Distributions on fat Cantor sets

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Suppose we have a probability distribution that is supported on the fat Cantor set. For details on the fat Cantor sets see here.

A have a few questions about these distributions.

  1. Is there a good reference that treats these distributions in a systematic way.
  2. Does such a distribution have a density?

I am very fascinated by this sets would really like to start reading about them. However, need a good starting place. I am particularly, interested in applications to probability theory.