Timeline for For $\mu$-a.e. $x \in X$, the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$. Then $(f_n)$ is Cauchy in $L^1 (X \times Y)$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 16, 2023 at 11:16 | vote | accept | Akira | ||
| Aug 16, 2023 at 11:01 | answer | added | Jochen Wengenroth | timeline score: 2 | |
| Aug 16, 2023 at 8:11 | comment | added | Akira | @JochenWengenroth From this Wikipedia page, there is a result If $\mu$ is $\sigma$-finite and $\left(f_n\right)$ converges (locally or globally) to $f$ in measure, there is a subsequence converging to $f$ almost everywhere. So you explanation makes a lot of sense. Could you please post it as an answer? | |
| Aug 16, 2023 at 8:04 | comment | added | Jochen Wengenroth | Apparently, Väth only claims that $y_n$ is Cauchy in the space of measurable functions with the (completely metrizable) topology of convergence in measure (which is given by the metric $d(y,z)=\int \min\{|y-z|,1\}d\mu$). | |
| Aug 16, 2023 at 7:55 | history | edited | Akira | CC BY-SA 4.0 |
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| Aug 16, 2023 at 7:50 | comment | added | Akira | @JochenWengenroth It's possible that I mis-understood the paragraphs in the book, so I have added the screenshots of the relevant ones. I think the correct hypothesis is that $f_n$ is $\lambda$-simple rather than just $\lambda$-measurable. Please have a check on my update. | |
| Aug 16, 2023 at 7:48 | history | edited | Akira | CC BY-SA 4.0 |
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| Aug 16, 2023 at 7:40 | history | edited | Akira | CC BY-SA 4.0 |
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| Aug 16, 2023 at 5:31 | comment | added | Jochen Wengenroth | Sure about the hypotheses? I do not even believe that they imply $f_n\in L^1(X\times Y)$. | |
| Aug 15, 2023 at 23:37 | history | asked | Akira | CC BY-SA 4.0 |