Let $f$ be a continuous function such that $$f:(0,1)\to[0,\infty).$$ Let $$S_f(y)=\text{the sum of the length of the disjoint intervals whose union}$$ $$\text{is the set} \{x\in (0,1): f(x)>y\}, \ y>0.$$ Let $1\leq p<\infty.$ Show that $$\int_0^1(f(x))^p\, dx=p\int_0^{\infty}y^{p-1}S_f(y)\, dy.$$
I begin with taking $f(x)=0$ and the result holds trivially. Then I take $f(x)=x$ but with the example I am not able to compute $S_f(y)$.
Basically, I have not understood how $S_f(y)$ will be computed. Further, how this helps in showing the integral.
Please help. Thank you in advance.