I would like to construct a random variable on a fractal set such as the Cantor set. In other words, I would like to construct a probability measure $\mu$ on $\mathbb{R}^N$ such that I have $\mathrm{Supp}\mu$ a set with a fractal nature, quantified with respect to (for instance) the Hausdorff dimension. I have two general questions:
- How can I concretely define such a probability measure? Can I keep properties like Borel sets being measurable still?
- How can I explicitly compute expectations of a real measurable function $f$ defined on this probability space (in other words, how can I integrate)?
One straightforward idea I had was to use the $s$ - dimensional Hausdorff (outer) measure restricted to the support set $A$ where $s = \mathrm{dim}_H A$ as the probability measure, assuming the regularity conditions $0 < s < \infty$ and the Hausdorff measure of the set $0 < \mathcal{H^s}(A) < \infty$. But it is not straightforward for me how to proceed with expectation computations, and also whether this yields an "interesting" measure.
I would also be interested in literature that has in the past proposed such generalisations and constructions.
In general I don't think it will be easy to compute expectations with respect to Hausdorff measure, but you may be able to use something based more on the way your fractal is obtained. Thus if your fractal is $E = \bigcap_{n=0}^\infty E_n$ where $E_n$ are compact sets and $E_{n+1} \subseteq E_n$, take probability measures $\mu_n$ on each $E_n$, and prove (if it is true) that the sequence $\mu_n$ has a weak-* limit $\mu$, i.e. $\int f \; d\mu = \lim_{n \to \infty} \int f d\mu_n$ exists for each continuous function $f$ on $E_0$. Then $\mu$ is your measure on $E$, and expectations with respect to it can be computed by taking a limit.