Let $G$ be an absolutely continuous function, $G:[a,b] \rightarrow [c,d]$ and $f \geq 0$ a Lebesegue measurable function in $[c,d]$. I managed to prove that if $f$ is just Borel measurable it holds $$\int_c^d f(t)dt= \int_a^b f(G(\tau) |G'(t)| d\tau$$
I know that every Lebesgue measurable function is a.e. equal to a Borel function so I think from that I can extend the theorem.
Now I am asked to do the following: Let $h$ be a function such that $h=0$ a.e. on $[c,d]$. Consider the sets $H=\lbrace{ \tau \in [a,b] : |G'(\tau) >0\rbrace}$ and $N \subset [c,d]$ such that $\lambda(N)=0$. Prove that $\lambda(G^{-1}(N) \cap H)=0$. Verify that this means that the function $(h\circ G )|G'|=0$ almost everywhere in $[a,b]$.