I have to prove the following important result.
Theorem. Let $(X,\mathcal{A},\mu)$ be a measure space such that:
$(1)\;$ the measurable space $(X,\mathcal{A})$ is separable.
$(2)\;$$\mu$ is sigma finite.
Then $L^p(X,\mathcal{A},\mu)$ is separable for $p\in [1,\infty)$.
We remember that a measurable space $(X,\mathcal{A})$ is said separable if exists a countable family $\mathcal{C}\subseteq \mathcal{P}(X)$ sich that $\mathcal{\sigma}_0(\mathcal{C})=\mathcal{A}$, where $\sigma_0$ denotes the generated sigma algebra.
First case$\mu(X)< \infty$.
We denote with $S(X,\mathcal{A},\mu)$ the set of all simple measurable function on $X$ a complex values.
We know that given a simple function $s\in S(X,\mathcal{A},\mu)$ we have that $$s\;\text{is $\mu-$integrable}\iff \mu(\{X\;:\; s\ne 0\})<\infty$$
We denote with $S_0(X,\mathcal{A},\mu)$ the subcollection of $S(X,\mathcal{A},\mu)$ that consists of all $\mu$ integrable simple funtion. It's simple to prove that
$(a)\; S_0(X,\mathcal{A},\mu)\subseteq L^p(X,\mathcal{A},\mu)$
$(b)\;$$S_0(X,\mathcal{A},\mu)$ is dense in $L^p(X,\mathcal{A},\mu)$.
We introduce the collection $$S_0^{\mathbb{Q}}(X,\mathcal{A},\mu)=\{s\in S_0(X,\mathcal{A},\mu)\;:\; s(X)\subseteq\mathbb{Q}+i\mathbb{Q}\}$$
First step$S_0^{\mathbb{Q}}$ is dense in $S_0$
We prove that $S_0^{\mathbb{Q}}$ is dense in $S_0(X,\mathcal{A},\mu)$. Let $\varepsilon > 0 $ and $s\in S_0$ fixed. Let $$s=\sum_{k=1}^n c_k \chi_{E_k}$$ the standard representation of $s$. Now for all $k$ let $q_k$ be a complex number in $\mathbb{Q}+i\mathbb{Q}$ whose real and imaginary parts are such that
$$\max_{k=1,\dots, n}{|c_k-q_k|}<\frac{\varepsilon}{[n\mu(X)]^{\frac{1}{p}}}.$$ Then the simple function $$s_{\mathbb{Q}}=\sum_{k=1}^nq_k\chi_{E_k}\in S_0^{\mathbb{Q}}$$ and results that
\begin{eqnarray*}\lVert s-s_{\mathbb{Q}} \rVert_p^p &=& \int_X \left | \sum_{k=1}^n(c_k-q_k)\chi_{E_k}\right |^p\; d\mu \\&\color{red}{=}& \int_X\sum_{k=1}^n\lvert c_k-q_k \rvert^p\chi_{E_k}\;d\mu \\&\color{blue}{=}& \sum_{k=1}^{n} \int_X\lvert c_k-q_k\rvert^p\chi_{E_k}\;d\mu \\&=&\sum_{k=1}^n\lvert c_k-q_k\rvert^p\mu(E_k) \\&\le& \sum_{k=1}^n\frac{\varepsilon^p}{n}\frac{\mu(E_k)}{\mu(X)}\\&\color{GREEN}{\le}& \sum_{k=1}^n\frac{\varepsilon^p}{n}=\varepsilon^p\end{eqnarray*}The red inequality arises from the fact that the $E_k$ are disjoint, the blue equality arises from the linearity of integral, and the green inequality arises from the fact that $\mu(E_k)/\mu(X)\le 1$.
To continue we need the following important
Lemma Let $(X,\mathcal{A},\mu)$ be a finite measure space. Let $\mathcal{C}\subseteq\mathcal{P}(X)$ a family such that $\sigma_0(\mathcal{C})=\mathcal{A}$; let $\mathcal{A}_0:=\mathcal{A}_0(\mathcal{C})$ the generated algebra of $\mathcal{C}$. Then for all $F\in \mathcal{A}$ e for all $\varepsilon >0$ exists $G\in\mathcal{A}_0$ such that $$\mu(F\setminus G)+\mu(G\setminus F)<\varepsilon.$$
Now, since the space $(X,\mathcal{A})$ is separable, exists a countable family $\mathcal{C}\subseteq \mathcal{P}(X)$ such that $\sigma_0(\mathcal{C})=\mathcal{A}$. Evidently the generate algebra $\mathcal{A}_0:=\mathcal{A}_0(\mathcal{C})$ is countable. Now, we introduce the collection $$S_0^{\mathbb{Q},\mathcal{A}_0}(X,\mathcal{A},\mu)=\left\{s=\sum_{k=1}^n d_k\chi_{G_k}\in S_0^{\mathbb{Q}}\;|\; G_k\in\mathcal{A}_0, n\in\mathbb{N}\right\}.$$
We observe that this collection is also countable. It remains to prove that $S_0^{\mathbb{Q},\mathcal{A}_0}$ is dense in $S_0^{\mathbb{Q}}$.
