Halmos Measure Theory section 39 Theorem D

Question

I have trouble explaining the remark "The function $\phi$ plays the role of Jacobian (or, rather, the absolute value of the Jacobian) in the theory of transformation of multiple integrals".

I know the Jacobian determinant in the multiple integrals, $$\iint_Df(x,y)dxdy=\iint_Df(x(u,v),y(u,v))|J(u,v)|dudv$$

How does the Theorem D relate to the above formula?

Here is the photo from the book.

Answer 1

Let $\mu=\nu=m$, the Lebesgue measure, $g(y)=g(u,v)$.

Then $\phi=\frac{d(\mu T^{-1})}{d\nu}=\frac{d(mT^{-1})}{dm}$. The function $\phi$ can also be seen roughly as $$\phi(y)=\lim_{y\in E,m(E)\rightarrow 0}\frac{mT^{-1}(E)}{m(E)}$$ for $y\in Y$. Obviously, it means the ratio before and after transformation for the aera of the set $E\subset Y$, i.e. the Jacobian determinant.