What are some concrete examples of functions either on the ternary Cantor set $C$ or on its subsets that could be measurable and/or integrable? I've been toying with the idea of trying $f(x) = x$ and $f(x) = x^2$ but I'm not sure if they fit the bill. Perhaps this might first of all require us to define exactly what a measurable function and an integrable function are when restricted to the Cantor set $C$ or its subsets. How should I proceed, and are there any useful research articles on this?
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The Cantor set is compact and therefore Borel measurable. Take any Borel measurable function defined on $[0,1]$ and restrict it to the Cantor set. This is Borel measurable. In fact, every Borel measurable subset of the Cantors set is some restriction of a Borel measurable function defined on [0,1]; just extend by defining it to be zero off of the Cantor set.
Notice I didn't say "Lebesgue measurable," because all subsets of the Cantor set are Lebesgue measurable.
When you say "integrable," I reply, "With respect to what measure?" The Cantor Singular function defines a measure supported on the Cantor set. Is this the measure you are thinking of?