Let $L_p$ be the complete, separable space with $p>0$. $\mathbf{J}=\{I = (r,s] \}$ where $r$ and $s$ are rational numbers. $\mathbf{A}$ is the algebra generated by $\mathbf{J}$, with $\mathbf{S}=\operatorname{span}(\mathbf{A})$.
a). Try to verify that $\mathbf{S}$ is dense in $L_p$ space with respect to $L_p$ metric.
b). Try to verify that for any $p>0$, $L_p$ is complete.
By definition of Lebesgue integral, simple functions (i.e. element of the vector space generated by the characteristic functions of measurable sets) are dense in $L^p$, $p>0$. So we have to show that each characteristic function of a Borel set of finite measure can be approximated by an element of $\bf S$ for the $L^p$ norm. Using this result, and the fact that finite disjoint unions of elements of $\bf J$ form an algebra which generates the Borel $\sigma$-algebra, we are done.
For $0<p<1$, see this question. For $1\leq p\leq \infty$, see this one (it deals with the case the function takes their values on Banach spaces).