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The idea is to show that $L^p$ is complete over a $\sigma$-finite measure space knowing that $L^1$ is complete. Let $(X,\mathcal{M},\mu)$ a measure space. If is a finite measure space it's not complicated to show that if $\{f_n\}_{n\in\mathbb{N}}\subseteq L^p$ is a Cauchy sequence, then is Cauchy in $L^1$ and since $L^1$ is complete, the sequence converges in $L^{1}$ to a function $f\in L^{1}$, also by Fatou's Lemma I can prove that $f$ is in $L^{p}$ (This is part of the steps to follow indicated by the problem). Now if $(X,\mathcal{M},\mu)$ is a $\sigma$-finite measure space, how I can prove that $L^p$ is complete using what has just been proved? $\textbf{The problem asks to use the diagonal argument}$, however I have only used this argument to prove the compactness of a set, not the completeness. Any hint?

JadMON2k1
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  • @OliverDíaz How? Does the $\sigma$-finitude have anything to do with the separability of the space $L^p$? I have not seen the concept of separable spaces in the context of $L^p$ spaces – JadMON2k1 Oct 11 '22 at 01:39
  • I think you need that the $\sigma$-algebra is countably generated by sets of finite measure, that is there is a sequence $\mathcal{C}={A_n:n\in\mathbb{N}}$ of integrable sets such that $\sigma(\mathcal{C})$ generates the $\sigma$-algebra $\mathcal{M}$. – Mittens Oct 11 '22 at 01:45

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