Matrix integration problem $\int g(Ax)dx=|A| \int g(y)dy$

Question

In p. 96 of Wand & Jones' (1995) book they asserted that the following equation is valid for linear changes of variables $$\int_{\mathbb{R^d}} g(A\mathbf{x})d\mathbf{x}=|A| \int_{\mathbb{R^d}} g(\mathbf{y})d\mathbf{y}$$ where $A$ is a $d \times d$ invertible matrix, $\mathbf{x},\mathbf{y}\in \mathbb{R^d}$ and $d\mathbf{x}, d\mathbf{y} $ denotes $dx_1 \dotsm dx_d$, $dy_1 \dotsm dy_d$, respectively.

Could someone explain how to obtain this specific result?

Is there a more general result related to the presented above?

Thanks in advance!

Answer 1

There are $2$ mistakes.

We put $y=Ax$; then $dy=|\det(A)|dx$ and

$\int_{\mathbb{R}^d} g(Ax)dx=\dfrac{1}{|\det(A)|}\int_{\mathbb{R}^d}g(y)dy$. (beware to the absolute value)