Given $F(y)=\int_0^y f(s)ds$ for some integrable function $f$ and $G$ a Lipschitz function, show that $G(F(x))=\int_0^x h(t)dt$ for some integrable $h$.
I can only show that G(F(x)) is absolutely continuous, but that only implies $G(F(x))-G(F(0))=\int_0^x h(t)dt$ for some $h$. Since $G(F(0))=G(0)$, which is not necessary to be $0$, I don't know how to continue.