Talk:Locally integrable function

I don't know much about this subject, which is what led me to this page. But the article gives a weaker definition than is apparently available, saying the more abstract setting is not used very often. Here is at least one reader who needed it! Maybe somebody who knows more about this subject than I do can improve the article. Preceding unsigned comment by 216.165.151.43

In this vein, here are some useful references:

• [https://www.google.com/books/edition/Counterexamples_in_Measure_and_Integrati/RVctEAAAQBAJ?hl=en&gbpv=1 Schilling & Kuhn (2021)] and [https://mathweb.ucsd.edu/~bdriver/240C-S2018/Lecture_Notes/2012%20Notes/240Lecture_Notes_Ver8.pdf Driver (2012)] define $\backslash mathcal\; L^1\_\backslash mathrm\{loc\}(\backslash mu)$ given an arbitrary topological space $(X,\backslash mathcal\; T)$ and an arbitrary measure $\backslash mu$ on the Borel $\backslash sigma$-algebra $\backslash sigma(\backslash mathcal\; T)$
• [https://www.google.com/books/edition/Littlewood_Paley_and_Multiplier_Theory/BO_vCAAAQBAJ?hl=en&gbpv=1&dq=topological+space+locally+integrable+function&pg=PA208&printsec=frontcover Edwards & Gaudry (1977)] and [https://www.google.com/books/edition/Integration_Theory/JDXfDwAAQBAJ?hl=en&gbpv=1&dq=topological+space+locally+integrable+function&pg=PA124&printsec=frontcover Filter & Weber (1997)] define $\backslash mathcal\; L^1\_\backslash mathrm\{loc\}(\backslash mu)$ given an arbitrary measure space $(X,\backslash mathcal\; A,\backslash mu):$ a function $f:X\backslash to\backslash bar\backslash mathbb\; R$ is in $\backslash mathcal\; L^1\_\backslash mathrm\{loc\}(\backslash mu)$ iff for every $A\backslash in\backslash mu^\backslash mathrm\{pre\}\backslash mathbb\; R,$ (i.e., $A$ is measurable and has finite measure), $fe\_A^X\backslash in\; L^1(\backslash mu\_A),$ where $\backslash mu\_A$ is the subspace measure induced on $A$. [ [https://www.google.com/books/edition/Measure_and_Integral/CqfiBQAAQBAJ?hl=en&gbpv=0 Jacobs (1978)] gives the definition for an arbitrary "measure"/"$\backslash sigma$-content" on an arbitrary "local $\backslash sigma$-ring" (a.k.a. $\backslash delta$-ring).] [Filter & Weber give a version for an arbitrary ring of sets $\backslash mathcal\; A$. At first glance, Filter & Weber may appear to say that any locally integrable function will be integrable; taking $A=X$ in their definition yields $f\backslash in\; L^1(\backslash mu)$. However, their definition corresponds to our $\backslash sigma$-algebra definition only when we consider the ring $\backslash mu^\backslash mathrm\{pre\}\backslash mathbb\; R$ of finite-measure measurable sets. In this case, it is obvious that their definition coincides with ours.]

Thatsme314 (talk) 10:16, 9 May 2022 (UTC)

In what sense is the final example a solution to the given differential equation on all of $\backslash mathbb\; R$? I get that the restriction of this function to $\{\backslash mathbb\; R\}\; -\; \backslash \{0\backslash \}$ does not extend to a distribution on $\backslash mathbb\; R$, but it's unclear how the function given (with $f(0)\; =\; 0$) can be interpreted as some kind of weak solution of the given equation across zero, since it's not integrable. Spireguy (talk) 15:05, 21 May 2023 (UTC)