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If $f$ is a function such that $f \in L^\infty \cap L^ {p_0}$ where $L^\infty$ is the space of essentially bounded functions and $ 0 < p_0 < \infty$. Show that $ || f|| _{L^p} \to ||f || _{L^\infty} $ as $ p \to \infty$. Where $|| f||_{L^\infty} $ is the least $M \in R$ such that $|f(x)| \le M$ for almost every $x \in X$.

The hint says to use the monotone convergence theorem, but i can't even see any pointwise convergence of functions. Any help is appreciated.

TZakrevskiy
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user112564
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    Hint: compactly supported functions with finite number of values (sometimes called ladder functions) are dense in $L_p$ for $p\in [1,\infty)$. For such functions you can apply the analogous result for $p$-norms in finite-dimensioned spaces. – TZakrevskiy Jan 16 '14 at 14:06
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    See this. MCT is overkill. – David Mitra Jan 16 '14 at 14:07

1 Answers1

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Hint: Let $M\lt\|f\|_{L^\infty}$ and consider $$ \int_{E_M}\left|\frac{f(x)}{M}\right|^p\,\mathrm{d}x $$ where $E_M=\{x:|f(x)|\gt M\}$. I believe the Monotone Convergence Theorem works here.

Further Hint: $M\lt\|f\|_{L^\infty}$ implies $E_M$ has positive measure. On $E_M$, $\left|\frac{f(x)}{M}\right|^p$ tends to $\infty$ pointwise. MCT says that for some $p$, the integral above exceeds $1$.

robjohn
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  • Sorry to bother: I'm having trouble applying MCT here. I understand the strategy that David linked to above, but I'd very much like to understand this approach as well. Would you mind giving me an additional nudge in the right direction? – Josh Keneda Jan 19 '14 at 05:17
  • @JoshKeneda: I have added a further hint. Let me know if you still have questions. – robjohn Jan 19 '14 at 19:38
  • Thank you for all of your help, robjohn. One last question: I see how this argument gives $|f|\infty \le \lim{p\rightarrow \infty}|f|_p$; we still need an argument in the other direction, though, right? Did I miss something? Is there a nice MCT approach there, or should I just follow David's argument in that direction? – Josh Keneda Jan 20 '14 at 04:23
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    @JoshKeneda: Again, use MCT. For any $M\gt|f|_\infty$, consider: $$\int\left|\frac{f(x)}{M}\right|^p,\mathrm{d}x$$ – robjohn Jan 20 '14 at 14:48
  • Got it. Thanks again for all of the help! – Josh Keneda Jan 20 '14 at 21:23
  • @robjohn: Would you please explain how we use the fact that $f\in L^{p_0}$ for some positive $p_0$? – RozaTh Aug 05 '18 at 01:06
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    @Soufa: We use it to assure the existence of the integral in my answer for some $p\lt\infty$. For example, on $\mathbb{R}^1$, $f(x)=1$ is in $L^\infty$, but in no $L^p$ for $p\lt\infty$, so the limit doesn't exist. – robjohn Aug 05 '18 at 02:25
  • This answer provides a more detailed exposition. – robjohn May 15 '22 at 17:08