(i) A stepfunction is any function $s$ on $\mathbb{R}$ which can be written in the form

$$s(x)=\sum_{k=1}^{n} c_{k} I_{\left(a_{k}, b_{k}\right]}(x), \quad x \in \mathbb{R},$$

where $a_{k}, b_{k}, c_{k}$ are real numbers, with $a_{k}<b_{k}$ for all $k$. Show that the set of all stepfunctions is dense in $L^{1}(\mathbb{R}, \mathcal{B}, \mu)$. Here, $\mathcal{B}$ denotes the Borel $\sigma$-algebra, and $\mu$ denotes Lebesgue measure.

[You may use without proof the fact that, for any Borel set $B$ of finite measure, and any $\varepsilon>0$, there exists a finite union of intervals $A$ such that $\mu(A \triangle B)<\varepsilon$.]

(ii) Show that the Fourier transform

$$\hat{s}(t)=\int_{\mathbb{R}} s(x) e^{i t x} d x$$

of a stepfunction has the property that $\hat{s}(t) \rightarrow 0$ as $|t| \rightarrow \infty$.

(iii) Deduce that the Fourier transform of any integrable function has the same property.