I am not an expert on measure theory. I am sorry if this question is too simple for some experts here.
Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. Are there good references about the calculus with respect this measure?
Here is one concrete question. Let $\mu$ be the Cantor measure, what is a general procedure to evaluate
$$ \int_0^1 f(x)\mu(d x) =? $$
for some continuous function $f(x)$. For some specific functions, such as $f(x)=x$, $f(x)=1-x$, $f(x)=1$, one can use integration by parts and symmetry to solve this problem. How about for general continuous function $f$?
Thanks a lot!