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.
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.