Timeline for Is strong convergence of measures equivalent to convergence in measure of the Radon Nikodym derivatives?
Current License: CC BY-SA 4.0
12 events
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| Jan 30, 2020 at 3:39 | vote | accept | James Baxter | ||
| Jan 29, 2020 at 14:58 | answer | added | Iosif Pinelis | timeline score: 2 | |
| Jan 29, 2020 at 14:17 | comment | added | Nate Eldredge | Note, however, that under your assumptions the Radon-Nikodym derivatives in fact converge to 1 in $L^1(\mu)$. If I'm not mistaken, that is equivalent to convergence in total variation. | |
| Jan 29, 2020 at 14:10 | comment | added | Nate Eldredge | Wikipedia's "strong convergence" is called "setwise convergence" in Bogachev's measure theory. It'd be helpful to have the definition in your question. | |
| Jan 29, 2020 at 10:45 | history | edited | James Baxter | CC BY-SA 4.0 |
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| Jan 29, 2020 at 10:41 | comment | added | James Baxter | It seems like it is indeed true, if I didn’t make any mistakes. | |
| Jan 29, 2020 at 10:38 | history | edited | James Baxter | CC BY-SA 4.0 |
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| Jan 29, 2020 at 10:38 | comment | added | James Baxter | If it’s meant in the other sense is it true? | |
| Jan 29, 2020 at 10:37 | comment | added | James Baxter | I did indeed mean convergence in the sense stated by wiki.. | |
| Jan 29, 2020 at 10:34 | comment | added | R W | @ Fedor Petrov - it depends on what the OP meant by strong convergence of measures. I was surprised to learn that wiki (apparently this is the article you quote) introduces a rather artificial notion of "strong convergence" of measures (I have never come across it in real life) and distinguishes it from the convergence in total variation ($\equiv$ convergence in the strong topology on the space of measures). This is not the first time I come across highly dubious claims in wiki. | |
| Jan 29, 2020 at 10:04 | comment | added | Fedor Petrov | From wiki: "For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ_n of measures on the interval [−1, 1] given by μ_n(dx) = (1+ sin(nx))dx converges strongly to Lebesgue measure." | |
| Jan 29, 2020 at 9:58 | history | asked | James Baxter | CC BY-SA 4.0 |