Lebesgue Integral-Question

Question

Hi guys, How can I evaluate Lebesgue Integral of this function. I think first I should show that is simple function ?

Answer 1

If you mean $C$ is the Cantor ternary set, then since $\lambda(C)=0, \lambda(C\cap\mathbb{Q})=0=\lambda(C\cap\mathbb{R}\setminus\mathbb{Q})$ where $\lambda$ is the Lebesgue measure, $C, \mathbb{R},\mathbb{Q}$ are Borel and hence measurable, $f(x)=\mathbb{1}_{C\cap\mathbb{Q}}(x)+2*\mathbb{1}_{C\cap\mathbb{R}\setminus\mathbb{Q}}(x)+3*\mathbb{1}_{[0,1]\setminus C}(x)$, hence $f$ is simple, and its integral is$ \lambda(C\cap\mathbb{Q})+2\lambda(C\cap\mathbb{R}\setminus\mathbb{Q})+3\lambda([0,1]\setminus C)=3$