\[ \newcommand{\C}{\mathbb{C}} \newcommand{\haar}{\mathsf{m}} \newcommand{\P}{\mathcal{P}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\d}{\mathsf{d}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \newcommand{\B}{\mathscr{B}} \]

In the context of the measure space $(\R,\mathsf{Bor}(\R),\lambda)$ where $\lambda$ is the Lebesgue measure, verify each of the following functions is $(\mathsf{Bor}(\R),\mathsf{Bor}(\R))$ measurable and calculate its integral. For optional fun, determine which of these functions is Riemann integrable.

1. $f(x) = 1_\mathbb{Q}(x)$
2. $f(x) = 1_{[\pi,\infty)}(x) \cdot \dfrac{|\sin(x)|}{x}$
3. $f(x) = 1_{[0,1]}(x) \cdot x^2$
4. $f(x) = 1_{(0,1)}(x) \cdot \dfrac{1}{2\sqrt{x}}$
5. $f(x) = \begin{cases} \tfrac{1}{q} & x = \tfrac{p}{q} \textsf{ in lowest terms} \\ 0 & x \in \R \setminus \mathbb{Q} \end{cases}$

6. The function $g$ defined to be zero except on $[0,1]$ where it is the pointwise limit of the sequence $n \mapsto f_n$ of functions on $[0,1]$ where $f_1(x) = x$ and \[ f_{n+1}(x) = \begin{cases} \dfrac{f_n(3x)}{2} & 0 \le x \le \tfrac{1}{3} \\ \dfrac{1}{2} & \tfrac{1}{3} \le x \le \tfrac{2}{3} \\ \dfrac{1}{2} + \dfrac{f_n(3x-2)}{2} & \tfrac{2}{3} \le x \le 1 \end{cases} \] for all $n \in \N$ and all $0 \le x \le 1$. (Take for granted that the sequence converges uniformly.)