In a book I am working with (Shao, Mathematical Statistics, p. 100) there is the following claim (I have slightly modified the original claim to make my question more focused):
$X:\mathbb R^n \to \mathbb R^n, x \mapsto \Sigma x$ where $\Sigma$ is a positive definite matrix $n \times n $. $P$ is a probability measure on $(\mathbb R^n, \mathcal B(\mathbb R^n))$ with the c.d.f. denoted by $F$. It is claimed that the c.d.f. of $P_X:=P \circ X^{-1}$ is given by $x \mapsto F(\Sigma^{-1}x)$.
Is it true, why?
It seems that for the claim to be true this should hold $X^{-1}\big((-\infty,x]\big)=(-\infty,\Sigma^{-1}(x)]$. But it must not be the case, must it? Does the fact that $\Sigma$ is positive definite guarantees this?