Let $$ f(x)= \begin{cases} \sin \pi x,& \text{if } x \in [0,1/2] -C\\ \cos \pi x , &\text{if } x\in (1/2, 1]-C\\ x^2, & \text{if } x \in C \\ \end{cases}$$
Where $C$ is a cantor set. Then the value of Lebesgue integral $\int_{[0, 1]} f d \mu$
How to find out? can it be solved by using simple function?
$\int_A f =0$ if $A$ is a null set and $f$ is integrable, so it suffices to just compute $$\int_0^{1/2} \sin(\pi x)\ d\mu + \int_{1/2}^1 \cos(\pi x) \ d\mu$$ which I trust you can do.