Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Fix $p \in [1, \infty)$. Let $L_p := L_p (X, \mu, E)$ and $\|\cdot\|_{L_p}$ be its norm. Here we use the Bochner integral. I'm trying to generalize Theorem 4.13. in Brezis's Functional Analysis, i.e.,
Theorem: If the $\sigma$-algebra $\mathcal A$ is countably generated and $E$ separable, then $L_p$ is separable.
Could you have a check on my below attampt?
Proof: Let $\mathcal F$ be the collection of all measurable sets with finite measure. We define a pseudometric metric $d_\mu$ on $\mathcal F$ by $d_{\mu}(A, B) := \mu(A \triangle B)$. Then $d_\mu$ becomes a metric when $\mathcal F$ is considered modulo the equivalence relation $\sim$ defined by $$ A \sim B \iff \mu(A \triangle B) = 0 \quad \forall A,B \in \mathcal F. $$
Then we have
- Lemma 1 If the $\sigma$-algebra $\mathcal A$ is countably generated, then $d_\mu$ is separable.
Also,
- Lemma 2: The space $\mathcal S := \mathcal S(X, \mu, E)$ of simple functions is dense in $\big (L_p, \|\cdot\|_{L_p} \big)$ for all $p \in [1, \infty)$. [Prop 4.8 in Amann's Analysis III]
By Lemma 2, it suffices to prove that $\mathcal S$ is separable. Let $D$ be a countable dense subset of $E$. By the finite form of a function in $\mathcal S$ and the triangle inequality, it suffices to prove that $\mathcal I := \{e 1_{A} : e\in D, A \in \mathcal F\}$ is separable. Fix some $e \in D$ such that $|e|=1$. Again, it suffices to prove that $\{e 1_{A} : A \in \mathcal F\}$ is separable. Consider the map $$ T:\mathcal F \to L_1, A \mapsto e1_A. $$
It follows from $1_{A \triangle B} = |1_A-1_B|$ that $T$ is an isometric embedding. By Lemma 1, $d_\mu$ is separable. So $T(\mathcal F)$ is also separable. This completes the proof.
