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Let $(X, \cal M, \mu)$ be a measure space with $\mu(X)<\infty$. Why \begin{align} \lim_{p\to\infty}\|f\|_p=\|f\|_\infty \end{align} for all $f\in L_\infty$?

Riccardo
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Jonathan
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1 Answers1

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You want to show that $lim_{p \to \infty} \|f\|_p \leq \|f\|_\infty$ and $\|f\|_\infty \leq lim_{p\to\infty}\|f\|_p$. There are two relevent estimates. The first is $\|f\|_p \leq \|f\|_\infty \mu(X)^{1/p}$. For the second, let $\epsilon > 0$, and pick $E \subset X$ with $\mu(E) > 0$ such that $|f(x)| > \|f\|_\infty - \epsilon$ for all $x \in E$. Then $\|f\|_p \geq (\|f\|_{\infty} - \epsilon)\mu(E)^{1/p}$. Try to justify each of these estimates, and then take $p \to \infty$ to complete the proof.

Justthisguy
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