From general measure theory, I think it's possible to create a measure space $(C, \mathcal{M_\phi}, m_\phi)$ where $C$ is the middle third Cantor set. Now the measure $m_\phi$ is defined as $m(\phi^{-1}(E))$ for which $\phi$ is the bijection from infinite binary sequences (on the closed unit interval?) to $C$, $m$ is the usual Lebesgue measure, and $E \subseteq C$. Also, $\mathcal{M_\phi}$ is a $\sigma$-algebra as follows: $\{E \subset C: \phi^{-1}(E) \in \mathcal{M}\}$ where $\mathcal{M}$ is the $\sigma$-algebra of Lebesgue measurable sets.
My question is how would you compute an integral in the following form: $\int_C x\;dm_\phi$? I already know that $m_\phi(C)=1$, but how could I go about computing the integral? Would approximation by simple functions work, and if so, what kind of functions should I work with? How would this generalize for $\int_C x^n\;dm_\phi$? Any input would be highly appreciated!