In this post How can I understand the tunneling problem by Euclidean path integral where the quadratic fluctuation has a negative eigenvalue? we have path integration variable $x$ is expanded around the $\bar{x}$ $\tag{2.5} x(t^E)~=~\bar{x}(t^E) + y(t^E), \qquad y(t^E)~:=~\sum_{n=0}^{\infty}c_n x_n(t^E),$
The integral measure is then defined as $\tag{2.7} [dx]~=~\prod_{n=0}^{\infty} \frac{dc_n}{\sqrt{2\pi\hbar}}.$
Should it not be
$[dx(t^E)]=[dy(t^E)]$
Why can we define the integral measure? Are the $c_n$ not constant or am i missing something?
