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Consider that a measure space $(X, \mu)$ is separable if there is a countable family of measurable subsets $\{ E_k \}_{k=1}^{\infty}$ such that if $E$ is any measurable set of finite measure, then $\mu(E \Delta E_{n_k}) \to 0$ as $k \to 0$, for an appropriate subsequence dependent on $E$. Moreover, $E \Delta E_{n_k} = (E - E_{n_k}) \cup (E_{n_k} - E)$. I want to show that the Lebesgue measure on $\mathbb{R}^n$ is separable and that if $X$ is a separable measure space, then $L^p(X)$ is separable in the usual sense, for $1 \leq p < \infty$.

So far, we consider that $$\mu(E \Delta E_{n_k}) = \mu(E - E_{n_k}) + \mu(E_{n_k} - E)$$ But I'm not sure where to go from here since we can't assume that $E_{n_k} \to E$...

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    I think the set of all finite unions of balls with rational center and radius might be a good shot for the sequence $\{E_k\}$. You might have to use the fact that the Lebesgue measure is regular somewhere along the way. P.S. I haven't worked out all the details, that's just my intuition about the problem – 2017-02-24
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    Related question: http://math.stackexchange.com/q/2137579/263364 – 2017-02-25

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