A countable dense set in $L^p[a,b]$

Question

I know $L^p[a,b]$ is separable for $1\leq p< \infty$, but I am not able to find a countable dense set in it. Please give some example.

Answer 1

(Assuming that the measure on the space is the Lebesgue Measure). Any measurable $f$ can be approximated pointwise almost everywhere by a sequence of simple functions $\{f_n\}$ such that $|f_n|\leq |f|$ everywhere. Hence, by the dominated convergence theorem, the space of simple functions is dense in $L^p$. Now, we prove that the space of step functions(linear combinations of indicator functions of rectangles) is dense in the space of simple functions. Since a simple function is a linear combination of indicator functions of sets of finite measure(characteristic function), we need to approximate a characteristic function $X_E$ by step functions. Since the Lebesgue measure is regular, we can find an open set $O$ such that $m(O-E)<\epsilon$. Now, we can write the open set as a countable union of disjoint open intervals. Truncating the countable union expansion at a large enough level $n$, we can approximate our set $E$ by the union of finitely many disjoint open intervals $O_i$, where $1\leq i \leq N$. Hence we get that the linear combinations of characteristic functions of open intervals are dense. Finally, we get that rational linear combinations of open intervals with rational end points are dense. Explicitly, this set is dense: $$\left \{\sum_{i=0}^{i=n}r_iX_{(a_i,b_i)}:r_i,a_i,b_i\in \Bbb Q, n\in \Bbb N\right \}$$ where each $r_i$ is rational and each $O_i$ is an open interval with rational end points.