The characteristic function of the Cantor set is Riemann integrable

Question

Show that $X_C(x)$ is Riemann integrable over [0,1] where $C\subset [0,1]$ is Cantor set annd find $\int$ $X_c(x)dx$ (from 0 to 1)

Guys could you help me about this question?

Answer 1

Hint:

Split the interval $[0,1]$ into $3^n$ equal subintervals. A subinterval is bad if it contains points of $C$ in its interior. Count the bad intervals $B_k$! Replace each $B_k$ by $B_k'$, where $B_k'$ is obtained by stretching $B_k$ from its center by a factor $1+\epsilon$, $\epsilon\ll1$. The $B_k'$ and the remaining spaces $S_k$ between successive $B_k'$ form a partition ${\cal P}$ of $[0,1]$. Now estimate the upper sum belonging to ${\cal P}$, using that on the $S_k$ the characteristic function of $C$ is $\equiv0$.