Integration of function using Hausdorff measure of Cantor Set I am learning geometry on fractal shapes, along with how fractional calculus can relate to said geometry. At the moment I am trying to understand integration over a Hausdorff measure.
According to Falconer's The Geometry of Fractal Sets, the middle-third Cantor set has a $\log_3 2$-Hausdorff measure of $1$. I would like to describe this relation as $\int_{C} 1 \; \mathrm{d}H^{s}(x)=1$, where $C$ is the Cantor set, and $s = \log_3 2$.
My question is: How would the integral $$\int_{C} (e^x + x)\; \mathrm{d}H^{s}(x)$$ be evaluated?
I'm not entirely sure where to begin in evaluating this, since this is new to me. I would appreciate resources as well, but I think an answer to this problem will give me the tools I need to evaluate other functions on other fractal sets.
Thank you.
 A: I highly recommend the paper Evaluating Integrals Using Self-similarity by Robert Strichartz. Since Hausdorff measure on the Cantor set happens to be a self-similar measure, the ideas in that paper are applicable. In that paper, we learn  that the basic properties of the integral yield a self-similar identity for integration over the Cantor set $C$ with Hausdorff measure $\mu$:
$$\int_C f(x) \, d\mu =  \frac{1}{2}\int_C f\left(\frac{1}{3}x\right)d\mu + \frac{1}{2} \int_C f\left(\frac{1}{3}x+\frac{2}{3}\right)d\mu.$$
We can use this to evaluate the second term in your integral exactly:
$$
\begin{align}
  \int_C x \, d\mu &= \frac{1}{2}\int_C \left(\frac{1}{3}x\right)d\mu + \frac{1}{2} \int_C \left(\frac{1}{3}x+\frac{2}{3}\right)d\mu \\
 &= \frac{1}{6} \int_C x \, d\mu + \frac{1}{6} \int_C x \, d\mu + \frac{1}{3} \int_C 1 \, d\mu
\end{align}
$$
We can now solve for $\int_C x \, d\mu$ in terms of $\int_C 1 \, d\mu=1$ to get
$$\int_C x \, d\mu = \frac{1}{2}.$$
It's worth mentioning that this same technique can be applied to $\int_C x^n \, d\mu$ to derive a recursive technique to integrate $x^n$ for every integer $n$ and, therefore, any  polynomial.
Unfortunately,  I do not believe that the integrals of the transcental functions can be expressed  in closed form. They can be estimated numerically easily enough, though. For example, the picture below should convince you that
$$\frac{1}{4} \left(1+e^{2/9}+e^{2/3}+e^{8/9}\right) < \int_C e^x \, d\mu < \frac{1}{4} \left(\sqrt[9]{e}+\sqrt[3]{e}+e^{7/9}+e\right).$$

Taking that out to 20 iterations, I estimate that
$$1.75327920434 < \int_C e^x \, d\mu < 1.75327920485.$$
