$\mathcal L_p([0,1],S,\lambda)$ is separable for $p\in(0,\infty)$ Let $\Omega=[0,1]$, $S$ be the Borel $\sigma$-algebra in $[0,1]$ and $\lambda$ be the Lebesgue measure over $S$. So I want to prove that:

$$\mathcal L_p(\Omega,S,\lambda) \text{  is separable for  }p\in(0,\infty)$$

My attempt:

I'm lost with this one since I can't see how to give a countable dense set in $\mathcal L_p$. What I got is that:

Let $f\in\mathcal L_p$, so $f$ is measurable then, by Luzin's theorem, we can assure that:
$$
\forall\epsilon>0\;\exists E_{\epsilon}\text{  open with }\lambda(E_{\epsilon})<\epsilon\text{  s.t.  }\;\;f\big|_{\Omega\setminus E}\text{  is continuous}
$$
and then, by Weierstrass aproximation theorem, we get that:
$$
\forall \tilde\epsilon>0\;\exists p_{\tilde\epsilon}(x)\text{  s.t. }|f(x)-p_{\tilde\epsilon}(x)|<\tilde\epsilon
$$
but got stucked here trying to figure out how this connects with the idea of finding a countable dense set. Any help would be appreciated.
 A: My approach suggested in the comments is to use functions like $$f = \sum^n_{i=1} \alpha_i \chi_{[a_i, b_i]}, \,\,\,\,\,\,\,\, (\text{I use } \chi \text{ to denote an indicator function)}$$ where $n \in \mathbb N, \alpha_i, a_i, b_i \in \mathbb Q$. The set of all such functions is countable since it can be identified with $\mathbb N \times \mathbb Q \times \mathbb Q \times \mathbb Q$ and it is dense in $\mathcal L^p([0,1])$ since any continuous function on $[0,1]$ can be uniformly approximated by such a function and any function in $\mathcal L^p([0,1])$ can be approximated by a continuous function in $\mathcal L^p$. 
That being said, it sounds like the approach that you are using is a bit different and I think it can still work. The family that you should prove dense is the set $\mathcal P_\mathbb Q$ of polynomials with rational coefficients. Indeed, for any compact set $K \subset R$, $\mathcal P_\mathbb Q$ forms unital subalgebra of $C(K)$ which separates points and so by the Weierstrass approximation theorem $\mathcal P_\mathbb Q$ is dense in $C(K)$ in the uniform norm. Also, the $\mathcal P_{\mathbb Q}$ is countable the set of polynomials over an infinite set has the same cardinality as the set itself (that is, $\mathcal P_\mathbb Q$ and $\mathbb Q$ have the same cardinality.
Take $f \in \mathcal L^p([0,1]).$ By integrability of $\lvert f \rvert^p$, we know that for every $\epsilon > 0$, there is $\delta > 0$ such that $$\lambda(E) < \delta \,\,\, \implies \,\,\, \int_E \lvert f \rvert^p d\lambda < \epsilon.$$ 
Take $\epsilon > 0$ and let $\delta > 0$ be as above. Notice, we can shrink $\delta$ later if necessary. By Lusin's theorem, we can find an open set $E = E(\delta) \subset [0,1]$ with $\lambda(E) < \delta$ and $f \,\big|_{[0,1]\setminus E}$ continuous. Since $[0,1] \setminus E$ is compact, we can find a polynomial $\tilde f$ with rational coefficients such that $$\sup_{x \in [0,1] \setminus E} \lvert f(x) - \tilde f(x) \rvert < \delta.$$ Then we see \begin{align*} \int_{[0,1]} \lvert f-\tilde f \rvert^p d\lambda &= \int_{[0,1] \setminus E} \lvert f - \tilde f \rvert ^p d\lambda + \int_{E} \lvert f - \tilde f \rvert d\lambda \\
&\le \int_{[0,1] \setminus E} \delta^p d\lambda + \int_{E} \lvert f - \tilde f \rvert d\lambda \\
&\le \delta^p + \int_{E} \lvert f - \tilde f \rvert d\lambda \end{align*}
We use some rather crude bounds for the other term. We see \begin{align*}\lvert f - \tilde f \rvert^p \le(\lvert f \rvert + \lvert \tilde f \rvert)^p \le 2^p \max\{\lvert f \rvert, \lvert \tilde f \rvert\}^p &= 2^{p} \max\{ \lvert f \rvert^p, \lvert \tilde f \rvert^p\} \\
&\le 2^p(\lvert f \rvert^p + \lvert \tilde f \rvert^p)
\end{align*} Then \begin{align*}\int_E \lvert f - \tilde f \rvert^p d\lambda &\le 2^p \int_E \lvert f \rvert^p d\lambda + 2^p \int_E \lvert \tilde f \rvert^p d\lambda \\ \le 2^p\epsilon + M \delta \end{align*} where $M = \max_{x\in [0,1]} \lvert \tilde f(x) \rvert < \infty$. This all gives $$\int_{[0,1]} \lvert f - \tilde f \rvert d\lambda < \delta^p+ 2^p \epsilon + 2^p M \delta$$ and the right hand side can be made arbitrarily small showing that $P_\mathbb Q$ is dense in $\mathcal L^p([0,1])$.
