I have started reading optimal transport from the book "Optimal Transport for Applied Mathematicians" and I have a question regarding change of variables in Monge's formulation.
We can reformulate the constraint $\mu(T^{-1}(A))=\nu(A)$ for every measureble $A$ as:
$$\int_{T^{-1}(A)}f(x)dx=\int_{A}g(y)dy.$$
And then by a change of variable $y=T(x)$ we can get
$$\int_{T^{-1}(A)}f(x)dx=\int_{T^{-1}(A)}g(T(x))det(DT(x))dx$$
which implies that $$f(x)=g(T(x))det(DT(x)).$$
However, the mentioned book says that this implication holds for injective $T$. I can't understand where the injective property is necessary
If $T$ is not injective, then there are multiple branches of $T^{-1}$, so the change-of-variable fails to capture the whole $T^{-1}(A)$ set.
Say we are working on $\mathbb{R}$, $T(x)=x^2$ and $A=[0,\infty)$. Then what you start with is $$\int_{-\infty}^{\infty}f(x)dx=\int_0^{\infty}g(y)dy.$$ When you do a change of variable, you are replacing $y=x^2$ with $x\in[0,\infty)$ instead of $x\in\mathbb{R}$. Therefore you only get $$\int_{-\infty}^{\infty}f(x)dx=\int_0^{\infty}g(x^2)det(D[x^2])dx$$ instead of $$\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{\infty}g(x^2)det(D[x^2])dx.$$