Integrability of The Indicator of a Perfect Set

Question

I am having trouble solving the following problem: Let $C$ denote the middle-third cantor set. i.e., $C = \bigcap_{i=1}^{\infty}C_n$ where \begin{align*} C_1 &= \left[0, \frac{1}{3}\right] \cup \left[\frac{2}{3}, 1\right] \\ C_2 &= \left[0, \frac{1}{9}\right] \cup \left[\frac{2}{9}, \frac{1}{3}\right] \cup \left[\frac{2}{3}, \frac{7}{9}\right] \cup \left[\frac{8}{9}, 1 \right] \\ & \vdots \\ C_n &= \bigcup_{(i_1,\dotsc,i_n) \in \{0,2\}^n} \left[ \frac{i_1}{3^1}+ \dotsc+ \frac{i_n}{3^n}, \frac{i_1}{3^1}+\dotsc + \frac{i_n+1}{3^n}\right] \\ & \vdots \end{align*} Let $f$ be defined by \begin{align*} f(x) = \begin{cases} 1 & \text{if $x \in C$} \\ 0 & \text{otherwise}\end{cases} \end{align*} How can I show that $f$ is Riemann integrable (i.e., $f \in \mathcal{R}$) on $[0,1]$, and what would be the value of \begin{align*} \int_{0}^{1} f(x) dx \end{align*}

Answer 1

Does the following answer look alright? Let $\epsilon > 0.$ We want to show that there exists a partition $P$ of $[0,1]$ such that \begin{align*} U(P,f) - L(P,f) < \epsilon \end{align*} where $U,L$ denote the upper and lower sums. We choose $P$ to be such that \begin{align*} \Delta x_i &= \frac{1}{n} \end{align*} for all $i \in [n]$. Let $M_i(f), m_i(f)$ denote the supremum and infimum of $f$ respectively on interval $[x_{i-1}, x_i].$ Thus, we observe that \begin{align*} U(P,f) - L(P,f) &= \frac{1}{n} \sum_{i=1}^{n} M_i(f) - m_i(f) \\ &\leq \frac{1}{n} \end{align*} We can choose $n$ to be arbitrarily large so that \begin{align*} U(P,f) - L(P,f) < \epsilon \end{align*}