Timeline for answer to Does every $L^1_{\text{loc}}$-function have a signed measure as a distributional derivative? by reuns
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| Aug 15, 2021 at 18:19 | comment | added | Maximilian Janisch | With my knowledge now I wish that I could accept this answer too (the fact that $\mu=\mathrm df$ follows from the uniqueness of the weak derivative.) | |
| Mar 23, 2021 at 12:44 | comment | added | Maximilian Janisch | Ok I have finally looked at your very nice construction. Can I still ask you the following: Suppose that $\mu$ is a measure satisfying (*). Is it then easy to prove that $\mu = \mathrm df$ (which would be some sort of uniqueness result) and how do you prove that $\int_A \,\mathrm df=\infty, \int_{]0,1[\setminus A} \,\mathrm df=-\infty$ ? | |
| Mar 23, 2021 at 12:42 | history | edited | Maximilian Janisch | CC BY-SA 4.0 |
slight notational changes
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| Mar 23, 2021 at 10:15 | comment | added | Maximilian Janisch | In fact, it turns out that with the proper definition of a signed measure, not even $f(x)=|x|$ has a measure-derivative (even though the heavy-side function is a weak derivative in $L^1_{\text{loc}}$ !). | |
| Mar 23, 2021 at 10:05 | comment | added | Maximilian Janisch | I have cleared up my confusion by editing the question to include a proper definition of a signed measure 🙂. Now, $\infty-\infty$ is avoided so $\int_{\mathbb R} 1_{[a,b]}$ is well-defined in $\mathbb R\cup\{\infty\}$ or $\mathbb R\cup\{-\infty\}$. | |
| Mar 23, 2021 at 1:52 | comment | added | reuns | Alternatively the spirit of my answer is that (with $d\mu=df$) $\mu([1/a,1/2])$ is well-defined for $a>2$ but $\lim_{a\to \infty}\mu([1/a,1/2])$ diverges, so $\mu$ is not sigma additive: we can't decompose $(0,1/2)$ as a countably infinite disjoint union $\bigcup U_n$ of Borel sets and get $\mu((0,1/2))=\sum_n \mu(U_n)$ independent of the chosen $U_n$. | |
| Mar 23, 2021 at 1:38 | history | edited | reuns | CC BY-SA 4.0 |
added 94 characters in body
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| Mar 23, 2021 at 1:36 | comment | added | Maximilian Janisch | I noticed only after asking the question how you run into terrible subtleties with signed measures: For example, the integral is defined through $$\int_{\mathbb R} \phi\,\mathrm d\mu=\int_{P} \phi\,\mathrm d\mu-\int_{N} \phi\,\mathrm d(-\mu),$$ where $(P, N)$ is the Hahn-decomposition of $\mu$. It seems to me that hence, even if $\int_{\mathbb R} \phi\,\mathrm d\mu$ is well-defined for all test functions $\phi$, the integral $\int_{\mathbb R} 1_{[a,b]} \,\mathrm d\mu$ doesn't even have to well-defined! In short, it seems that signed measures have some crazy properties. | |
| Mar 23, 2021 at 1:31 | history | answered | reuns | CC BY-SA 4.0 |