I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows:
Consider a finite time, time series $X_t$. One can define finite time Fourier transform of this vector using the unitary circular matrix $F$ which its multiplication by $X$ returns the Fourier transform $\hat{X_\omega}$, i.e. $$\hat{X}=FX$$
since $F$ is unitary its obvious that $\det(A^{-1})$=1 and therefore if $X\sim p(x)$ then:
$$p_{\hat{X}}(\hat{X})=p(X)\det(dX/d\hat{X})=p_X(X).$$
Where $p_{\hat{X}}$ transformed density after applying Fourier transform. But in general Fourier transformation is not only for finite domain. The question is that whether this argument is true for infinite domain, i.e. when when $X$ is a process.