Integral over Fractals with respect to fractal dimension

Question

I understand that there is type of integral with respect to measures that can return values when evaluated over an integral. But is there an Integral that returns d dimensional volume where d is the fractal dimension of the fractal. Basically, I'm asking about finding an integral that can find the volume of say the cantor set in terms of $m^{ln2/ln3}$. This question is mostly motivated by the fact that the area of 2-d objects can vary a lot, yet they all have units of $m^2$. I'd assume this would be the case for fractal objects as well. As an example, if I wanted to take the integral of the siernpinski triangle and get a $ln3/ln2$ dimensional volume instead of taking the area of a square, could I? If so, could you provide a link or an explanation?

Answer 1

If $\mu$ is a measure and $1_E$ is the characteristic function of a set $E$ then, in principle, $$\int 1_E \, d\mu$$ returns the measure $\mu(E)$. If $\mu_s$ is an $s$-dimensional measure (like the Hausdorff or Packing measure), then I guess the integral returns what you're looking for. I don't think that makes computing the exact value any easier, though.

If you are interested in an introduction to integration using fractal measures, I highly recommend the paper Evaluating Integrals Using Self-similarity by Bob Strichartz. The paper gives an excellent introduction to integration with respect to self-similar measures.