$$$$

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

1. $f(x)=1_{\mathbb{Q}}(x)$
2. $f(x)=1_{[\pi ,\mathrm{\infty })}(x)\cdot {\displaystyle \frac{|\sin (x)|}{x}}$
3. $f(x)=1_{[0,1]}(x)\cdot x^2$
4. $f(x)=1_{(0,1)}(x)\cdot {\displaystyle \frac{1}{2\sqrt{x}}}$
5. $f(x)=\{\begin{array}{ll}\frac{1}{q}& x=\frac{p}{q}\mathsf{\text{ in lowest terms}}\\ 0& x\in \mathbb{R}\setminus \mathbb{Q}\end{array}$

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 $$f_{n+1}(x)=\{\begin{array}{ll}{\displaystyle \frac{f_n(3x)}{2}}& 0\le x\le \frac{1}{3}\\ {\displaystyle \frac{1}{2}}& \frac{1}{3}\le x\le \frac{2}{3}\\ {\displaystyle \frac{1}{2}}+{\displaystyle \frac{f_n(3x-2)}{2}}& \frac{2}{3}\le x\le 1\end{array}$$ for all $n\in \mathbb{N}$ and all $0\le x\le 1$. (Take for granted that the sequence converges uniformly.)