Approximation property of integrable function Q44. Let $f$ be integrable over $\mathbb{R}$ and $\epsilon>0$. Establish the following three approximation properties.
(i )There is a simple function $\eta$ on $\mathbb{R}$ which has finite support and $\int_\mathbb{R}|f-\eta|<\epsilon$.
(ii) There is a step function s on $\mathbb{R}$ which vanishes outside a closed, bounded interval and
$\int_\mathbb{R} |f - s| < \epsilon.$
(iii) There is a continuous function $g$ on $\mathbb{R}$ which vanishes outside a bounded set and
$\int_\mathbb{R}|f - g| < \epsilon.$
I am good with (i), but I am having some trouble with proving (ii),(iii), so I would appreciate any thought or hints about them, thank you in advance.
 A: Step (i) follows easily from the Simple Approximation Lemma discussed in virtually every book on Lebesgue measure and integration.  
Once you find the step function $s$ in (ii), it is easy to approximate the step function with a continuous function $g$ for (iii) by a straightforward piecewise linear construction. 
The difficult part is finding a step function $s$ that approximates the simple function $\eta$ such that
$$\int_{\mathbb{R}}|f - s| \leqslant \int_{\mathbb{R}}|f - \eta| + \int_{\mathbb{R}}|\eta - s| < \epsilon$$
Here is a sketch of the proof.
Let $E = \text{supp }(\eta)$. It is enough to prove that for a characteristic function $\chi_A$ on a measurable set $A \subset E$, there is a step function $\phi$ such that $\int_E| \chi_A - \phi| < \epsilon$. For any $\delta > 0$, there is an open set $O$ containing $A$ with $m(O \setminus A) < \delta$. As an open set $O$ is a countable union of disjoint open intervals
$$O = \bigcup_{j=1}^\infty(a_j,b_j),$$ 
we have
$$m(O) = \sum_{j=1}^\infty(b_j-a_j) < m(A) + \delta$$
Construct $\phi$ as the characteristic function of a finite union of a sufficiently large number of the intervals $(a_1,b_1), \ldots ,(a_m,b_m)$. This is a step function with the desired properties.
