Timeline for Where does the Lebesgue differentiation theorem fail?
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22 events
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| Feb 3, 2017 at 20:46 | comment | added | Christian Remling | @Squark: Stein's book actually does not have any details, the result is just quoted there. Saks's book mentions it too (again without details). The original paper seems to be Jessen, Marcinkiewicz, Zygmund, Fund. Math. 1935. | |
| Feb 3, 2017 at 19:09 | comment | added | Vanessa | @ChristianRemling: This is interesting, unfortunately I don't have access to Stein's book so I can't see the details. I think that the apparent contradiction might come from the fact that Theorem 2.3 in Haus and Pauc requires the Vitali property both wrt $\mu$ ad wrt $f \mu$, whereas in Stein's example the density theorem is satisfied for the Lebesgue measure but (maybe?) not for other measures absolutely continuous wrt the Lebesgue measure? | |
| Feb 1, 2017 at 14:25 | comment | added | Christian Remling | @Squark: Thanks for these references. I'm not doubting any of this, but let me just point out that if you consider more sets, then there are situations where the theorem holds for $f\in L^p$, $p>1$, but not for general $f\in L^1$. A concrete example is given by rectangles with sides parallel to the coordinate axes in $\mathbb R^2$. See Stein, Singular integrals... . So there is a difference between the differentiation and density theorems in general. | |
| Jan 31, 2017 at 22:20 | comment | added | Vanessa | The case of $X = \mathbb{R}^n$ and arbitrary $\mu$ is also covered in "Measure Theory vol. I" (Bogachev), see Theorem 5.8.8. | |
| Jan 31, 2017 at 19:59 | comment | added | Vanessa | For $\mathbb{R}^n$ and arbitrary Borel measure, the theorem appears in "The Metric Entropy of Diffeomorphisms: Part I" (Ledrappier and Young, 1985), see Lemma 4.1.2 there. I think that the generalization to arbitrary Riemannian manifolds shouldn't be hard because for a Riemannian metric the balls have locally bounded eccentricity w.r.t. the Euclidean metric. For ultrametric spaces I think the Vitali property should follow from the fact that any two balls are either disjoint or one contains the other. | |
| Jan 31, 2017 at 19:29 | comment | added | Vanessa | @ChristianRemling: In "Derivations and Martingales" (Hayes and Pauc, 1970), page 20 we have Theorem 2.3 which says that differentiation works as long as our family of sets satisfies the "Vitali property." On page 30 we find Theorem 1.2 which says that the Vitali property is satisfied iff the density theorem holds. So, unless I misunderstand something, the density and differentiation theorems are more or less equivalent. | |
| Jan 31, 2017 at 17:25 | comment | added | Christian Remling | @Squark: Thanks for the clarification, this is very helpful I think. I'm still not sure about the other point I raised: do you have a reference for the claims in your (now) second paragraph? (As I said, if it's taken from the wikipedia article, I think you are in fact misquoting it.) | |
| Jan 31, 2017 at 15:41 | history | edited | Vanessa | CC BY-SA 3.0 |
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| Jan 31, 2017 at 15:00 | answer | added | Gerald Edgar | timeline score: 0 | |
| Jan 31, 2017 at 8:55 | history | edited | Vanessa | CC BY-SA 3.0 |
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| Jan 31, 2017 at 8:49 | history | edited | Vanessa | CC BY-SA 3.0 |
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| Jan 31, 2017 at 7:14 | comment | added | Vanessa | @ChristianRemling, see recent edit. | |
| Jan 31, 2017 at 7:13 | history | edited | Vanessa | CC BY-SA 3.0 |
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| Jan 31, 2017 at 1:42 | comment | added | Christian Remling | @Squark: I think we are having some basic misunderstandings here and in the comments to my answer below. The point I'm trying to make (here and below) is that already on $\mathbb R^2$, it is certainly not true that $|A|^{-1} \int_A f(x-t)\, dt \to f(x)$ for a.e. $x$ as $\textrm{diam}(A)\to 0$, $0\in A$, if the sets $A$ get too general, and eccentricity is the problem. For example, you can't allow general rectangles. Maybe it would be helpful if you stated precisely what statement exactly you're interested in rather than just refer to it as the Lebesgue differentiation theorem. | |
| Jan 30, 2017 at 22:46 | comment | added | Vanessa | Which part do you find dubious? For Riemannian manifolds I think that the theorem follows quite easily from the Euclidean case by considering a local diffeomorphism with $\mathbb{R}^n$ (resulting eccentricity will be bounded on compact sets). | |
| Jan 30, 2017 at 22:03 | comment | added | Christian Remling | Could it be that you misquoted something in your opening paragraph? If I compare it with the wikipedia article for example, then the corresponding claims are only made about the Lebesgue density theorem there. | |
| Jan 30, 2017 at 21:05 | history | edited | Vanessa | CC BY-SA 3.0 |
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| Jan 30, 2017 at 20:58 | history | edited | Vanessa | CC BY-SA 3.0 |
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| Jan 30, 2017 at 16:20 | answer | added | Christian Remling | timeline score: 3 | |
| Jan 29, 2017 at 20:43 | comment | added | Aryeh Kontorovich | and mathoverflow.net/questions/218457/… | |
| Jan 29, 2017 at 20:42 | comment | added | Aryeh Kontorovich | These questions are relevant: mathoverflow.net/questions/244665/… | |
| Jan 29, 2017 at 19:57 | history | asked | Vanessa | CC BY-SA 3.0 |