It is a famous discovery by M Shishikura that the Hausdorff dimension of the Mandelbrot Set's boundary is 2.
I would like to computationally approximate the following integral:
$$I=\int_{\partial M} \frac{d\mathcal{H}^2(s)}{2s+1}.$$
The integral is in respect to $\mathcal{H}^2(s)$, which is the 2-dimensional Hausdorff measure.
Would I be able to approximate this with something akin to (where $B$ is some plane which contains $\partial M$):
$$I_p = \int_{B} \frac{1_{\partial M}}{2s+1} d\mathcal{H}^2_p(s).$$
If so, how could I approximate $1_{\partial M}$?
What would people suggest for computationally approximating this.