Compute the Lebesgue-Stieltjes integral:
$\displaystyle\int_{0}^{1}\sin(\pi x)\mathrm{dF(x)}$
Where $F(x)$ is the Cantor function on $[0,1]$.
The standard construction of the Cantor function involves the functions $F_{0}(x)=x$ and $F_{n+1}(x)=(1/2)F_{n}(3x)$ for $x \in [0,1/3]$, $F_{n+1}(x)=1/2$ for $x \in (1/3,2/3)$ and finally $F_{n+1}(x)=\frac{1}{2}+\frac{1}{2}F_{n}(3(x-\frac{2}{3}))$ for $x \in [2/3,1]$, and defining the limit as $n \to \infty$ as $F$ (the convergence of $F_{n}$ to $F$ will be uniform).
For computing this integral, I was thinking of splitting the integral into $[0,1/3]$ and $[2/3,1]$ (intuitively, I think that the integral vanishes on the interval $[1/3,2/3]$ as $F$ is constant) and then obtaining a recursive definition integrating with respect to $\mathrm{dF_{n}(x)}$. However, I haven't been able to do much from here. What would be an intuitive way to compute the integral in this case? Any help would be appreciated!