In his book Measure and Integration Theory Bauer proves the following theorem:
Here $\text{GL}(d,\mathbb{R})$ denotes the set of all invertible linear transformations from $\mathbb{R}^d$ to $\mathbb{R}^d$, $\lambda^d$ denotes the Lebesgue-Borel measure on the Borel sets $\mathcal{B}^d$, and $T(\lambda^d)$ is the image measure defined by $T(\lambda^d)(A):=\lambda^d(T^{-1}(A))$ for all $A\in \mathcal{B}^d$.
Bauer then makes the following remark:
Am not sure how to interpret equation $(8.16)$. If I understand correctly, for each $x \in G$, $\text{D}\varphi(x):\mathbb{R}^d\to\mathbb{R}^d$ is a linear map, and so $|\text{det}\text{D}\varphi|$ is really a function on $G$. But then how do we compute the right side of $(8.16)$? The left side means $\varphi(\lambda^d)(A):=\lambda^d(\varphi^{-1}(A))$ for all $A\in \mathcal{B}^d$.
EDIT: From the comment of WoolierThanThou I believe that the correction interpretation of $(8.16)$ is
$$\lambda_G^d(\varphi^{-1}(A))=\int_A \frac{1}{|\text{det}\text{D}\varphi|\circ \varphi^{-1}} \textrm{d}\lambda_{G'}^d$$ for all $A\in \mathcal{B}^d$.
Bauer also mentions that $(8.16)$ is equivalent to the statement
$$\varphi^{-1}(\lambda_{G'}^d)=|\text{det}\text{D}\varphi| \lambda_G^d$$
but why is this so?
Thanks a lot for your help.