Cantor Set : Characteristic function

Question

What is the characteristic function of the C Cantor set?

I already found that the function is Integrable but i couldn't find the exact result of the charateristic function of the C Cantor set.

Answer 1

I assume you already know what the cantor set is. In that case, the characteristic function is the function $f$ with $f(x)=1$ for $x \in C$ and $f(x)=0$ otherwise.

Answer 2

You must employ a Lebesgue integration as $\chi_C$, the characteristic function of $C$, is not classically or even Riemann integrable because its set of discontinuities,which is $ C$ , is uncountable.The measure $\mu (I \backslash C)=\mu([0,1]\backslash C)= 1$ so $\mu (C)=0$ so $\chi_C (x)=0$ almost everywhere (which means except on a set of measure 0 ).If Lebesgue-integrable functions $f ,g$ are equal a.e. (almost everywhere) then $ \int_0^1 f(x) \mu( dx)=\int_0^1 g(x) \mu (dx)$.