Separability of $L^p(\mathbb R^n ,\mu)$

Question

Let $\mu$ be a Radon measure in $\mathbb R^n$ it is true that $L^p(\mathbb R^n,\mu)$ is separable? I not find a proof of these fact, I only would know that these is true.

Answer 1

Assume $1 \leq p < \infty$. Yes if you have a $\sigma$-finite measure space $(X, \mathcal{F}, \mu)$ whose $\sigma$-algebra $\mathcal{F}$ is generated by a countable set $\mathcal{C}$, then $L^p(X, \mu)$ is separable. The proof for when $\mu$ is finite is simple: Let $\mathcal{A}$ be the algebra generated by $\mathcal{C}$. You can show that $\mathcal{A}$ is countable, and then using the $\pi$-$\lambda$ theorem or the monotone class theorem, you can show that closure of the set $\{1_A : A \in \mathcal{A} \}$ in $L^p$ contains $\{1_{E} : E \in \mathcal{F}\}$. Thus the span of this set over $\mathbb{Q}$ is dense in $L^p(X, \mu)$, proving separability.

You can easily reduce the $\sigma$-finite case to the finite case.