13, 2016 at 19:09 | comment | added | Bill Johnson | For a "natural" extension of $f$ when $f$ is bounded, in finite dimensions you can apply a Banach limit to $\lambda(B_\epsilon(x))^{-1} \, \int_{B_\epsilon(x)} f \, d\lambda$. | |
| May 13, 2016 at 19:05 | comment | added | Bill Johnson | If you want to understand the basics of null sets (a replacement for "measure zero") in infinite dimensional normed spaces, read chapter 6 in the book of Benyamini and Lindenstrauss. This concept was pretty well understood in the 1970s. Much later Lindenstrauss and Preiss developed another notion of null set that allowed them to treat some very difficult differentiation questions. It may be that to get a reasonable answer to your question you must specify a specific notion of null set. | |
| May 13, 2016 at 17:42 | history | asked | Bogdan Grechuk | CC BY-SA 3.0 |