Uniform Distibution Over Middle-Third Cantor Set (and Its Approximates)

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S1 Q2.7

Above is my question. I have done part 1, but I am unsure how to do the next part(s). Part 2 at least doesn't look that difficult, but I can't get the specifics. My main issue is that I can't get a particularly clean representation of $C_n$. A slight hint would be most appreciated! Thanks! :)