$L^\infty$ norm as limit of $L^p$-norm

Asked Jul 13 '23 at 08:47

Active Jul 13 '23 at 08:47

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Assume $X$ a measurable space, with $\mu(X) = +\infty$. I want to proof that if $|f|_p \le c$ for every $p$ then $f$ belongs to $L^{\infty}$ and $|f|_{\infty} \le c$. Here it's already been asked a similar question, but I'm little bit confused if the answer marked as "right answer" really worked with a general measurable set, without assumption of finite measure.

Any help is very appreciated.

asked Jul 13 '23 at 08:47

Rodolfo Ferreira de Oliveira

The answer in that link does work. You only need $|f|_p<\infty$ for some $p$ for that proof to work. – Kavi Rama Murthy Jul 13 '23 at 08:51
@geetha290krm so, the result still holds even if $\mu(X) = +\infty$? Did you have any reference? – Rodolfo Ferreira de Oliveira Jul 13 '23 at 08:59
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The accepted answer in your link does not assume that $\mu (X) <\infty$. Why do you need another reference? – Kavi Rama Murthy Jul 13 '23 at 09:01
Yes, but I need use this result in my thesis, but my advisor don't seems believe in the result. i think that if i shown it in a book... sorry @geetha290krm – Rodolfo Ferreira de Oliveira Jul 13 '23 at 09:36
So, I show for him this proof and he said that it isn't good. For this, I want to find this result in some book, paper, something like that – Rodolfo Ferreira de Oliveira Jul 13 '23 at 09:50
What they proved in that accepted answer is an exercise in Rudin's RCA. (The 4th question in the chapter on $L^{p}$ spaces in the old edition that I have). – Kavi Rama Murthy Jul 13 '23 at 09:54
Than you so much. Sorry me again! – Rodolfo Ferreira de Oliveira Jul 13 '23 at 09:59
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@RodolfoFerreiradeOliveira: The answer I posted in the link you are referring to works whether $\mu$ is a finite measure or not. – Mittens Jul 13 '23 at 13:24

$L^\infty$ norm as limit of $L^p$-norm

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