Show that $L^1_\mathcal{F}$ is separable

Question

Let $(\Omega, P)$ be a separable, perfect probability space. Let $\mathcal{F}$ be a closed $\sigma$-algebra on $\Omega$. Let $L^1_\mathcal{F}$ be the subspace of $L^1(\Omega)$ consisting of all the $\mathcal{F}$-measurable random variables. Show that $L^1_\mathcal{F}$ is a separable space.

I know that $L^1(\Omega)$ is separable since $(\Omega, P)$ is strongly isomorphic to $(S, \mu)$ for some $S \subset \mathbb{R}^1$ and some $K$-regular probability measure $\mu$. We can take $$ \sum_{i = 0}^n a_i 1_{[\alpha_i, \beta_i)} $$ for $a_i, \alpha_i, \beta_i \in \mathbb{Q}$ on $L^1(S)$. I wonder how we can apply this idea for $L^1_\mathcal{F}$, or if there is any other ideas.

Answer 1

Here, $L_\mathcal{F}^1$ is a subspace of $L^1$. Therefore, we can use the following fact.

Subspace of a separable space is separable