Background information:
Theorem 2.10 - Let $(X,M)$ be a measurable space.
a.) If $f:X\rightarrow [0,\infty]$ is measurable, there is a sequence $\{\phi_n\}$ of simple functions such that $0 \leq \phi_1 \leq \phi_2 \leq \ldots \leq f$, $\phi_n \rightarrow f$ pointwise, and $\phi_n\rightarrow f$ uniformly on any set on which $f$ is bounded.
b.) If $f:X\rightarrow \mathbb{C}$ is measurable, there is a sequence $\{\phi_n \}$ of simple functions such that $0 \leq |\phi_1| \leq |\phi_2| \leq \ldots \leq |f|$, $\phi_n\rightarrow f$ pointwise, and $\phi_n\rightarrow f$ uniformly on any set on which $f$ is bounded.
Proposition 1.20 - If $E\in M_\mu$ and $\mu(E) < \infty$, then for every $\epsilon > 0$ there is a set $A$ that is a finite union of open intervals such that $\mu(E \ \triangle \ A) < \epsilon$.
Question:
Theorem 2.26 - If $f\in L^1(\mu)$ and $\epsilon > 0$, then
a.) there is an integrable simple function $\phi = \sum_{1}^{n}a_j\chi_{E_j}$ such that $\int |f - \phi|d\mu < \epsilon$.
b.) If $\mu$ is a Lebesgue-Stieltjes measure on $\mathbb{R}$, the sets $E_j$ in the definition of $\phi$ can be taken to be finite unions of open intervals.
c.) Moreover, in situation b.), there is a continuous function $g$ that vanished outside a bounded interval such that $\int |f - g|d\mu < \epsilon$.
Attempted Proof a.) - By theorem 2.10 we can find a sequence of simple functions $\{\phi_j\}$ with $\phi_j\rightarrow f$ pointwise and $|\phi_1|\leq |\phi_2|\leq \ldots \leq |f|$. Now, $|\phi_j - f|\rightarrow 0$ pointwise and $$|\phi_j - f|\leq |\phi_j| + |f| \leq 2|f|$$ Applying the Dominated Convergence Theorem, $$\lim_{j\rightarrow \infty}\int |\phi_j - f| = \int 0 = 0$$
Attempted Proof b.) - From a.) we can find a simple function $\phi = \sum_{1}^{n}a_j\chi_{E_j}$ within distance of $\epsilon/2$ from $f$. Since the simple sum is integrable, all of the $E_j$'s have finite measure. So by Theorem 1.20, for each $E_j$ there is a finite union of open intervals with $\mu(E \ \triangle \ F_j) < \frac{\epsilon}{2|a_j|n}$. Now, \begin{align*}\int_n\left|\sum_{1}^{n}a_j\chi_{E_j} - \sum_{1}^{n}a_j \chi_{F_j}\right| &\leq \sum_{1}^{n}|a_j|\int |\chi_{E_j} - \chi_{F_j}|\\ &= \sum_{1}^{n}|a_j|\mu(E_j \ \triangle \ F_j)\\ &\leq \sum_{1}^{n}\frac{\epsilon}{2n}\\ &= \frac{\epsilon}{2} \end{align*}
I am not sure if this correct and I can't figure out c.) yet, I will post it once I figure it out.