Timeline for Completeness of exponentials $\mathcal{E} = \{ e^{ist} : s \in \mathbb{R} \}$ in $L^p(\mu)$
Current License: CC BY-SA 4.0
8 events
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| Dec 9, 2019 at 4:39 | comment | added | Nate Eldredge | @LSpice: What is true is that if $f_n$ are continuous and converge in $L^\infty(\mu)$ to a function $g$, then $g$ is $\mu$-a.e. equal to a function whose restriction to the support of $\mu$ is continuous. (And by Tietze, it is thus also $\mu$-a.e. equal to a function which is continuous everywhere.) But I updated my answer to be more explicit - the claim I had before wasn't quite right. | |
| Dec 9, 2019 at 4:31 | history | edited | Nate Eldredge | CC BY-SA 4.0 |
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| Dec 9, 2019 at 3:36 | comment | added | Nate Eldredge | @LksRheil: I am unfortunately away from my bookshelf, so I can't say for sure what books would have it. But the Fourier transform fact would be everywhere (e.g. Folland's Real Analysis) and the rest is at the level of an exercise. | |
| Dec 9, 2019 at 0:30 | comment | added | LSpice | It's obvious that the uniform closure of the exponentials contains only continuous functions, but is it obvious that the $\mu$-a.e. uniform closure of the exponentials contains only continuous functions? | |
| Dec 8, 2019 at 19:54 | comment | added | PhoemueX | @LksRheil: If you are interested in the Fourier-analytic part, one reference is "Probability theory" by Bauer. I only have the German version right now, but there the precise statement you want is Theorem 23.4. | |
| Dec 8, 2019 at 18:24 | vote | accept | Jamie Mathews | ||
| Dec 8, 2019 at 15:21 | comment | added | Jamie Mathews | Thank you very much for your answer! Could you give me a reference in which books I can find such kind of theorems? | |
| Dec 8, 2019 at 14:13 | history | answered | Nate Eldredge | CC BY-SA 4.0 |