Change of variable rule for multidimensional integral

Question

Consider a real-valued function $g: \mathbb{R}^q\rightarrow \mathbb{R}$. I want to show that $\int g(x)dx=\frac{1}{h^q}\int g(\frac{x}{h})dx$. I guess I should use the change of variable rule but I don't see how to get $\frac{1}{h^q}$. Any hint would be really appreciated.

Answer 1

Try rewriting that as $$ \int g(u)~du~=~\frac{1}{h^q}\int g(\frac{x}{h})~dx. $$ Now the substitution $u = \frac{x}{h}$ (which really means $u_i = \frac{x_i}{h}$ for $i = 1, \ldots, q$), plus the change of variable theorem, does the trick.

Answer 2

$dx=dx_1\cdots dx_q$. Change each of the $x_i$ as $x_i =t_i/h$. Therefore $$ dx=dx_1\cdots dx_q=\frac{dt_1}{h}\cdots\frac{dt_q}{h}=\frac{1}{h^q}dt_1\cdots dt_q\ . $$