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Whether a $L^p(X,\mu)$ space is separable? I understand that the answer depends on $p$ and $X$.

It seems to me that it is separable when $1\leq p < \infty, X=\mathbb{R}^n$ or $X=\mathbb{N}$. Also, $L^\infty(X,\mu) $ is seperable iff $X$ has only finite points.

Is there any general results? For example, does the separability of $(X,\mu)$ implies the separability of $L^p(X,\mu)$ when $p<\infty$?

Zach
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    Wikipedia http://en.wikipedia.org/wiki/Separable_space#Non-separable_spaces: The Lebesgue spaces Lp, over a separable measure space, are separable for any 1 ≤ p < ∞. – Urgje Jun 27 '14 at 22:30
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    If $p=\infty$, then $L_p(X,\Sigma,\mu)$ is separable iff $X$ consist of finite amount atoms. If $1\leq p<\infty$, then $L_p(X,\mu)$ is separable iff $(X,\Sigma,\mu)$ is separable. This means that there exists a countable family $\mathcal{F}\subset\Sigma$ such that, for any $\varepsilon>0$ and $A\in\Sigma$ one can find $B\in\mathcal{F}$ with $\mu(A\triangle B)<\varepsilon$. – Norbert Jun 28 '14 at 09:42
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    Thank you so much for your help! Norbert, your answer is very comprehensive! The definition of the separability of $(X,\Sigma,\mu)$ is quite new to me. Could you give me a reference of the proof for your latter statement? Thank you! – Zach Jun 28 '14 at 15:45
  • @Norbert : When you say "$X$ consist of finite amount atoms", does this means that $X$ is the union of a finite number of atoms (and not that the number of atoms is finite)? – Watson Mar 05 '16 at 18:56
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    [By the way, I think that you could post your comment as an answer. Don't you think so?] – Watson Mar 05 '16 at 18:57
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    @Watson, exactly. I don't think my answer is good enough, because it will take too long to write down the proofs, and I don't have a good reference for these facts. – Norbert Mar 05 '16 at 20:57
  • @Zach : Related: http://math.stackexchange.com/questions/1447055 – Watson Jun 15 '16 at 19:57

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