Second step$S_0^{\mathbb{Q}, \mathcal{A}_0}$ is dense in $S_0^{\mathbb{Q}}$.
Let $t\in S_0^{\mathbb{Q}}$, then $$t=\sum_{k=1}^nc_k\chi_{F_k},$$ where $F_k\in\mathcal{A}$ and $c_k\in \mathbb{Q}+i\mathbb{Q}$ ($k=1,\dots, n$). From the above lemma for all $k=1,\dots, n$ exists $G_k\in\mathcal{A}_0$ such that $$\mu(F_k\setminus G_k)+\mu(G_k\setminus F_k)<\left(\frac{\varepsilon}{nM} \right)^p,$$ where $M:=\max _{k=1,\dots, n}|c_k|$. Define $$s=\sum_{k=1}^nc_k\chi_{G_k},$$ then $s\in S_0^{\mathbb{Q},\mathcal{A}_0}$ and results that
\begin{eqnarray*}\lVert t-s \rVert_p &=& \left\lVert \sum_{k=1}^n c_k(\chi_{F_k}-\chi_{G_k})\right\rVert_p \\& \stackrel{Minkowski}{\leq}& \sum_{k=1}^{n} \left\lVert c_k (\chi_{F_k}-\chi_{G_k})\right \rVert_p \\&=& \sum_{k=1}^{n} \lvert c_k \rvert \left\lVert \chi_{F_k}-\chi_{G_k}\right\rVert_p \\&=& \sum_{k=1}^{n} \lvert c_k \rvert \{\mu(F_k\setminus G_k)+\mu(G_k\setminus F_k)\}^{1/p}< nM\left(\frac{\varepsilon}{nM} \right)=\varepsilon.\end{eqnarray*}
Second case$\mu(X)=\infty$ Let $f\in L^p(X)$ and let $s\in S_0$. In this case exists an increasing sequence $\{E_n\}\subseteq\mathcal{A}$ such that
$$X=\bigcup_{n=1}^\infty E_n\quad\text{end}\quad \mu(E_n)<\infty\;\forall n$$
We observe that $$\lVert s \rVert_p^p=\int_X \lvert s \rvert\; d\mu \le \sum_{n=1}^\infty\int_{E_n}\lvert s\rvert^p\;d\mu<\infty,$$ then exists $n_0\in\mathbb{N}$ such that $$\sum_{n=n_0}^\infty\int_{E_n} |s|^p\;d\mu< \varepsilon.$$
Now we have that $$\int_{X\setminus E_{n_0-1}}|s|^p\;d\mu=\int_{{\bigcup_{n=n_0}^\infty}E_n} |s|^p\; d\mu\le \sum_{n=n_0}^\infty\int_{E_n} |s|^p\;d\mu< \varepsilon.$$
We define now $t:=s\chi_{E_{n_0-1}}$ and we observe that $t\in S_0(E_{n_0-1})$, observe that $\mu(t\ne 0)\le \mu(E_{n_0-1})<\infty$, and $$\lVert s-t\rVert_p^p= \int_{X\setminus E_{n_0-1}} |s|^p\;d\mu<\varepsilon.$$
Now we can conclude:
Let $f\in L^p(X)$, then for $(b)$ exists $s\in S_0(X)$ such that $\lVert f-s \rVert_p <\varepsilon/4$.
For the second case exists $t=s\chi_{Y}\in S_0(Y)$ with $\mu(Y)<\infty$ such that $\lVert s-t\rVert_{L^p(X)}<\varepsilon/4$. Now for the first step exists $t_{\mathbb{Q}}\in S_0^{\mathbb{Q}}(Y)$ such that $\lVert t-t_{\mathbb{Q}} \rVert_{L^p(Y)}<\varepsilon/4$ and for the second step exists $t_{\mathbb{Q},\mathcal{A}_0}\in S_0^{\mathbb{Q},\mathcal{A}_0}$ such that $\lVert t_{\mathbb{Q}}-t_{\mathbb{Q},\mathcal{A}_0}\lVert_{L^p(Y)}<\varepsilon/4 $
Then redefining equal to zero on $X\setminus Y$: $t_{\mathbb{Q}}$ and $t_{\mathbb{Q},\mathcal{A}_0}$ we have that $$\lVert f -t_{\mathbb{Q},\mathcal{A}_0} \rVert_p<\varepsilon.$$
Is this procedure correct? Or is there a need for more details?