How to shows that $f(x)=x, x\in[0,1]-C,f(x)=1,x\in C$ is an upper function and is Lebesgue integrable where $C$ is a Cantor set?

Question

How to show that $f(x)=x, x\in[0,1]\setminus C,f(x)=1,x\in C$ is an upper function and is Lebesgue integrable where $C$ is a Cantor set?

Answer 1

Since $m(C)=0$ and $f \geq 0$ we have that $f=x$ a.e on

Thus $\int_0^1 f=\int_0^1 x dx =\frac{1}{2}<+\infty$

Thus the function is Lebesgue integrable.

Also as a simpler proof note that $f$ is measurable and non-negative and bounded by $1$ so it is integrable.